Causal Reasoning with Bipartite Graphical Causal Models

# Causal Reasoning with Bipartite Graphical Causal Models
Source: [https://arxiv.org/html/2608.19831](https://arxiv.org/html/2608.19831)
[Joris M\. Mooij](mailto:<[email protected]>)Affiliation:Korteweg\-De Vries Institute for Mathematics University of Amsterdam Amsterdam, the Netherlands

###### Abstract

Causal Bayesian networks \(CBNs\) and structural causal models \(SCMs\) are the dominant frameworks for graphical causal reasoning, but they cannot adequately represent all real\-world causal systems\. In particular, systems at equilibrium—where feedback mechanisms create cyclic causal dependencies—can exhibit causal semantics that are fundamentally incompatible with these frameworks: different interventions that enforce the same variable value may have different effects, rendering the standard “perfect intervention”do⁡\(X=x\)\\mathrm\{do\}\(X\\\!=\\\!x\)ambiguous\. We propose*bipartite graphical causal models*\(BGCMs\), in which the structure of a system of equations is encoded by a bipartite graph with variable and equation nodes\. In this framework, a hard interventiondo\(fj:Xv=ξv\)\\mathrm\{do\}\(f\_\{j\}:X\_\{v\}\\\!=\\\!\\xi\_\{v\}\)specifies which equation is replaced, which variable is targeted, and at what value—resolving the ambiguity of the standard notion\. We demonstrate, through a detailed case study of a physical system, that this representation naturally corresponds to distinct real\-world interventions\. We formulate a Markov property in terms of a new graphical separation criterion \(BB\-separation\) that exploits the functional determinism inherent in the equations, and we extend it to settings with non\-random inputs\. We show how this gives rise to a do\-calculus for reasoning about domain invariances\. BGCMs strictly generalize CBNs and SCMs while retaining the ability to perform graphical causal reasoning\.

## 1Introduction

In many scientific disciplines—physics, engineering, economics, biology—complex systems are naturally described by systems of equations relating endogenous and exogenous variables\. Each equation represents an independent mechanism or physical law; the exogenous variables represent external inputs or noise\. Causal Bayesian networks \(CBNs\)\[[20](https://arxiv.org/html/2608.19831#bib.bib19)\]and structural causal models \(SCMs\)\[[20](https://arxiv.org/html/2608.19831#bib.bib19),[4](https://arxiv.org/html/2608.19831#bib.bib20)\]can represent many such systems, but not all\. In particular, systems at equilibrium—where feedback mechanisms create cyclic causal dependencies—can exhibit causal semantics that are fundamentally incompatible with these frameworks\.

A canonical example, due to[14](https://arxiv.org/html/2608.19831#bib.bib18), is a bathtub at equilibrium: the equilibrium relations between inflow, outflow, pressure, and depth form a system of equations whose causal interpretation depends on*which equation*is changed by an intervention\. Different interventions that set the same variable to the same value can have different effects on other variables, rendering the standard notion of a “perfect intervention”do⁡\(X=x\)\\mathrm\{do\}\(X=x\)ambiguous\[[3](https://arxiv.org/html/2608.19831#bib.bib21)\]\. Neither CBNs nor SCMs can express this distinction, since they associate each variable with a unique structural equation or Markov kernel\.

The key observation underlying our approach is that the structure of a system of equations is naturally encoded by a*bipartite graph*, with two types of nodes—variable nodes and equation nodes—connected by an edge whenever a variable appears in an equation\. By retaining both variable and equation nodes as first\-class citizens, the bipartite graph preserves information that is lost when projecting onto a directed graph over variables alone\. Building on Simon’s causal ordering algorithm\[[22](https://arxiv.org/html/2608.19831#bib.bib16)\], which derives a partial causal ordering of variables by analyzing the bipartite graph, we develop a full\-fledged causal modeling framework\. This approach is closely related to how engineers already reason about causality, for example in the equation\-based modeling language Modelica, where systems are specified as sets of “acausal” equations and causality is derived automatically through symbolic analysis\[[5](https://arxiv.org/html/2608.19831#bib.bib8)\]\.

In particular, the bipartite representation makes it possible to represent a hard intervention asdo\(fj:Xv=ξv\)\\mathrm\{do\}\(f\_\{j\}:X\_\{v\}=\\xi\_\{v\}\)—specifying which equationfjf\_\{j\}is replaced, which variableXvX\_\{v\}is targeted, and at what valueξv\\xi\_\{v\}\. While this intervention notion was already proposed by[3](https://arxiv.org/html/2608.19831#bib.bib21), it appeared ad hoc, and it remained unclear to what extent it provides a proper and natural mathematical abstraction of real\-world interventions\.

The contributions of this paper are:

1. 1\.We formally define*bipartite graphical causal models*\(BGCMs\) and show how they strictly extend CBNs, acyclic SCMs, simple SCMs, and general SCMs\.
2. 2\.We demonstrate, through a complete analysis of all hard interventions on the bathtub system, that the BGCM intervention notiondo\(fj:Xv=ξv\)\\mathrm\{do\}\(f\_\{j\}:X\_\{v\}=\\xi\_\{v\}\)provides a*natural and physically meaningful*representation of real\-world interventions—each corresponding to a distinct physical procedure with distinct causal effects\.
3. 3\.We formulate a*Markov property*for BGCMs using a graphical separation criterion \(BB\-separation\) that generalizesdd\-separation to partially oriented bipartite graphs\. By encoding both the causal structure and the conditional independence structure in a single partially oriented bipartite graph—rather than in separate graphs as in[3](https://arxiv.org/html/2608.19831#bib.bib21)—we make the connection between causal and Markov semantics transparent\. BecauseBB\-separation exploits the functional determinism induced by the equations, the resulting Markov property is strictly stronger than the one obtainable from the Markov ordering graph of[3](https://arxiv.org/html/2608.19831#bib.bib21)viadd\-separation\.
4. 4\.We establish an*extended*Markov property for the case where some exogenous variables are treated as non\-random inputs, phrased in terms of transitional conditional independence\[[10](https://arxiv.org/html/2608.19831#bib.bib17)\]\.
5. 5\.We develop a*do\-calculus*for BGCMs by exploiting this connection: the Markov property yields domain invariances—relationships between observational and interventional distributions—that go beyond Pearl’s three rules\[[20](https://arxiv.org/html/2608.19831#bib.bib19)\]\.

## 2Background

### 2\.1Modeling Cyclic Causal Relations

Feedback mechanisms in dynamical systems may induce cyclic causal relationships at equilibrium\. Fast dynamical interactions can lead to effectively “instantaneous” causal cycles\. Examples arise across many disciplines: coupled oscillators in physics, supply–demand–price feedback loops in economics, gene regulatory networks in biology, and climate feedback mechanisms\. In many such applications, the ability to model causal cycles is essential, and acyclic models are insufficient\.

We briefly review some existing causal modeling frameworks and their relationships\. A*causal Bayesian network*\(CBN\) consists of a directed acyclic graph \(DAG\) together with a collection of Markov kernels, one for each variable given its parents in the DAG\[[20](https://arxiv.org/html/2608.19831#bib.bib19)\]\. An*acyclic structural causal model*\(acyclic SCM\) consists of a set of structural equations of the formXi:=fi​\(pa⁡\(Xi\),Ui\)X\_\{i\}:=f\_\{i\}\(\\mathrm\{pa\}\(X\_\{i\}\),U\_\{i\}\), wherepa⁡\(Xi\)\\mathrm\{pa\}\(X\_\{i\}\)denotes the parents ofXiX\_\{i\}andUiU\_\{i\}is an exogenous noise variable, together with an acyclic causal graph\[[20](https://arxiv.org/html/2608.19831#bib.bib19)\]\. An*SCM*extends this to allow cyclic causal graphs, with each equation having a unique “dependent variable”\[[4](https://arxiv.org/html/2608.19831#bib.bib20)\]\. While general SCMs can be complicated to work with, the subclass of*simple SCMs*allows for \(sufficiently weak\) cycles and retains most of the convenient mathematical properties of acyclic SCMs\[[4](https://arxiv.org/html/2608.19831#bib.bib20)\]\.

These frameworks form a hierarchy of increasing generality:

CBNs⊂acyclic SCMs⊂simple SCMs⊂SCMs⊂BGCMs⊂CCMs,\\begin\{split\}\\text\{CBNs\}&\\subset\\text\{acyclic SCMs\}\\subset\\text\{simple SCMs\}\\\\ &\\subset\\text\{SCMs\}\\subset\\text\{BGCMs\}\\subset\\text\{CCMs\},\\end\{split\}where BGCMs are bipartite graphical causal models \(introduced in this paper\) and CCMs are causal constraint models\[[1](https://arxiv.org/html/2608.19831#bib.bib22)\]\. BGCMs occupy a position in this hierarchy that balances model flexibility with the ability to perform causal reasoning\.

[22](https://arxiv.org/html/2608.19831#bib.bib16)introduced the*causal ordering algorithm*, which derives a causal interpretation of a system of equations from its structural properties\. The key idea is that given a system of equations with designated exogenous variables, one can determine a partial ordering on the endogenous variables by analyzing which subsets of equations can be solved for which subsets of variables, and in what order\. The original version of the algorithm as proposed by[22](https://arxiv.org/html/2608.19831#bib.bib16)required solving NP\-hard subproblems\. Later,[19](https://arxiv.org/html/2608.19831#bib.bib14)proposed a computationally efficient version based on perfect matchings\.

[3](https://arxiv.org/html/2608.19831#bib.bib21)expanded upon Simon’s approach to causality by using his algorithm to construct two different graphs out of a set of equations: the*causal ordering graph*\(a directed cluster graph that represents causal effects of certain interventions\) and the*Markov ordering graph*\(a directed graph on variable nodes, obtained by declustering and marginalizing out equation nodes, from which conditional independences can be read off viadd\-separation\)\.

## 3Bipartite Graphical Causal Models

### 3\.1Systems of Equations and Bipartite Graphs

We consider a system of equations involving a set of variablesXV=\(Xv\)v∈VX\_\{V\}=\(X\_\{v\}\)\_\{v\\in V\}taking values in standard Borel spaces\(𝒳v\)v∈V\(\\mathcal\{X\}\_\{v\}\)\_\{v\\in V\}, and partitioned into*endogenous*variablesXV∖UX\_\{V\\setminus U\}and*exogenous*variablesXUX\_\{U\}, for someU⊆VU\\subseteq V\. The equations\{fj\}j∈F\\\{f\_\{j\}\\\}\_\{j\\in F\}are of the form0=ϕj​\(Xnb⁡\(fj\)\)0=\\phi\_\{j\}\(X\_\{\\mathrm\{nb\}\(f\_\{j\}\)\}\), whereϕj\\phi\_\{j\}is a measurable function andnb⁡\(fj\)⊆V\\mathrm\{nb\}\(f\_\{j\}\)\\subseteq Vdenotes the set of variables appearing in equationfjf\_\{j\}\.

###### Definition 1\(Bipartite graph of a system of equations\)\.

The*bipartite graph*of a system of equations is the undirected bipartite graphG=\(V,F,E\)G=\(V,F,E\), whereVVis the set of variable nodes,FFis the set of equation nodes, andE⊆V×FE\\subseteq V\\times Fcontains an edge\(v,f\)\(v,f\)if and only if variableXvX\_\{v\}appears in equationff\.

We illustrate this with a running example, a simplification of the bathtub system in\[[14](https://arxiv.org/html/2608.19831#bib.bib18)\]\.

![Refer to caption](https://arxiv.org/html/2608.19831v1/Gemini_bathtub.jpg)Figure 1:The bathtub system \(rendered by Google Gemini\)\.###### Example 2\(Bathtub at equilibrium\)\.

Consider a bathtub with constant inflow of water at equilibrium \(Figure[1](https://arxiv.org/html/2608.19831#S3.F1)\)\. The endogenous variables are:XOX\_\{O\}\(water outflow through the drain\),XDX\_\{D\}\(water depth\), andXPX\_\{P\}\(pressure at the drain\)\. The exogenous variables are:XIX\_\{I\}\(water inflow from faucet\),XKX\_\{K\}\(drain area\), andXgX\_\{g\}\(gravitational acceleration\)\. The equilibrium is described by three independent mechanisms:

f1:\\displaystyle f\_\{1\}:0=XI−XO\\displaystyle\\quad 0=X\_\{I\}\-X\_\{O\}\(1\)f2:\\displaystyle f\_\{2\}:0=XK​XP−XO\\displaystyle\\quad 0=X\_\{K\}\\sqrt\{X\_\{P\}\}\-X\_\{O\}\(2\)f3:\\displaystyle f\_\{3\}:0=Xg​XD−XP\\displaystyle\\quad 0=X\_\{g\}X\_\{D\}\-X\_\{P\}\(3\)Equation \([1](https://arxiv.org/html/2608.19831#S3.E1)\) states that at equilibrium, outflow equals inflow\. Equation \([2](https://arxiv.org/html/2608.19831#S3.E2)\) is Torricelli’s law: outflow is proportional to the drain area and the square root of the pressure\. Equation \([3](https://arxiv.org/html/2608.19831#S3.E3)\) is Stevin’s law: pressure is proportional to depth and gravitational acceleration\.

The bipartite graph of this system has variable nodes\{XO,XP,XD,XI,XK,Xg\}\\\{X\_\{O\},X\_\{P\},X\_\{D\},X\_\{I\},X\_\{K\},X\_\{g\}\\\}and equation nodes\{f1,f2,f3\}\\\{f\_\{1\},f\_\{2\},f\_\{3\}\\\}, with edges connecting each equation to the variables appearing in it\. We use squares for equation nodes, circles for endogenous nodes, while exogenous variables\{XI,XK,Xg\}\\\{X\_\{I\},X\_\{K\},X\_\{g\}\\\}are shown without circles:

XOX\_\{O\}XPX\_\{P\}XDX\_\{D\}f1f\_\{1\}f2f\_\{2\}f3f\_\{3\}XIX\_\{I\}XKX\_\{K\}XgX\_\{g\}

### 3\.2Causal Ordering and Partial Orientation

Given a bipartite graphG=\(V,F,E\)G=\(V,F,E\)and a setU⊆VU\\subseteq Vof exogenous variables, Simon’s causal ordering algorithm produces a partial orientation ofGGthat encodes the causal structure\.

The algorithm first finds a*perfect matching*MMof the subgraphG\(V∖U\)∪FG\_\{\(V\\setminus U\)\\cup F\}, i\.e\., a subset of edges such that each endogenous variable node and each equation node is incident to exactly one edge inMM\.111For simplicity of exposition, we assume throughout that there is such a perfect matching\. If this is not the case, one can make use of the Dulmage\-Mendelsohn decomposition\[[3](https://arxiv.org/html/2608.19831#bib.bib21)\]\.This matching associates each equation with a unique endogenous variable, which can be thought of as the variable that the equation “solves for\.”[19](https://arxiv.org/html/2608.19831#bib.bib14)showed that the perfect matching can be found efficiently using the Hopcroft\-Karp algorithm\[[13](https://arxiv.org/html/2608.19831#bib.bib11)\]\.

We then define an equivalence relation on the nodes ofGGthat identifies nodes belonging to the same cluster\.

###### Definition 3\(Equivalence relation and clusters\)\.

Given a bipartite graphG=\(V,F,E\)G=\(V,F,E\), subsetU⊆VU\\subseteq V, and perfect matchingMMofG\(V∖U\)∪FG\_\{\(V\\setminus U\)\\cup F\}, define∼\\simas the equivalence relation onV∪FV\\cup Fgenerated by the following:a∼ba\\sim bifa\-⁣\-b∈M\\text\{$a\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}b$\}\\in M, or ifaaandbblie on a closedMM\-alternating walk \(i\.e\., a walk that alternates between matched and unmatched edges and returns to its starting node\)\. The equivalence class of a nodeaais denoted\[a\]\[a\], and we refer to this as a “cluster”\.

###### Lemma 4\([7](https://arxiv.org/html/2608.19831#bib.bib15)\)\.

The equivalence relation∼\\simdepends only on the bipartite graphGGand the set of exogenous variablesUU, not on the choice of perfect matchingMM\.

Note that each exogenous node forms a singleton cluster\. Using this equivalence relation, we define the partial orientation of the bipartite graph\.

###### Definition 5\(Partial orientation\)\.

The*partial orientation*\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilofGGis obtained by orienting each edgev\-⁣\-f∈Ev\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}f\\in E\(withv∈Vv\\in V,f∈Ff\\in F\) as follows:

v\-⁣\-f↦\{v→fif​v≁f,v=⁣=fif​v∼f\.v\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}f\\mapsto\\begin\{cases\}v\\to f&\\text\{if \}v\\not\\sim f,\\\\ v\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}f&\\text\{if \}v\\sim f\.\\end\{cases\}

The mappingG↦\#�GG\\mapsto\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}is equivalent to Simon’s causal ordering algorithm\[[22](https://arxiv.org/html/2608.19831#bib.bib16)\]\. Directed edgesv→fv\\to findicate that variablevvis an input to equationfffrom a different \(earlier\) cluster\. Double\-undirected edgesv=⁣=fv\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}findicate thatvvandffbelong to the same cluster\. Note that since matched pairs always satisfyv∼fv\\sim f, matched edges are always oriented asv=⁣=fv\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}f\.222In Simon’s original formulation, matched edges are directed asf→vf\\to v, suggesting thatff“solves for”vv\. We usev=⁣=fv\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}funiformly because solvability is not guaranteed by the graph structure alone, but is an additional assumption \(Assumption[10](https://arxiv.org/html/2608.19831#Thmtheorem10)\)\.

###### Example 6\(Bathtub: causal ordering\)\.

For the bathtub system \(Example[2](https://arxiv.org/html/2608.19831#Thmtheorem2)\), the \(unique\) perfect matching associatesf1f\_\{1\}withXOX\_\{O\},f2f\_\{2\}withXPX\_\{P\}, andf3f\_\{3\}withXDX\_\{D\}\. The partially oriented graph is:

XOX\_\{O\}XPX\_\{P\}XDX\_\{D\}f1f\_\{1\}f2f\_\{2\}f3f\_\{3\}XIX\_\{I\}XKX\_\{K\}XgX\_\{g\}

This encodes the causal ordering: first solvef1f\_\{1\}forXOX\_\{O\}in terms ofXIX\_\{I\}, yieldingXO=XIX\_\{O\}=X\_\{I\}; then solvef2f\_\{2\}forXPX\_\{P\}in terms ofXOX\_\{O\}andXKX\_\{K\}, yieldingXP=XI2/XK2X\_\{P\}=X\_\{I\}^\{2\}/X\_\{K\}^\{2\}; finally solvef3f\_\{3\}forXDX\_\{D\}in terms ofXPX\_\{P\}andXgX\_\{g\}, yieldingXD=XI2/\(XK2​Xg\)X\_\{D\}=X\_\{I\}^\{2\}/\(X\_\{K\}^\{2\}X\_\{g\}\)\. The clusters are\{XI\},\{XK\},\{Xg\},\{f1,XO\},\{f2,XP\},\{f3,XD\}\\\{X\_\{I\}\\\},\\\{X\_\{K\}\\\},\\\{X\_\{g\}\\\},\\\{f\_\{1\},X\_\{O\}\\\},\\\{f\_\{2\},X\_\{P\}\\\},\\\{f\_\{3\},X\_\{D\}\\\}\.

We now formally define what we mean by “causal ordering”\. We define a*walk*in\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilas an alternating sequence of nodes and edgesn0,e1,n1,e2,…,ek,nkn\_\{0\},e\_\{1\},n\_\{1\},e\_\{2\},\\ldots,e\_\{k\},n\_\{k\}where each edgeeie\_\{i\}connectsni−1n\_\{i\-1\}andnin\_\{i\};k=0k=0corresponds with a trivial walk\. A*path*is a walk in which no node repeats\.

###### Definition 7\(Anterior\)\.

A nodeaais*anterior*to a nodebbin\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilif there is a walk fromaatobbconsisting only of→\\toand=⁣=\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}edges\. That is, a walk of the formn0,e1,n1,e2,…,ek,nkn\_\{0\},e\_\{1\},n\_\{1\},e\_\{2\},\\ldots,e\_\{k\},n\_\{k\}wheren0=an\_\{0\}=a,nk=bn\_\{k\}=b, and each edgeeie\_\{i\}is eitherni−1→nin\_\{i\-1\}\\to n\_\{i\}orni−1=⁣=nin\_\{i\-1\}\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}n\_\{i\}\. We writeant\#�G​\(B\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(B\)for the set of all nodes anterior to some node inB⊆V∪FB\\subseteq V\\cup F\.

Mathematically, the relation “is anterior to” is a*partial order*on the clusters \(it is reflexive and transitive; it is antisymmetric because any two mutually anterior nodes lie in the same cluster\)\. Simon’s insight was that this mathematical relationship captures what we perceive as causes and their effects: we consideraaa cause ofbbprecisely ifa∈ant\#�G​\(b\)a\\in\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(b\)\.

### 3\.3Solutions, Distributions, and Markov Kernels

By solving the system of equations according to the causal ordering, we obtain*solution functions*that express all endogenous variables in terms of the exogenous variables\. When a cluster contains more than one equation, the equations in the cluster must be solved simultaneously for the variables in that cluster\.

###### Definition 8\(Solution function\)\.

A*solution function*for a system of equations with exogenous variablesXUX\_\{U\}and endogenous variablesXV∖UX\_\{V\\setminus U\}is a functionΦ:𝒳U→𝒳V\\Phi:\\mathcal\{X\}\_\{U\}\\to\\mathcal\{X\}\_\{V\}such thatΦU​\(xU\)=xU\\Phi\_\{U\}\(x\_\{U\}\)=x\_\{U\}, andΦ⁡\(xU\)\\Phi\(x\_\{U\}\)satisfies all equations for everyxU∈𝒳Ux\_\{U\}\\in\\mathcal\{X\}\_\{U\}\.

If we assume that all exogenous variables are mutually independent random variables with distributionsXu∼ℙ⁡\(Xu\)X\_\{u\}\\sim\\mathbb\{P\}\(X\_\{u\}\)foru∈Uu\\in U, then the*joint distribution*ℙ⁡\(XV\)\\mathbb\{P\}\(X\_\{V\}\)of all variables is obtained as the pushforward of the product distribution⨂u∈Uℙ⁡\(Xu\)\\bigotimes\_\{u\\in U\}\\mathbb\{P\}\(X\_\{u\}\)through the solution functionΦ\\Phi\. This corresponds to the distribution ofXVX\_\{V\}under the sampling scheme

Xu∼ℙ⁡\(Xu\)for​u∈U,XV=Φ⁡\(XU\)\.X\_\{u\}\\sim\\mathbb\{P\}\(X\_\{u\}\)\\ \\ \\text\{for \}u\\in U,\\qquad X\_\{V\}=\\Phi\(X\_\{U\}\)\.More generally, we can treat some exogenous variables as random and others as non\-random\. This yields a*Markov kernel*rather than a distribution\. If one only assigns independent distributions to exogenous variables in subsetU∖JU\\setminus JwithJ⊆UJ\\subseteq U, one obtains the Markov kernelℙ\(XV∥XJ\)\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\), corresponding to the sampling scheme

Xu∼ℙ⁡\(Xu\)for​u∈U∖J,XV=Φ⁡\(XJ,XU∖J\)X\_\{u\}\\sim\\mathbb\{P\}\(X\_\{u\}\)\\ \\ \\text\{for \}u\\in U\\setminus J,\\qquad X\_\{V\}=\\Phi\(X\_\{J\},X\_\{U\\setminus J\}\)where the*input variables*XJX\_\{J\}are left unconstrained and we make no assumptions about their distribution\. For instance, treatingXIX\_\{I\}as a non\-random input andXK,XgX\_\{K\},X\_\{g\}as random yields the Markov kernelℙ\(XK,Xg,XO,XP,XD∥XI\)\\mathbb\{P\}\(X\_\{K\},X\_\{g\},X\_\{O\},X\_\{P\},X\_\{D\}\\,\\\|\\,X\_\{I\}\), which specifies the joint distribution ofXK,Xg,XO,XP,XDX\_\{K\},X\_\{g\},X\_\{O\},X\_\{P\},X\_\{D\}for every possible valuexIx\_\{I\}\.

###### Definition 9\(Unique solvability of a cluster\)\.

A cluster\[c\]\[c\]in the partially oriented graph\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilis called*uniquely solvable*if the equations inF∩\[c\]F\\cap\[c\]can be solved for the variablesV∩\[c\]V\\cap\[c\]in terms ofpa\#�G​\(\[c\]\)\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[c\]\), and the local solution functionΦ\[c\]:𝒳pa\#�G​\(\[c\]\)→𝒳\[c\]∩V\\Phi^\{\[c\]\}:\\mathcal\{X\}\_\{\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[c\]\)\}\\to\\mathcal\{X\}\_\{\[c\]\\cap V\}is unique\.

If all local unique solvability assumptions are met, this guarantees existence and uniqueness of a global solution function\. We collect our assumptions so far:

###### Assumption 10\.

For a system of equations

0=ϕj​\(Xnb⁡\(fj\)\),j∈F0=\\phi\_\{j\}\(X\_\{\\mathrm\{nb\}\(f\_\{j\}\)\}\),\\quad j\\in Fwith corresponding bipartite graphG=\(V,F,E\)G=\(V,F,E\), exogenous variablesU⊆VU\\subseteq V, standard Borel spaces\(𝒳v\)v∈V\(\\mathcal\{X\}\_\{v\}\)\_\{v\\in V\}:

1. 1\.The functionsϕj:𝒳nb⁡\(fj\)→ℝ\\phi\_\{j\}:\\mathcal\{X\}\_\{\\mathrm\{nb\}\(f\_\{j\}\)\}\\to\\mathbb\{R\}are measurable\.
2. 2\.The subgraphG\(V∖U\)∪FG\_\{\(V\\setminus U\)\\cup F\}has a perfect matching\.
3. 3\.The exogenous variables are*variation independent*: their joint value space is a Cartesian product∏u∈U𝒳u\\prod\_\{u\\in U\}\\mathcal\{X\}\_\{u\}\.
4. 4\.The system is*clusterwise uniquely solvable*: each endogenous cluster\[v\]\[v\]\(forv∈V∖Uv\\in V\\setminus U\) is uniquely solvable\.

###### Proposition 11\.

Under Assumption[10](https://arxiv.org/html/2608.19831#Thmtheorem10), there exists a unique solution functionΦ:𝒳U→𝒳V\\Phi:\\mathcal\{X\}\_\{U\}\\to\\mathcal\{X\}\_\{V\}\.

###### Proof\.

The local solution functions\(Φ\[v\]\)v∈V∖U\(\\Phi^\{\[v\]\}\)\_\{v\\in V\\setminus U\}combine into a system of equations

Xv=Φv\[v\]​\(Xpa\#�G​\(\[v\]\)\)v∈V\\displaystyle X\_\{v\}=\\Phi\_\{v\}^\{\[v\]\}\(X\_\{\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[v\]\)\}\)\\qquad v\\in Vthat has an acyclic structure\. Recursive substitution of the local solution functions of each cluster along the causal ordering then yields the global solution function\. ∎

This unique \(global\) solution function then induces a unique joint distributionℙ⁡\(XV\)\\mathbb\{P\}\(X\_\{V\}\)and unique Markov kernelsℙ\(XV∥XJ\)\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\)forJ⊆UJ\\subseteq U\.

## 4Markov Property

In this section, we formulate a Markov property for bipartite graphical causal models that relates the conditional independence structure of the joint distribution to the graphical structure of the partially oriented bipartite graph\.

### 4\.1BB\-Separation

We define a graphical separation criterion for partially oriented bipartite graphs, called*BB\-separation*\(for “bipartite”\), an analog of classicaldd\-separation that is appropriate for our setting\. It combines ideas from the segment\-based formulation ofσ\\sigma\-separation\[[8](https://arxiv.org/html/2608.19831#bib.bib23)\]to deal with cycles \(clusters with more than a single variable\), and fromDD\-separation\[[12](https://arxiv.org/html/2608.19831#bib.bib2)\]to take into account deterministic relations \(each endogenous cluster is a deterministic function of its parents\)\.

###### Definition 12\(Segments, exits, collider/non\-collider segments\)\.

A walksson\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilcan be partitioned into*segments*: consecutive maximal subwalkss1,…,sms\_\{1\},\\dots,s\_\{m\}of the formsi,1=⁣=…=⁣=si,kis\_\{i,1\}\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}\\dots\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}s\_\{i,k\_\{i\}\}\(possiblyki=1k\_\{i\}=1\), i\.e\., with all edges double\-undirected \(=⁣=\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}\)\. We call a boundary node of a segment an*exit*if its bounding edge points out of the segment or it is an end node of the walk\. That is,si,1s\_\{i,1\}is an exit if the edge onssto the left of it is…←si,1\\ldots\\mathrel\{\\leftarrow\}s\_\{i,1\}or if it is the first node ofss\(i=1i=1\), andsi,kis\_\{i,k\_\{i\}\}is an exit if the edge onssto the right of it issi,ki→…s\_\{i,k\_\{i\}\}\\to\\ldotsor if it is the last node ofss\(i=mi=m\)\. A segment with no exit is a*collider segment*; a segment with one or two exits is a*non\-collider segment*\.

The following notion tracks deterministic relations that are imposed by the structure of the bipartite graph\.

###### Definition 13\(Functionally determined\)\.

LetC⊆VC\\subseteq Vbe a subset of variable nodes in\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil\. DefineC0:=CC\_\{0\}:=Cand

Cn\+1:=Cn∪\{v∈V∖U:pa\#�G​\(\[v\]\)⊆Cn\}\.C\_\{n\+1\}:=C\_\{n\}\\cup\\\{v\\in V\\setminus U:\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[v\]\)\\subseteq C\_\{n\}\\\}\.We definefdet\#�G​\(C\):=⋃n≥0Cn\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\):=\\bigcup\_\{n\\geq 0\}C\_\{n\}and refer to those as the variable nodes that are*functionally determined byCC*\.

Note that exogenous variable nodes are only functionally determined byCCif they are inCC\.

With these definitions in place, we define:

###### Definition 14\(BB\-blocking\)\.

ForC⊆VC\\subseteq V, the walk is called*BB\-blocked byCC*if it contains:

1. 1\.a*collider segment*that does not intersectant\#�G​\(C\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\), or
2. 2\.a*non\-collider segment*with an exit infdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\.

Otherwise, the walk is called*BB\-open givenCC*\.

###### Definition 15\(BB\-separation\)\.

LetA,B,C⊆VA,B,C\\subseteq Vbe sets of variable nodes\. We say that*AAandBBareBB\-separated givenCCin\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil*, written

A​⟂\#�GB​B\|C,A\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,C,if every walk from a node inAAto a node inBBisBB\-blocked byCCin\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil\.333It is equivalent to require only that every*path*fromAAtoBBbeBB\-blocked byCC\(Lemma[43](https://arxiv.org/html/2608.19831#Thmtheorem43)\); the path formulation is usually more convenient to check by hand\.

To build intuition, it is helpful to see howBB\-separation adaptsdd\-separation to the two features that distinguish partially oriented bipartite graphs from DAGs: clusters and determinism\.

*Clusters\.*The variables and equations of a cluster are solved jointly, so segments along a walk behave as a single indivisible unit\. A walk can enter or leave a segment only through an*exit*, and conditioning therefore interacts with a segment only through its exits\. This is the bipartite\-graph counterpart of collapsing a strongly connected component inσ\\sigma\-separation\[[8](https://arxiv.org/html/2608.19831#bib.bib23)\]: whether a segment blocks or transmits dependence is decided by the segment as a whole rather than node by node\. As indd\-separation, a*non\-collider*segment \(a chain, a fork, or an endpoint of the walk\) transmits dependence unless it is “pinned down” by the conditioning set, whereas a*collider*segment \(a common effect, entered by arrows from both sides\) blocks association unless it is “activated” by the conditioning set\.

*Determinism\.*Since each endogenous cluster is a deterministic function of its parents, conditioning onCCfixes not onlyXCX\_\{C\}but every variable that is functionally determined by it, i\.e\.,Xfdet\#�G​\(C\)X\_\{\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\}\. This is what “pinned down” means here: a non\-collider segment is already blocked once one of its exits lies infdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\), even if that exit is not itself inCC\. Activation of colliders, on the other hand, is governed byant\#�G​\(C\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\): a collider segment transmits dependence only if the common effect, or one of its descendants, actually belongs to the conditioning setCC\. It is precisely this extra blocking granted byfdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)that makesBB\-separation stronger than criteria that ignore determinism\. This intuition is made precise in Appendix[C](https://arxiv.org/html/2608.19831#A3)\.

### 4\.2Global Markov Property

###### Theorem 16\(Global Markov property\)\.

Suppose that Assumption[10](https://arxiv.org/html/2608.19831#Thmtheorem10)holds\. When assigning independent distributions to all exogenous variables, the resulting joint distributionℙ⁡\(XV\)\\mathbb\{P\}\(X\_\{V\}\)satisfies: for allA,B,C⊆VA,B,C\\subseteq V,

A​⟂\#�GBB\|C⟹XA​⟂⁣⟂ℙ⁡\(XV\)⁡XB\|XC\.A\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,C\\implies X\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\)\}X\_\{B\}\\,\|\\,X\_\{C\}\.

The Markov property “propagates” the independence of the exogenous variables through the equations along the partial ordering, yielding conditional independences among endogenous variables\.

###### Example 17\(Bathtub: Markov property\)\.

In the partially oriented bathtub graph \(Example[6](https://arxiv.org/html/2608.19831#Thmtheorem6)\), every path fromXDX\_\{D\}toXOX\_\{O\}must pass throughXPX\_\{P\}\(via the equation nodes\)\. One can verify thatXD​⟂\#�GBXO\|XPX\_\{D\}\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}X\_\{O\}\\,\|\\,X\_\{P\}, which implies the conditional independence:

XD⟂⁣⟂XO\|XP\.X\_\{D\}\\Indep X\_\{O\}\\,\|\\,X\_\{P\}\.This means the joint distribution factorizes as

ℙ⁡\(XD,XO,XP\)=ℙ⁡\(XD\|XP\)⊗ℙ⁡\(XO,XP\)\.\\mathbb\{P\}\(X\_\{D\},X\_\{O\},X\_\{P\}\)=\\mathbb\{P\}\(X\_\{D\}\\,\|\\,X\_\{P\}\)\\otimes\\mathbb\{P\}\(X\_\{O\},X\_\{P\}\)\.

### 4\.3Extended Global Markov Property

A more general version of the Markov property allows treating some exogenous variables as non\-random, using an extended notion of conditional independence\[[10](https://arxiv.org/html/2608.19831#bib.bib17)\]\.

###### Theorem 18\(Extended Global Markov property\)\.

Suppose Assumption[10](https://arxiv.org/html/2608.19831#Thmtheorem10)holds\. Treat exogenous variablesJ⊆UJ\\subseteq Uas non\-random, and assign independent distributions to exogenous variables inU∖JU\\setminus J, yielding Markov kernelℙ\(XV∥XJ\)\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\)\. Then for allA,B,C⊆VA,B,C\\subseteq Vsuch thatJ⊆B∪CJ\\subseteq B\\cup C:444One can replace the assumptionA​⟂\#�GB​B\|CA\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,CbyA​⟂\#�GBB∪J\|CA\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\cup J\\,\|\\,Cto obtain a result that is valid for*all*choices ofA,B,CA,B,C\.

A⟂B\#�GB\|C⟹XA⟂⁣⟂ℙ\(XV∥XJ\)XB\|XC\.A\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,C\\implies X\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\)\}X\_\{B\}\\,\|\\,X\_\{C\}\.

Concretely, the conditional independenceXA⟂⁣⟂ℙ\(XV∥XJ\)XB\|XCX\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\)\}X\_\{B\}\\,\|\\,X\_\{C\}for a Markov kernelℙ\(XV∥XJ\)\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\)means that there*exists*a Markov kernelQ\(XA∥XC\)Q\(X\_\{A\}\\,\\\|\\,X\_\{C\}\)\(not depending onXBX\_\{B\}\) such that

ℙ\(XA,XB,XC∥XJ\)=Q\(XA∥XC\)⊗ℙ\(XB,XC∥XJ\)\.\\mathbb\{P\}\(X\_\{A\},X\_\{B\},X\_\{C\}\\lVert X\_\{J\}\)=Q\(X\_\{A\}\\lVert X\_\{C\}\)\\otimes\\mathbb\{P\}\(X\_\{B\},X\_\{C\}\\lVert X\_\{J\}\)\.This is the notion of*transitional conditional independence*introduced by[10](https://arxiv.org/html/2608.19831#bib.bib17); it is asymmetric \(the roles ofAAandBBare not interchangeable\)\.555Our notion ofBB\-separation does not distinguish non\-random and random variables explicitly, which is why the conditionJ⊆B∪CJ\\subseteq B\\cup Cis needed here; see Appendix[D](https://arxiv.org/html/2608.19831#A4)\.

###### Example 19\(Bathtub: extended Markov property\)\.

In the bathtub model, treatingXIX\_\{I\}as non\-random yields the Markov kernelℙ\(XK,Xg,XO,XP,XD∥XI\)\\mathbb\{P\}\(X\_\{K\},X\_\{g\},X\_\{O\},X\_\{P\},X\_\{D\}\\,\\\|\\,X\_\{I\}\)\. SinceXD​⟂\#�GBXI\|XPX\_\{D\}\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}X\_\{I\}\\,\|\\,X\_\{P\}, the extended Markov property impliesXD⟂⁣⟂XI\|XPX\_\{D\}\\Indep X\_\{I\}\\,\|\\,X\_\{P\}, which means there exists a Markov kernelℙ\(XD∥XP\)\\mathbb\{P\}\(X\_\{D\}\\,\\\|\\,X\_\{P\}\)such that:

ℙ\(XD,XP∥XI\)=ℙ\(XD\|XP\)⊗ℙ\(XP∥XI\)\.\\mathbb\{P\}\(X\_\{D\},X\_\{P\}\\,\\\|\\,X\_\{I\}\)=\\mathbb\{P\}\(X\_\{D\}\\,\|\\,X\_\{P\}\)\\otimes\\mathbb\{P\}\(X\_\{P\}\\,\\\|\\,X\_\{I\}\)\.

## 5Interventions in Bipartite Graphical Causal Models

Causality is fundamentally about change: how does a system react to externally imposed modifications? In the BGCM framework, there are two types of elementary interventions:

1. 1\.Changing the distribution of an exogenous variable:replacingℙ⁡\(Xu\)\\mathbb\{P\}\(X\_\{u\}\)by another distributionℙ~​\(Xu\)\\tilde\{\\mathbb\{P\}\}\(X\_\{u\}\)\.
2. 2\.Replacing an equation:substituting equationfjf\_\{j\}by a different equationf~j\\tilde\{f\}\_\{j\}\.

A particularly important class of the second type is the*hard intervention*do\(fj:Xv=ξv\)\\mathrm\{do\}\(f\_\{j\}:X\_\{v\}=\\xi\_\{v\}\), which replaces equationfjf\_\{j\}by the equation0=Xv−ξv0=X\_\{v\}\-\\xi\_\{v\},666For𝒳v≠ℝ\\mathcal\{X\}\_\{v\}\\neq\\mathbb\{R\}one can more generally take0=𝟙\{ξv\}​\(Xv\)−10=\\mathbbm\{1\}\_\{\\\{\\xi\_\{v\}\\\}\}\(X\_\{v\}\)\-1\.thereby fixing variableXvX\_\{v\}to valueξv\\xi\_\{v\}via the intervened mechanismfjf\_\{j\}\. This notion naturally corresponds to concrete physical procedures: different choices offjf\_\{j\}lead to genuinely different real\-world implementations, even when targeting the same variableXvX\_\{v\}at the same valueξv\\xi\_\{v\}\(see Example[20](https://arxiv.org/html/2608.19831#Thmtheorem20)and Appendix[E](https://arxiv.org/html/2608.19831#A5)for detailed examples\)\.

### 5\.1Ambiguity of Perfect Interventions

A key insight of the BGCM framework is that the standard notiondo⁡\(Xv=ξv\)\\mathrm\{do\}\(X\_\{v\}=\\xi\_\{v\}\)for a “perfect intervention” can be ambiguous: different equations can be replaced to achieveXv=ξvX\_\{v\}=\\xi\_\{v\}, leading to different causal effects on other variables\[[3](https://arxiv.org/html/2608.19831#bib.bib21)\]\.

do\(f1:XD=ξD\)\\mathrm\{do\}\(f\_\{1\}:X\_\{D\}=\\xi\_\{D\}\)![Refer to caption](https://arxiv.org/html/2608.19831v1/Gemini_Generated_Image_djpsr5djpsr5djps.jpg)do\(f2:XD=ξD\)\\mathrm\{do\}\(f\_\{2\}:X\_\{D\}=\\xi\_\{D\}\)![Refer to caption](https://arxiv.org/html/2608.19831v1/Gemini_Generated_Image_76lv5076lv5076lv.jpg)do\(f3:XD=ξD\)\\mathrm\{do\}\(f\_\{3\}:X\_\{D\}=\\xi\_\{D\}\)![Refer to caption](https://arxiv.org/html/2608.19831v1/Gemini_Generated_Image_b1qmbob1qmbob1qm.jpg)Figure 2:Three different interventions on the bathtub that all set the water depth to a fixed value \(Example[20](https://arxiv.org/html/2608.19831#Thmtheorem20)\)\.###### Example 20\(Bathtub: ambiguous interventions\)\.

Consider setting the water depth to a fixed valueξD\\xi\_\{D\}in the bathtub model\. There are at least three distinct interventions that achieve this \(see Figure[2](https://arxiv.org/html/2608.19831#S5.F2)\)\. The partially oriented bipartite graphs for these and all other hard interventions are shown in Figure[3](https://arxiv.org/html/2608.19831#A6.F3)in Appendix[F](https://arxiv.org/html/2608.19831#A6)\.

\(i\)do\(f1:XD=ξD\)\\mathrm\{do\}\(f\_\{1\}:X\_\{D\}=\\xi\_\{D\}\):Replace the equilibrium condition \([1](https://arxiv.org/html/2608.19831#S3.E1)\) by0=XD−ξD0=X\_\{D\}\-\\xi\_\{D\}\. Physically, this may correspond to cutting the bathtub at heightξD\\xi\_\{D\}and ensuring it overflows\. The causal ordering*reverses*:f~1\\tilde\{f\}\_\{1\}determinesXDX\_\{D\}, thenf3f\_\{3\}determinesXPX\_\{P\}fromXDX\_\{D\}andXgX\_\{g\}, and finallyf2f\_\{2\}determinesXOX\_\{O\}fromXPX\_\{P\}andXKX\_\{K\}\. The solution isXO=XK​Xg​ξDX\_\{O\}=X\_\{K\}\\sqrt\{X\_\{g\}\\xi\_\{D\}\},XP=Xg​ξDX\_\{P\}=X\_\{g\}\\xi\_\{D\},XD=ξDX\_\{D\}=\\xi\_\{D\}\.

\(ii\)do\(f2:XD=ξD\)\\mathrm\{do\}\(f\_\{2\}:X\_\{D\}=\\xi\_\{D\}\):Replace Torricelli’s law \([2](https://arxiv.org/html/2608.19831#S3.E2)\) by0=XD−ξD0=X\_\{D\}\-\\xi\_\{D\}\. Physically, this could involve disabling the drain and rerouting the inflow to the outflow once the water has reached heightξD\\xi\_\{D\}\. The causal ordering changes:f1f\_\{1\}determinesXOX\_\{O\},f~2\\tilde\{f\}\_\{2\}determinesXDX\_\{D\}, andf3f\_\{3\}now determinesXPX\_\{P\}fromXDX\_\{D\}\. The solution isXO=XIX\_\{O\}=X\_\{I\},XP=Xg​ξDX\_\{P\}=X\_\{g\}\\xi\_\{D\},XD=ξDX\_\{D\}=\\xi\_\{D\}\.

\(iii\)do\(f3:XD=ξD\)\\mathrm\{do\}\(f\_\{3\}:X\_\{D\}=\\xi\_\{D\}\):Replace Stevin’s law \([3](https://arxiv.org/html/2608.19831#S3.E3)\) by0=XD−ξD0=X\_\{D\}\-\\xi\_\{D\}\. Physically, this can be achieved by sealing the bathtub at heightξD\\xi\_\{D\}and ensuring it is completely filled\. The causal ordering is preserved:f1f\_\{1\}still determinesXOX\_\{O\},f2f\_\{2\}still determinesXPX\_\{P\}, andf~3\\tilde\{f\}\_\{3\}determinesXDX\_\{D\}\. The solution isXO=XIX\_\{O\}=X\_\{I\},XP=XI2/XK2X\_\{P\}=X\_\{I\}^\{2\}/X\_\{K\}^\{2\},XD=ξDX\_\{D\}=\\xi\_\{D\}\.

These three interventions all setXD=ξDX\_\{D\}=\\xi\_\{D\}but have different effects onXOX\_\{O\}andXPX\_\{P\}\. Therefore, the notiondo⁡\(XD=ξD\)\\mathrm\{do\}\(X\_\{D\}=\\xi\_\{D\}\)is ambiguous and must be refined todo\(fj:XD=ξD\)\\mathrm\{do\}\(f\_\{j\}:X\_\{D\}=\\xi\_\{D\}\)\.

### 5\.2Hard Interventions for the Bathtub

Table[1](https://arxiv.org/html/2608.19831#S5.T1)summarizes all possible hard interventions for the bathtub model\. In Appendix[E](https://arxiv.org/html/2608.19831#A5)we propose possible physical implementations of all feasible hard interventions\. That analysis shows that this is more than a purely mathematical exercise\. Not every combination of target equation and target variable yields a uniquely solvable system \(signaled by the corresponding intervened graph having no perfect matching\); indeed, some interventions \(marked with∅\\emptyset\) generically lead to systems with no solution\. In those cases it would be futile to attempt to implement such interventions\.

Table 1:Feasibility of hard interventionsdo\(fj:Xv=ξv\)\\mathrm\{do\}\(f\_\{j\}:X\_\{v\}=\\xi\_\{v\}\)for the bathtub\. Checkmarks indicate uniquely solvable systems;∅\\emptysetindicates the system is not uniquely solvable\.Table[2](https://arxiv.org/html/2608.19831#S5.T2)shows the solution functions for all well\-defined hard interventions, illustrating how different interventions lead to different causal effects\.

Table 2:Solution functions for all hard interventions on the bathtub model\.XOX\_\{O\}XPX\_\{P\}XDX\_\{D\}observationalXIX\_\{I\}XI2XK2\\frac\{X\_\{I\}^\{2\}\}\{X\_\{K\}^\{2\}\}XI2XK2​Xg\\frac\{X\_\{I\}^\{2\}\}\{X\_\{K\}^\{2\}X\_\{g\}\}do⁡\(XI=ξI\)\\mathrm\{do\}\(X\_\{I\}=\\xi\_\{I\}\)ξI\\xi\_\{I\}ξI2XK2\\frac\{\\xi\_\{I\}^\{2\}\}\{X\_\{K\}^\{2\}\}ξI2XK2​Xg\\frac\{\\xi\_\{I\}^\{2\}\}\{X\_\{K\}^\{2\}X\_\{g\}\}do⁡\(XK=ξK\)\\mathrm\{do\}\(X\_\{K\}=\\xi\_\{K\}\)XIX\_\{I\}XI2ξK2\\frac\{X\_\{I\}^\{2\}\}\{\\xi\_\{K\}^\{2\}\}XI2ξK2​Xg\\frac\{X\_\{I\}^\{2\}\}\{\\xi\_\{K\}^\{2\}X\_\{g\}\}do⁡\(Xg=ξg\)\\mathrm\{do\}\(X\_\{g\}=\\xi\_\{g\}\)XIX\_\{I\}XI2XK2\\frac\{X\_\{I\}^\{2\}\}\{X\_\{K\}^\{2\}\}XI2XK2​ξg\\frac\{X\_\{I\}^\{2\}\}\{X\_\{K\}^\{2\}\\xi\_\{g\}\}do\(f1:XO=ξO\)\\mathrm\{do\}\(f\_\{1\}:X\_\{O\}=\\xi\_\{O\}\)ξO\\xi\_\{O\}ξO2XK2\\frac\{\\xi\_\{O\}^\{2\}\}\{X\_\{K\}^\{2\}\}ξO2XK2​Xg\\frac\{\\xi\_\{O\}^\{2\}\}\{X\_\{K\}^\{2\}X\_\{g\}\}do\(f1:XP=ξP\)\\mathrm\{do\}\(f\_\{1\}:X\_\{P\}=\\xi\_\{P\}\)ξP​XK\\sqrt\{\\xi\_\{P\}\}\\,X\_\{K\}ξP\\xi\_\{P\}ξPXg\\frac\{\\xi\_\{P\}\}\{X\_\{g\}\}do\(f1:XD=ξD\)\\mathrm\{do\}\(f\_\{1\}:X\_\{D\}=\\xi\_\{D\}\)XK​Xg​ξDX\_\{K\}\\sqrt\{X\_\{g\}\\xi\_\{D\}\}Xg​ξDX\_\{g\}\\xi\_\{D\}ξD\\xi\_\{D\}do\(f2:XP=ξP\)\\mathrm\{do\}\(f\_\{2\}:X\_\{P\}=\\xi\_\{P\}\)XIX\_\{I\}ξP\\xi\_\{P\}ξPXg\\frac\{\\xi\_\{P\}\}\{X\_\{g\}\}do\(f2:XD=ξD\)\\mathrm\{do\}\(f\_\{2\}:X\_\{D\}=\\xi\_\{D\}\)XIX\_\{I\}Xg​ξDX\_\{g\}\\xi\_\{D\}ξD\\xi\_\{D\}do\(f3:XD=ξD\)\\mathrm\{do\}\(f\_\{3\}:X\_\{D\}=\\xi\_\{D\}\)XIX\_\{I\}XI2XK2\\frac\{X\_\{I\}^\{2\}\}\{X\_\{K\}^\{2\}\}ξD\\xi\_\{D\}
### 5\.3Interventions Change the Causal Structure

The bathtub system cannot be modeled as a CBN or an SCM, becausedo⁡\(XD=ξD\)\\mathrm\{do\}\(X\_\{D\}=\\xi\_\{D\}\)does not have a unique meaning\. In a CBN or SCM, a perfect interventiondo⁡\(Xv=ξv\)\\mathrm\{do\}\(X\_\{v\}=\\xi\_\{v\}\)replaces a unique structural equation \(the one withXvX\_\{v\}as its dependent variable\), but in the bathtub there is no such unique association\.

An important caveat is that hard interventions can change the bipartite graph and its partial orientation, and hence the conditional independence structure\. For example, the interventiondo\(f1:XD=ξD\)\\mathrm\{do\}\(f\_\{1\}:X\_\{D\}=\\xi\_\{D\}\)on the bathtub reverses the causal ordering entirely: instead ofXO→XP→XDX\_\{O\}\\to X\_\{P\}\\to X\_\{D\}, the ordering becomesXD→XP→XOX\_\{D\}\\to X\_\{P\}\\to X\_\{O\}\(in terms of the directed part of the partially oriented graph\), as can be seen in Figure[3](https://arxiv.org/html/2608.19831#A6.F3)\. This is another phenomenon that has no counterpart in standard CBN or SCM frameworks\.

## 6Domain Invariances

A central application of causal models is reasoning about what changes—and what remains invariant—across different experimental conditions or “domains\.” In CBNs, Pearl’s three rules of the do\-calculus\[[20](https://arxiv.org/html/2608.19831#bib.bib19)\]formalize such invariances for observational versus interventional distributions\. We now develop an analogous theory for BGCMs\.

### 6\.1The General Recipe

To relate the distributions in two domains \(e\.g\., an observational domain and an interventional domain\), we employ the following procedure:

1. 1\.Construct the joint model:Introduce an exogenous*domain indicator*input variableRRand write the equations of both domains as a single system, where the equations that differ between domains depend onRR\.
2. 2\.Construct the bipartite graphGRG^\{R\}:Build the bipartite graph of the joint model, which includesRRas an exogenous variable node connected to the equations in whichRRoccurs\.
3. 3\.Run causal ordering:Compute the partial orientation\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrGR\\hfil\\textstyle G^\{R\}\\hfilofGRG^\{R\}\.
4. 4\.Check solvability:Verify that Assumption[10](https://arxiv.org/html/2608.19831#Thmtheorem10)holds for the joint model\.
5. 5\.Apply the Markov property:Use Theorem[18](https://arxiv.org/html/2608.19831#Thmtheorem18)on\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrGR\\hfil\\textstyle G^\{R\}\\hfilto derive conditional independences involvingRR, which translate into invariances across domains\.

Except for the solvability check \(step 4\), this is a purely graphical procedure\. When applying the conditional invariances \(Examples[22](https://arxiv.org/html/2608.19831#Thmtheorem22)and[24](https://arxiv.org/html/2608.19831#Thmtheorem24)\), one must carefully handle the null sets arising from conditioning on continuous variables\. Tracking the null sets rigorously requires additional bookkeeping that we do not spell out here\[[9](https://arxiv.org/html/2608.19831#bib.bib24), see\]\.

### 6\.2Bathtub Examples

###### Example 21\(Observational vs\.do⁡\(Xg=ξg\)\\mathrm\{do\}\(X\_\{g\}=\\xi\_\{g\}\)\)\.

Consider comparing the observational setting \(domain A\) with the setting where gravitational acceleration is fixed toξg\\xi\_\{g\}\(domain B\), for instance by “moving the bathtubs to Mars\.” We introduce an exogenous variableUgU\_\{g\}and write the joint model:

f1:\\displaystyle f\_\{1\}:0=XI−XO,\\displaystyle\\hskip 9\.24994pt0=X\_\{I\}\-X\_\{O\},f2:\\displaystyle f\_\{2\}:0=XK​XP−XO,\\displaystyle\\hskip 9\.24994pt0=X\_\{K\}\\sqrt\{X\_\{P\}\}\-X\_\{O\},f3:\\displaystyle f\_\{3\}:0=Xg​XD−XP,\\displaystyle\\hskip 9\.24994pt0=X\_\{g\}X\_\{D\}\-X\_\{P\},f4:\\displaystyle f\_\{4\}:0=Xg−\{Ugif​R=A,ξgif​R=B\.\\displaystyle\\hskip 9\.24994pt0=X\_\{g\}\-\\begin\{cases\}U\_\{g\}&\\text\{if \}R=A,\\\\ \\xi\_\{g\}&\\text\{if \}R=B\.\\end\{cases\}The bipartite graphGRG^\{R\}has an additional equation nodef4f\_\{4\}connected toXgX\_\{g\},UgU\_\{g\}, andRR\. In the partial orientation\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrGR\\hfil\\textstyle G^\{R\}\\hfil\(see Figure[4](https://arxiv.org/html/2608.19831#A7.F4)\), the variablesXPX\_\{P\}andXOX\_\{O\}areBB\-separated fromRR\(unconditionally\)\. By the Markov property:

XP,XO​⟂\#�GRBR⟹XP,XO⟂⁣⟂R⟹ℙA​\(XP,XO\)=ℙB​\(XP,XO\)\.\\begin\{split\}X\_\{P\},X\_\{O\}\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G^\{R\}\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G^\{R\}\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G^\{R\}\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G^\{R\}\\hfil$\\crcr\}\}\}\}R&\\implies X\_\{P\},X\_\{O\}\\Indep R\\\\ &\\implies\\mathbb\{P\}\_\{A\}\(X\_\{P\},X\_\{O\}\)=\\mathbb\{P\}\_\{B\}\(X\_\{P\},X\_\{O\}\)\.\\end\{split\}Equivalently,ℙ\(XP,XO\)=ℙ\(XP,XO∥do\(Xg=ξg\)\)\\mathbb\{P\}\(X\_\{P\},X\_\{O\}\)=\\mathbb\{P\}\(X\_\{P\},X\_\{O\}\\,\\\|\\,\\mathrm\{do\}\(X\_\{g\}=\\xi\_\{g\}\)\)\. Hence, the joint distribution of pressure and outflow at equilibrium is invariant under changes in gravitational acceleration\.

###### Example 22\(Observational vs\.do\(f3:XD=ξD\)\\mathrm\{do\}\(f\_\{3\}:X\_\{D\}=\\xi\_\{D\}\)\)\.

Now compare the observational setting with the interventiondo\(f3:XD=ξD\)\\mathrm\{do\}\(f\_\{3\}:X\_\{D\}=\\xi\_\{D\}\)\(sealing the bathtub\)\. The joint model replacesf3f\_\{3\}by a domain\-dependent equation:

f1:\\displaystyle f\_\{1\}:0=XI−XO,\\displaystyle\\hskip 9\.24994pt0=X\_\{I\}\-X\_\{O\},f2:\\displaystyle f\_\{2\}:0=XK​XP−XO,\\displaystyle\\hskip 9\.24994pt0=X\_\{K\}\\sqrt\{X\_\{P\}\}\-X\_\{O\},f3:\\displaystyle f\_\{3\}:0=\{Xg​XD−XPif​R=A,XD−ξDif​R=B\.\\displaystyle\\hskip 9\.24994pt0=\\begin\{cases\}X\_\{g\}X\_\{D\}\-X\_\{P\}&\\text\{if \}R=A,\\\\ X\_\{D\}\-\\xi\_\{D\}&\\text\{if \}R=B\.\\end\{cases\}In the partial orientation\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrGR\\hfil\\textstyle G^\{R\}\\hfil\(Figure[4](https://arxiv.org/html/2608.19831#A7.F4)\), we haveXO​⟂\#�GRBR\|XD,XPX\_\{O\}\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G^\{R\}\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G^\{R\}\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G^\{R\}\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G^\{R\}\\hfil$\\crcr\}\}\}\}R\\,\|\\,X\_\{D\},X\_\{P\}\. By the Markov property:

ℙA​\(XO\|XD=ξD,XP\)=ℙB\(XO∥do\(f3:XD=ξD\)\|XP\)\.\\begin\{split\}&\\mathbb\{P\}\_\{A\}\(X\_\{O\}\\,\|\\,X\_\{D\}=\\xi\_\{D\},X\_\{P\}\)\\\\ &\\quad=\\mathbb\{P\}\_\{B\}\(X\_\{O\}\\,\\\|\\,\\mathrm\{do\}\(f\_\{3\}:X\_\{D\}=\\xi\_\{D\}\)\\,\|\\,X\_\{P\}\)\.\\end\{split\}This means that the conditional distribution of outflow given pressure is the same whether we*observe*depthξD\\xi\_\{D\}or*intervene*to set it toξD\\xi\_\{D\}by sealing the bathtub\.

###### Example 23\(Observational vs\.do\(f1:XD=ξD\)\\mathrm\{do\}\(f\_\{1\}:X\_\{D\}=\\xi\_\{D\}\)\)\.

Comparing the observational setting with the interventiondo\(f1:XD=ξD\)\\mathrm\{do\}\(f\_\{1\}:X\_\{D\}=\\xi\_\{D\}\)\(cutting the bathtub and letting it overflow\) yields a joint model wheref1f\_\{1\}is domain\-dependent:

f1:\\displaystyle f\_\{1\}:0=\{XI−XOif​R=A,XD−ξDif​R=B,\\displaystyle\\hskip 9\.24994pt0=\\begin\{cases\}X\_\{I\}\-X\_\{O\}&\\text\{if \}R=A,\\\\ X\_\{D\}\-\\xi\_\{D\}&\\text\{if \}R=B,\\end\{cases\}f2:\\displaystyle f\_\{2\}:0=XK​XP−XO,\\displaystyle\\hskip 9\.24994pt0=X\_\{K\}\\sqrt\{X\_\{P\}\}\-X\_\{O\},f3:\\displaystyle f\_\{3\}:0=Xg​XD−XP\.\\displaystyle\\hskip 9\.24994pt0=X\_\{g\}X\_\{D\}\-X\_\{P\}\.In the partial orientation\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrGR\\hfil\\textstyle G^\{R\}\\hfilof this joint model \(Figure[4](https://arxiv.org/html/2608.19831#A7.F4)\), all endogenous variables and equations belong to a single cluster \(all edges are double\-undirected\)\. The Markov property does not yield non\-trivial conditional independences involvingRR\. Thus, we cannot use it to relate the observational and interventional distributions in this case—which is consistent with the fact that this intervention fundamentally changes the entire causal structure of the system\.

###### Example 24\(do\(f1:XD=ξD\)\\mathrm\{do\}\(f\_\{1\}:X\_\{D\}=\\xi\_\{D\}\)vs\.do\(f1:XD=ξD′\)\\mathrm\{do\}\(f\_\{1\}:X\_\{D\}=\\xi\_\{D\}^\{\\prime\}\)\)\.

While we cannot relate the observational distribution todo\(f1:XD=ξD\)\\mathrm\{do\}\(f\_\{1\}:X\_\{D\}=\\xi\_\{D\}\), we*can*relate two interventional distributions with different parameter values\. Consider the joint model where both domains have the same structural form but different intervention values:

f1:\\displaystyle f\_\{1\}:0=\{XD−ξDif​R=A,XD−ξD′if​R=B,\\displaystyle\\hskip 9\.24994pt0=\\begin\{cases\}X\_\{D\}\-\\xi\_\{D\}&\\text\{if \}R=A,\\\\ X\_\{D\}\-\\xi\_\{D\}^\{\\prime\}&\\text\{if \}R=B,\\end\{cases\}f2:\\displaystyle f\_\{2\}:0=XK​XP−XO,\\displaystyle\\hskip 9\.24994pt0=X\_\{K\}\\sqrt\{X\_\{P\}\}\-X\_\{O\},f3:\\displaystyle f\_\{3\}:0=Xg​XD−XP\.\\displaystyle\\hskip 9\.24994pt0=X\_\{g\}X\_\{D\}\-X\_\{P\}\.Note that in both domains, the causal ordering is the same \(reversed compared to the observational setting\):XD→XP→XOX\_\{D\}\\to X\_\{P\}\\to X\_\{O\}\. In the partial orientation\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrGR\\hfil\\textstyle G^\{R\}\\hfil\(Figure[4](https://arxiv.org/html/2608.19831#A7.F4)\), we haveXO​⟂\#�GRBR\|XPX\_\{O\}\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G^\{R\}\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G^\{R\}\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G^\{R\}\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G^\{R\}\\hfil$\\crcr\}\}\}\}R\\,\|\\,X\_\{P\}\. By the Markov property:

ℙA\(XO∥do\(f1:XD=ξD\)\|XP\)=ℙB\(XO∥do\(f1:XD=ξD′\)\|XP\)\.\\begin\{split\}&\\mathbb\{P\}\_\{A\}\(X\_\{O\}\\,\\\|\\,\\mathrm\{do\}\(f\_\{1\}:X\_\{D\}=\\xi\_\{D\}\)\\,\|\\,X\_\{P\}\)\\\\ &\\quad=\\mathbb\{P\}\_\{B\}\(X\_\{O\}\\,\\\|\\,\\mathrm\{do\}\(f\_\{1\}:X\_\{D\}=\\xi\_\{D\}^\{\\prime\}\)\\,\|\\,X\_\{P\}\)\.\\end\{split\}Hence, overflowing bathtubs yield the same conditional distribution of outflow given pressure, regardless of their height\. This conclusion might not be intuitively obvious but can easily be derived using our \(mostly\) graphical causal reasoning calculus\.

## 7Discussion and Related Work

The BGCM framework extends several existing causal modeling frameworks\. Every CBN, acyclic SCM, simple SCM, and general SCM can be represented as a BGCM\. Conversely, the bathtub example demonstrates that BGCMs can represent systems whose causal semantics are not captured by any of these frameworks\.

Our approach builds on Simon’s causal ordering algorithm\[[22](https://arxiv.org/html/2608.19831#bib.bib16)\], theσ\\sigma\-separation criterion for cyclic SCMs\[[8](https://arxiv.org/html/2608.19831#bib.bib23)\], theDD\-separation criterion\[[12](https://arxiv.org/html/2608.19831#bib.bib2)\]and the framework of[3](https://arxiv.org/html/2608.19831#bib.bib21)\. The latter framework uses two distinct graphs that serve complementary purposes:[3](https://arxiv.org/html/2608.19831#bib.bib21)show that their Markov ordering graph does not correctly represent causal effects of interventions, while their causal ordering graph does not directly encode conditional independences\.

The notion that perfect interventionsdo⁡\(X=x\)\\mathrm\{do\}\(X=x\)can be ambiguous was identified by[3](https://arxiv.org/html/2608.19831#bib.bib21), who proposed the refined notiondo\(fj:Xv=ξv\)\\mathrm\{do\}\(f\_\{j\}:X\_\{v\}=\\xi\_\{v\}\)to resolve the ambiguity\. This refinement is essential for systems like the bathtub, where the same target variable value can be achieved through different mechanisms with different causal consequences\. By performing a*complete*analysis of the causal semantics of the bathtub system under such interventions, we lend further credibility to their claim that this refined notion is a natural representation of “elementary” interventions\.

A key contribution of the present paper is that the partially oriented bipartite graph\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilencodes*both*the causal structure and the conditional independence structure in a single object: the cluster structure and edge directions encode the causal ordering, whileBB\-separation encodes conditional independences\. This is more convenient, as it avoids the need to construct and switch between multiple graphs, and it retains the equation nodes so that the intervention structure remains directly visible\. Furthermore, ourBB\-separation Markov property is more powerful than the Markov properties derived by[3](https://arxiv.org/html/2608.19831#bib.bib21): by also exploiting the functional determinism among the variables, further conditional independences are obtained\.777Our results also imply that these could alternatively be obtained by usingDD\-separation in the Markov ordering graph\.Finally, our extended Markov property, which handles Markov kernels with non\-random inputs through transitional conditional independence\[[10](https://arxiv.org/html/2608.19831#bib.bib17)\], has no counterpart in that work\. We believe that these features may also facilitate future extensions and applications\.

The BGCM framework is closely related to how engineers reason about causality\[[11](https://arxiv.org/html/2608.19831#bib.bib9),[5](https://arxiv.org/html/2608.19831#bib.bib8),[16](https://arxiv.org/html/2608.19831#bib.bib10)\]\. In modeling languages such as Modelica, systems are specified as sets of “acausal” equations, and causality is derived automatically through symbolic analysis—precisely the kind of analysis formalized by Simon’s causal ordering algorithm and the BGCM framework\.

When the bipartite graph does not admit a perfect matching, the Dulmage\-Mendelsohn decomposition\[[7](https://arxiv.org/html/2608.19831#bib.bib15)\]provides a useful generalization that can represent overcomplete subsystems \(more equations than variables\) and incomplete subsystems \(more variables than equations\)\.[3](https://arxiv.org/html/2608.19831#bib.bib21)demonstrate how marginal Markov properties for the complete and overcomplete subsystems can still be derived in this general setting\.

To our knowledge, there is little other work using bipartite graphs for causal inference\.[25](https://arxiv.org/html/2608.19831#bib.bib6)introduce a framework for estimating causal effects under interference when the treated units are distinct from the observed units\.[21](https://arxiv.org/html/2608.19831#bib.bib7)prove that every valid graph in the observational equivalence class of linear Gaussian cyclic SCMs corresponds to a perfect matching\.

## 8Conclusion

We have proposed bipartite graphical causal models \(BGCMs\) as a causal modeling framework that uses bipartite graphs with equation nodes and variable nodes\. This framework offers several advantages\. First, it reduces the ambiguity inherent in specifying interventions by requiring that hard interventions explicitly reference the equation being replaced\. Second, Simon’s causal ordering algorithm provides a principled method for deriving the partial orientation of the bipartite graph, which encodes the causal structure\. Third, theBB\-separation criterion and the resulting Markov property propagate conditional independences along the partial ordering\. Fourth, the Markov property facilitates causal reasoning about domain invariances, providing a generalization of Pearl’s do\-calculus to BGCMs\.

BGCMs naturally model equilibrium systems such as the bathtub, and can be applied to a wide range of other systems, including equilibrated economic markets \(e\.g\., supply–demand systems, see Appendix[H](https://arxiv.org/html/2608.19831#A8)\), electronic circuits, and biochemical reaction networks\[[2](https://arxiv.org/html/2608.19831#bib.bib25)\]\. Directions for future work include dynamical extensions incorporating \(stochastic\) differential equations, and the development of structure learning algorithms for BGCMs\.

###### Acknowledgements\.

I thank Claude Code \(Opus 4\.6–4\.8\) for assistance in the writing process, and Kathy Molenaar for useful discussions\.

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Causal Reasoning with Bipartite Graphical Causal Models \(Supplementary Material\)

This Supplementary Material contains proofs of the main results and additional details\.

## Appendix AMeasurability of Solution Functions

We use the following standard measurable\-graph fact to justify that the uniquely defined solution functions appearing in the main text are measurable\.

###### Lemma 25\(Measurability of uniquely defined solution maps\)\.

Let𝒳\\mathcal\{X\}and𝒴\\mathcal\{Y\}be standard Borel spaces, let𝒵\\mathcal\{Z\}be a measurable space, letz0∈𝒵z\_\{0\}\\in\\mathcal\{Z\}be such that\{z0\}\\\{z\_\{0\}\\\}is measurable, and letH:𝒳×𝒴→𝒵H:\\mathcal\{X\}\\times\\mathcal\{Y\}\\to\\mathcal\{Z\}be measurable\. Suppose that for everyx∈𝒳x\\in\\mathcal\{X\}there exists a uniquey=:ψ⁡\(x\)∈𝒴y=:\\psi\(x\)\\in\\mathcal\{Y\}such thatH⁡\(x,y\)=z0H\(x,y\)=z\_\{0\}\. Thenψ:𝒳→𝒴\\psi:\\mathcal\{X\}\\to\\mathcal\{Y\}is measurable\.

###### Proof\.

The graph ofψ\\psiis

Γψ=\{\(x,y\)∈𝒳×𝒴:H⁡\(x,y\)=z0\}\.\\Gamma\_\{\\psi\}=\\\{\(x,y\)\\in\\mathcal\{X\}\\times\\mathcal\{Y\}:H\(x,y\)=z\_\{0\}\\\}\.This set is measurable because it is the inverse image of\{z0\}\\\{z\_\{0\}\\\}under the measurable map\(x,y\)↦H⁡\(x,y\)\(x,y\)\\mapsto H\(x,y\)\. Hence, by\[[15](https://arxiv.org/html/2608.19831#bib.bib1), 14\.12\],ψ\\psiis measurable\. ∎

###### Corollary 26\(Measurability of cluster solution functions\)\.

Under Assumption[10](https://arxiv.org/html/2608.19831#Thmtheorem10), suppose an endogenous cluster\[c\]\[c\]is uniquely solvable in the sense of Definition[9](https://arxiv.org/html/2608.19831#Thmtheorem9)\. Then its local solution function

Φ\[c\]:𝒳pa\#�G​\(\[c\]\)→𝒳\[c\]∩V\\Phi^\{\[c\]\}:\\mathcal\{X\}\_\{\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[c\]\)\}\\to\\mathcal\{X\}\_\{\[c\]\\cap V\}is measurable\.

###### Proof\.

Write𝒳:=𝒳pa\#�G​\(\[c\]\)\\mathcal\{X\}:=\\mathcal\{X\}\_\{\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[c\]\)\}and𝒴:=𝒳\[c\]∩V\\mathcal\{Y\}:=\\mathcal\{X\}\_\{\[c\]\\cap V\}\. For each equationfj∈F∩\[c\]f\_\{j\}\\in F\\cap\[c\], the variables occurring infjf\_\{j\}are contained in\(\[c\]∩V\)∪pa\#�G​\(\[c\]\)\(\[c\]\\cap V\)\\cup\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[c\]\)\. Hence the measurable equation mapϕj\\phi\_\{j\}induces a measurable function of𝒳×𝒴→ℝ\\mathcal\{X\}\\times\\mathcal\{Y\}\\to\\mathbb\{R\}\. Collect these equations into the measurable map

H:𝒳×𝒴→ℝF∩\[c\]:\(x,y\)↦\(ϕj​\(x,y\)\)j∈F∩\[c\]\.H:\\mathcal\{X\}\\times\\mathcal\{Y\}\\to\\mathbb\{R\}^\{F\\cap\[c\]\}:\(x,y\)\\mapsto\(\\phi\_\{j\}\(x,y\)\)\_\{j\\in F\\cap\[c\]\}\.Unique solvability says that for every parent valuex∈𝒳x\\in\\mathcal\{X\}there is a uniquey=Φ\[c\]​\(x\)y=\\Phi^\{\[c\]\}\(x\)such thatH⁡\(x,y\)=0H\(x,y\)=0\. Lemma[25](https://arxiv.org/html/2608.19831#Thmtheorem25)therefore implies thatΦ\[c\]\\Phi^\{\[c\]\}is measurable\. ∎

Since the graph has finitely many clusters, recursive substitution of the measurable local solution functions along the causal ordering shows that the global solution functionΦ:𝒳U→𝒳V\\Phi:\\mathcal\{X\}\_\{U\}\\to\\mathcal\{X\}\_\{V\}of Proposition[11](https://arxiv.org/html/2608.19831#Thmtheorem11)is measurable as well\.

## Appendix BPreliminaries ondd\-Separation andDD\-Separation

We recall the standard notion ofdd\-separation, introduced by[24](https://arxiv.org/html/2608.19831#bib.bib3)\(see also[18](https://arxiv.org/html/2608.19831#bib.bib5),[20](https://arxiv.org/html/2608.19831#bib.bib19)\), and, for graphs with deterministic relations, that ofDD\-separation\[[12](https://arxiv.org/html/2608.19831#bib.bib2)\]\.

###### Definition 27\(dd\-blocking\)\.

A walkv1​…​vkv\_\{1\}\\dots v\_\{k\}on a DAGGGisdd\-blocked byC⊆VC\\subseteq V, if it contains:

- •a*collider*vi−1→vi←vi\+1v\_\{i\-1\}\\to v\_\{i\}\\mathrel\{\\leftarrow\}v\_\{i\+1\}withvi∉ancG​\(C\)v\_\{i\}\\notin\\mathrm\{anc\}\_\{G\}\(C\), or
- •a*non\-collider*\(possibly endpoint\)vi∈Cv\_\{i\}\\in C\.

Otherwise, the walk is called*dd\-open givenCC*\.

###### Definition 28\.

LetA,B,C⊆VA,B,C\\subseteq Vbe sets of variable nodes in a DAGGG\. We say that*AAandBBaredd\-separated givenCCinGG*, written

A​⟂Gd​B\|C,A\\Perp^\{d\}\_\{G\}B\\,\|\\,C,if every walk from a node inAAto a node inBBisdd\-blocked byCCinGG\.888It suffices if every path from a node inAAto a node inBBisdd\-blocked byCCinGG, yielding an equivalent formulation ofdd\-separation that is easier to check manually\.

###### Definition 29\(Functionally determined in an acyclic SCM\)\.

LetC⊆VC\\subseteq Vbe a subset of nodes in an acyclic SCM with graphGG, with exogenous nodesUUand endogenous nodesV∖UV\\setminus U\. DefineC0:=CC\_\{0\}:=Cand

Cn\+1:=Cn∪\{v∈V∖U:paG​\(v\)⊆Cn\}\.C\_\{n\+1\}:=C\_\{n\}\\cup\\\{v\\in V\\setminus U:\\mathrm\{pa\}\_\{G\}\(v\)\\subseteq C\_\{n\}\\\}\.We definefdetG​\(C\):=⋃n≥0Cn\\mathrm\{fdet\}\_\{G\}\(C\):=\\bigcup\_\{n\\geq 0\}C\_\{n\}and refer to those as the nodes that are*functionally determined byCC*\.

The following definition is inspired by\[[12](https://arxiv.org/html/2608.19831#bib.bib2)\]\.

###### Definition 30\(DD\-separation\)\.

LetA,B,C⊆VA,B,C\\subseteq Vbe sets of nodes in an acyclic SCM with graphGG\(exogenous nodesUU, endogenousV∖UV\\setminus U\)\. We say that*AAandBBareDD\-separated givenCCinGG*, writtenA​⟂GD​B\|CA\\Perp^\{D\}\_\{G\}B\\,\|\\,C, ifA​⟂GdB\|fdetG​\(C\)A\\Perp^\{d\}\_\{G\}B\\,\|\\,\\mathrm\{fdet\}\_\{G\}\(C\)\(formulation\(2\)\(2\)of Lemma[31](https://arxiv.org/html/2608.19831#Thmtheorem31)\)\.

This formulation ofDD\-separation \(which corresponds with formulation \(2\) in the following lemma\) is equivalent to two other formulations:

###### Lemma 31\.

LetA,B,CA,B,Cbe sets of nodes in an acyclic SCM with graphGG, with exogenous nodesUUand endogenous nodesV∖UV\\setminus U\. LetfdetG​\(C\)\\mathrm\{fdet\}\_\{G\}\(C\)be the nodes in the DAG that are functionally determined byCC\(as in Definition[29](https://arxiv.org/html/2608.19831#Thmtheorem29)\)\. The following three formulations ofDD\-separation are equivalent:

1. 1\.all walks between a node inAAand a node inBBcontain 1. \(a\)a collider not inancG​\(C\)\\mathrm\{anc\}\_\{G\}\(C\) 2. \(b\)a non\-endpoint non\-collider infdetG​\(C\)\\mathrm\{fdet\}\_\{G\}\(C\) 3. \(c\)an end node infdetG​\(C\)\\mathrm\{fdet\}\_\{G\}\(C\)
2. 2\.all walks between a node inAAand a node inBBcontain 1. \(a\)a collider not inancG​\(fdetG​\(C\)\)\\mathrm\{anc\}\_\{G\}\(\\mathrm\{fdet\}\_\{G\}\(C\)\) 2. \(b\)a non\-endpoint non\-collider infdetG​\(C\)\\mathrm\{fdet\}\_\{G\}\(C\) 3. \(c\)an end node infdetG​\(C\)\\mathrm\{fdet\}\_\{G\}\(C\) \(⇔\\iffAA,BBaredd\-separated givenfdetG​\(C\)\\mathrm\{fdet\}\_\{G\}\(C\)\)
3. 3\.all walks between a node inAAand a node inBBcontain 1. \(a\)a collider not inancG​\(C\)\\mathrm\{anc\}\_\{G\}\(C\) 2. \(b\)a non\-endpoint non\-collider inCC 3. \(c\)an end node infdetG​\(C\)\\mathrm\{fdet\}\_\{G\}\(C\) 4. \(d\)a fork infdetG​\(C\)\\mathrm\{fdet\}\_\{G\}\(C\)

###### Proof\.

WriteC¯:=fdetG​\(C\)\\bar\{C\}:=\\mathrm\{fdet\}\_\{G\}\(C\)\. From Definition[29](https://arxiv.org/html/2608.19831#Thmtheorem29)we haveC⊆C¯C\\subseteq\\bar\{C\}and

n∈C¯∖C⟹n∈V∖U​and​paG​\(n\)⊆C¯;n\\in\\bar\{C\}\\setminus C\\;\\Longrightarrow\\;n\\in V\\setminus U\\text\{ and \}\\mathrm\{pa\}\_\{G\}\(n\)\\subseteq\\bar\{C\};\(4\)in particularancG​\(C\)⊆ancG​\(C¯\)\\mathrm\{anc\}\_\{G\}\(C\)\\subseteq\\mathrm\{anc\}\_\{G\}\(\\bar\{C\}\)\. On a walkπ\\piwe call an internal node a*collider*if both incident edges point into it, a*fork*if both point out of it, and a*chain*if one points in and one out; the two end nodes are treated separately\. A*parent\-neighbor*of a nodennonπ\\piis a neighborpponπ\\piwithp→np\\to ninGG; thus a collider has two parent\-neighbors, a chain has one, and a fork has none\. Callπ\\pi*jj\-active*if it is not blocked according to formulation\(j\)\(j\)\. We prove that the three notions of “active” coincide on every walkπ\\pibetweenAAandBB; the equivalence of the three “all walks are blocked” statements is then immediate\.

*\(1\)⇔\(2\)\(1\)\\Leftrightarrow\(2\)\.*The two criteria differ only in the collider clause, andancG​\(C\)⊆ancG​\(C¯\)\\mathrm\{anc\}\_\{G\}\(C\)\\subseteq\\mathrm\{anc\}\_\{G\}\(\\bar\{C\}\), so a collider outsideancG​\(C¯\)\\mathrm\{anc\}\_\{G\}\(\\bar\{C\}\)is also outsideancG​\(C\)\\mathrm\{anc\}\_\{G\}\(C\); as the other clauses coincide, every walk blocked under\(2\)\(2\)is blocked under\(1\)\(1\), i\.e\., every11\-active walk is22\-active\.

Conversely, letπ\\pibe22\-active: every collider lies inancG​\(C¯\)\\mathrm\{anc\}\_\{G\}\(\\bar\{C\}\), and no node ofC¯\\bar\{C\}occurs onπ\\pias a non\-collider or as an end node\. Letkkbe a collider ofπ\\pi; we showk∈ancG​\(C\)k\\in\\mathrm\{anc\}\_\{G\}\(C\)\. Ifk∈C¯k\\in\\bar\{C\}then in factk∈Ck\\in C: otherwise \([4](https://arxiv.org/html/2608.19831#A2.E4)\) givespaG​\(k\)⊆C¯\\mathrm\{pa\}\_\{G\}\(k\)\\subseteq\\bar\{C\}, so the two parent\-neighbors ofkkwould be nodes ofC¯\\bar\{C\}occurring as non\-colliders or end nodes, contradicting22\-activity; hencek∈C⊆ancG​\(C\)k\\in C\\subseteq\\mathrm\{anc\}\_\{G\}\(C\)\. Ifk∉C¯k\\notin\\bar\{C\}, pick a shortest directed pathk→n1→…→ntk\\to n\_\{1\}\\to\\dots\\to n\_\{t\}withnt∈C¯n\_\{t\}\\in\\bar\{C\}\(one exists sincek∈ancG​\(C¯\)k\\in\\mathrm\{anc\}\_\{G\}\(\\bar\{C\}\)\), so thatn1,…,nt−1∉C¯n\_\{1\},\\dots,n\_\{t\-1\}\\notin\\bar\{C\}\. Werent∈C¯∖Cn\_\{t\}\\in\\bar\{C\}\\setminus C, then \([4](https://arxiv.org/html/2608.19831#A2.E4)\) would place its predecessor on the path—nt−1n\_\{t\-1\}, orkkift=1t=1—inpaG​\(nt\)⊆C¯\\mathrm\{pa\}\_\{G\}\(n\_\{t\}\)\\subseteq\\bar\{C\}, contradicting the choice of that predecessor outsideC¯\\bar\{C\}\. Hencent∈Cn\_\{t\}\\in Candk∈ancG​\(C\)k\\in\\mathrm\{anc\}\_\{G\}\(C\)\. So every collider ofπ\\pilies inancG​\(C\)\\mathrm\{anc\}\_\{G\}\(C\)andπ\\piis11\-active\.

*\(1\)⇔\(3\)\(1\)\\Leftrightarrow\(3\)\.*The collider clause and the end\-node clause are identical in the two formulations\. Ifπ\\piis11\-active it has no internal non\-collider inC¯\\bar\{C\}; in particular it has no internal non\-collider inCCand no fork inC¯\\bar\{C\}, soπ\\piis33\-active\.

Conversely, letπ\\pibe33\-active\. To show it is11\-active it suffices to rule out internal non\-colliders inC¯\\bar\{C\}\. Forks inC¯\\bar\{C\}are excluded by33\-activity, and chains inCCare excluded as well, so the only remaining possibility is a chain nodem∈C¯∖Cm\\in\\bar\{C\}\\setminus C; suppose one occurs\. Defineq0:=mq\_\{0\}:=mand letqi\+1q\_\{i\+1\}be the parent\-neighbor ofqiq\_\{i\}, continuing as long asqiq\_\{i\}is a chain inC¯∖C\\bar\{C\}\\setminus C\(which by \([4](https://arxiv.org/html/2608.19831#A2.E4)\) guarantees a parent\-neighbor inC¯\\bar\{C\}\)\. Eachqiq\_\{i\}lies inC¯\\bar\{C\}, andqi\+1→qi→…→q0q\_\{i\+1\}\\to q\_\{i\}\\to\\dots\\to q\_\{0\}is a directed path in the DAGGG, so theqiq\_\{i\}are distinct and the process stops, at someqk∈C¯q\_\{k\}\\in\\bar\{C\}\. Sinceqkq\_\{k\}has an edge pointing out of it \(towardqk−1q\_\{k\-1\}\), it is not a collider\. Ifqkq\_\{k\}is an end node, then an end node lies inC¯\\bar\{C\}; ifqkq\_\{k\}is a fork, then a fork lies inC¯\\bar\{C\}; and ifqkq\_\{k\}is a chain, then—the process having stopped—qk∈Cq\_\{k\}\\in C, so an internal non\-collider lies inCC\. Each case contradicts33\-activity\. Hence no chain node ofπ\\pilies inC¯∖C\\bar\{C\}\\setminus C, soπ\\pihas no internal non\-collider inC¯\\bar\{C\}and is11\-active\.

Thus the three notions of “active” \(equivalently, of “blocked”\) coincide on every walk, and the three formulations ofDD\-separation are equivalent\. By definition, formulation\(2\)\(2\)isdd\-separation ofAAandBBgivenC¯\\bar\{C\}\. ∎

[12](https://arxiv.org/html/2608.19831#bib.bib2)definedDD\-separation \(restricted to disjointA,B,CA,B,C\) with formulation\(3\)\(3\), and showed that it is equivalent to formulation\(1\)\(1\)\. What Lemma[31](https://arxiv.org/html/2608.19831#Thmtheorem31)adds is the equivalence with formulation\(2\)\(2\):DD\-separation givenCCcoincides with ordinarydd\-separation given the enlarged conditioning setC¯=fdetG​\(C\)\\bar\{C\}=\\mathrm\{fdet\}\_\{G\}\(C\)\.999This equivalence was observed by Claude Code\.This reduction is useful because it lets us fall back on the theory ofdd\-separation, which is considerably wider in scope than that ofDD\-separation: it extends to cyclic systems throughσ\\sigma\-separation\[[8](https://arxiv.org/html/2608.19831#bib.bib23)\]and underpins a broad range of Markov\-property, completeness, and algorithmic results that have no directDD\-separation counterpart\.

## Appendix CProof of the Markov Property

Our strategy to prove the Markov property for BGCMs \(Theorem[16](https://arxiv.org/html/2608.19831#Thmtheorem16)\) will be as follows\.

We will first ignore deterministic relations and prove a weaker Markov property using a separation notion that we callbb\-separation \(lowercasebbfor “bipartite”\)\. This separation notion is designed to correspond todd\-separation on the*acyclification*, a directed acyclic graph constructed from the partially ordered bipartite graph\. This mimics the acyclification strategy for cyclic SCMs\[[23](https://arxiv.org/html/2608.19831#bib.bib12),[8](https://arxiv.org/html/2608.19831#bib.bib23),[4](https://arxiv.org/html/2608.19831#bib.bib20)\]\. However, we do not make the clusters \(corresponding to strongly connected components in SCMs\) fully connected, because we typically work with the augmented graph that contains all nodes, including exogenous random variable nodes, and there is no reason to assume a latent noise source feeds into a cycle\. From the equivalence ofbb\-separation on the partially oriented bipartite graph anddd\-separation on its acyclification \(Lemma[38](https://arxiv.org/html/2608.19831#Thmtheorem38)\), we then prove abb\-separation Markov property \(Theorem[39](https://arxiv.org/html/2608.19831#Thmtheorem39)\) by reduction to the standard Markov property for acyclic SCMs\.

We then strengthen the Markov property by taking into account determinism, analogous to howDD\-separation\[[12](https://arxiv.org/html/2608.19831#bib.bib2)\]strengthensdd\-separation in Bayesian networks\. The key intuition is: once every parent of a cluster is fixed by the conditioning information, all variables in the cluster are fixed too\. This leads directly to the notion ofBB\-separation\. Similarly to howDD\-separation is related todd\-separation,BB\-separation can be expressed in terms ofbb\-separation \(Lemma[41](https://arxiv.org/html/2608.19831#Thmtheorem41)\)\. This observation allows us to obtain Theorem[16](https://arxiv.org/html/2608.19831#Thmtheorem16)as a Corollary of Theorem[39](https://arxiv.org/html/2608.19831#Thmtheorem39)\.

### C\.1bb\-Separation Markov Property

We repeatedly make use of the following elementary consequences of the definitions:

- •Directed edges in\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilalways point from variable to equation;
- •Parents of a cluster are variables;
- •For a walk between two variable nodes, all segment exits will be variable nodes;
- •The end nodes of the walk are always exits of their segments\.

We will also use that clusters are connected by double\-undirected edges\.

###### Lemma 32\(Double\-edge connectivity of clusters\)\.

Leta,b∈V∪Fa,b\\in V\\cup F\. Ifa∼ba\\sim b, then there exists a possibly trivial walk fromaatobbin\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilthat uses only=⁣=\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}edges and whose nodes all lie in\[a\]=\[b\]\[a\]=\[b\]\.

###### Proof\.

It suffices to prove the claim for each generating relation in Definition[3](https://arxiv.org/html/2608.19831#Thmtheorem3), since walks can then be concatenated along a finite chain of such relations\. Ifa\-⁣\-ba\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}bis a matched edge, thena∼ba\\sim b, so this edge is oriented asa=⁣=ba\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}bin\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilby Definition[5](https://arxiv.org/html/2608.19831#Thmtheorem5)\. Ifaaandbblie on a common closedMM\-alternating walk, take the subwalk of that closed walk fromaatobb\. Every edge on this subwalk has both endpoints on the same closedMM\-alternating walk, hence its endpoints are equivalent; by Definition[5](https://arxiv.org/html/2608.19831#Thmtheorem5), each such edge is therefore oriented as=⁣=\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}\. The resulting walk stays inside the common equivalence class\. ∎

We first define the appropriate acyclification\.

###### Definition 33\.

For a partially oriented bipartite graph\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilwith variable nodesVVand equation nodesFF, we define its*acyclification*as the directed acyclic graph\#�Gacy=\(V,Eacy\)\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}=\(V,E^\{\\mathrm\{acy\}\}\)with nodesVVand with a directed edgev→v′v\\to v^\{\\prime\}forv,v′∈Vv,v^\{\\prime\}\\in Vif and only ifv∈pa\#�G​\(\[v′\]\)v\\in\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[v^\{\\prime\}\]\)\.

First we show that “anterior in\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil” \(at the node level\) corresponds to “ancestral in\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}” \(at the cluster level\)\.

###### Lemma 34\.

For nodesa,b∈Va,b\\in V:

a∈ant\#�G​\(b\)⇔\[a\]∩anc\#�Gacy​\(b\)≠∅\.a\\in\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(b\)\\iff\[a\]\\cap\\mathrm\{anc\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}\(b\)\\neq\\emptyset\.

###### Proof\.

Letπ\\pibe an anterior walk in\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil, that is, a walk of the form

s1,1=⁣=…=⁣=s1,k1→s2,1=⁣=…=⁣=s2,k2→…→sm,1=⁣=…=⁣=sm,kms\_\{1,1\}\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}\\dots\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}s\_\{1,k\_\{1\}\}\\to s\_\{2,1\}\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}\\dots\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}s\_\{2,k\_\{2\}\}\\to\\dots\\to s\_\{m,1\}\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}\\dots\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}s\_\{m,k\_\{m\}\}which we partitioned into maximal subwalkssis\_\{i\}of equivalent nodes, eachsis\_\{i\}being of the formsi,1=⁣=…=⁣=si,kis\_\{i,1\}\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}\\dots\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}s\_\{i,k\_\{i\}\}\(with possiblyki=1k\_\{i\}=1\)\. We project it onto a directed walk in\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}by picking from each segmentsis\_\{i\}the outgoing nodesi,kis\_\{i,k\_\{i\}\}:

s1,k1→s2,k2→…→sm,km\.s\_\{1,k\_\{1\}\}\\to s\_\{2,k\_\{2\}\}\\to\\dots\\to s\_\{m,k\_\{m\}\}\.Hence, ifs1,1s\_\{1,1\}is anterior tosm,kms\_\{m,k\_\{m\}\}in\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil, thens1,k1s\_\{1,k\_\{1\}\}is an ancestor ofsm,kms\_\{m,k\_\{m\}\}in\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\. Sinces1,k1∈\[s1,1\]s\_\{1,k\_\{1\}\}\\in\[s\_\{1,1\}\], the claim follows\.

Vice versa, letπacy\\pi^\{\\mathrm\{acy\}\}be a directed walk in\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}:

v1→…→vm\.v\_\{1\}\\to\\dots\\to v\_\{m\}\.By definition, each edge inπacy\\pi^\{\\mathrm\{acy\}\}connects variables in different clusters of\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil\. The edgevi→vi\+1v\_\{i\}\\to v\_\{i\+1\}in\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\(withvi∈pa\#�G​\(\[vi\+1\]\)v\_\{i\}\\in\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[v\_\{i\+1\}\]\)\) can be lifted to a walk in\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilas follows\. Choose an equationfi∈F∩\[vi\+1\]f\_\{i\}\\in F\\cap\[v\_\{i\+1\}\]withvi→fiv\_\{i\}\\to f\_\{i\}in\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil, and connectfif\_\{i\}tovi\+1v\_\{i\+1\}by a=⁣=\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}\-walk within\[vi\+1\]\[v\_\{i\+1\}\]using Lemma[32](https://arxiv.org/html/2608.19831#Thmtheorem32), resulting in the liftvi→fi=⁣=⋯=⁣=vi\+1v\_\{i\}\\to f\_\{i\}\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}\\cdots\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}v\_\{i\+1\}in\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil\. Concatenating these lifts yields an anterior walk in\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilfromv1v\_\{1\}tovmv\_\{m\}\. By concatenating this with the double\-edge walk from Lemma[32](https://arxiv.org/html/2608.19831#Thmtheorem32), we may obtain an anterior walk in\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilfrom any node in\[v1\]\[v\_\{1\}\]tovmv\_\{m\}\. Hence, if a node in\[a\]\[a\]is ancestral tobbin\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}, thenaaitself is anterior tobbin\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil\. ∎

The following notion is related to the segment\-based version ofσ\\sigma\-separation\[[8](https://arxiv.org/html/2608.19831#bib.bib23)\], but strengthens it by adding another way in which non\-collider segments can block \(the third rule\)\.

###### Definition 35\(bb\-blocking\)\.

ForC⊆VC\\subseteq V, the walk is called*bb\-blocked byCC*if it contains:

1. 1\.a collider segment that does not intersectant\#�G​\(C\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\), or
2. 2\.a non\-collider segment that has an exit inCC, or
3. 3\.a non\-collider segment with two distinct exits whose cluster has all its\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil\-parents inCC\.

Otherwise, the walk is called*bb\-open givenCC*\.

Note: Since the two end nodes qualify as exits, an end node inCCalwaysbb\-blocks the walk\.

###### Definition 36\(bb\-separation\)\.

LetA,B,C⊆VA,B,C\\subseteq Vbe sets of variable nodes\. We say that*AAandBBarebb\-separated givenCCin\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil*, written

A​⟂\#�Gb​B\|C,A\\Perp^\{b\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,C,if every walk from a node inAAto a node inBBisbb\-blocked byCCin\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil\.101010Thanks to rule 3 of Definition[35](https://arxiv.org/html/2608.19831#Thmtheorem35), it is equivalent to require only that every*path*fromAAtoBBbebb\-blocked byCC\(Lemma[37](https://arxiv.org/html/2608.19831#Thmtheorem37)\); the path formulation is usually more convenient to check by hand\.

The three rules in whichbb\-separation blocks a walk mirror, segment by segment, the waydd\-separation blocks the corresponding structure in the acyclification\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\(Lemma[38](https://arxiv.org/html/2608.19831#Thmtheorem38)\): a collider segment projects to a collider whose center is a common child; a non\-collider segment with a single exitvv\(a chain, an endpoint, or a one\-node fork←v→\\mathrel\{\\leftarrow\}v\\to\) projects to a chain/fork centered atvv, blocked iffv∈Cv\\in C; and a non\-collider segment with two*distinct*exitsa≠ba\\neq bprojects to a forka←p→ba\\mathrel\{\\leftarrow\}p\\to bthrough a common parentpp, blocked iffa∈Ca\\in Corb∈Cb\\in Cor every suchpplies inCC\.111111Note that a one\-node fork←v→\\mathrel\{\\leftarrow\}v\\tohas a single \(distinct\) exit and is thus governed by rule 2 only, not rule 3\.

We added the third rule to makebb\-separation via walks coincide withbb\-separation via paths\.

###### Lemma 37\(bb\-separation via walks or paths\)\.

For allA,B,C⊆VA,B,C\\subseteq V, every walk fromAAtoBBisbb\-blocked byCCif and only if every path fromAAtoBBisbb\-blocked byCC\.

###### Proof\.

Since paths are walks, “all walksbb\-blocked” implies “all pathsbb\-blocked”\.

Conversely, suppose there exists abb\-open walk fromAAtoBB, and among all such walks with the same end nodes choose one, sayπ\\pi, of minimal length\. We show thatπ\\pihas no repeated node, and hence is abb\-open path\.

First note that no segment ofπ\\picontains the same node twice\. Indeed, if a segment contains two occurrences of a nodeyy, deleting the closed=⁣=\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}\-subwalk between these two occurrences gives a strictly shorter walk with the same end nodes\. Only this segment is changed; its bounding directed edges, exits, and cluster remain the same\. Hence rules 2 and 3 of Definition[35](https://arxiv.org/html/2608.19831#Thmtheorem35)have the same truth value before and after the deletion\. If the segment is a collider, then, sinceπ\\piisbb\-open, it meetsant\#�G​\(C\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\. But all nodes in a segment are connected by=⁣=\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}\-walks, so if one node of the segment is anterior toCC, then every node of the segment is anterior toCC\. Thus the shortened collider segment is still activated\. The shortened walk is thereforebb\-open, contradicting the minimality ofπ\\pi\.

Now suppose, for contradiction, thatπ\\pinevertheless visits some node twice\.

Writeπ=n0,e1,…,ek,nk\\pi=n\_\{0\},e\_\{1\},\\dots,e\_\{k\},n\_\{k\}, letx=nμ=nνx=n\_\{\\mu\}=n\_\{\\nu\}withμ<ν\\mu<\\nuand letπ′\\pi^\{\\prime\}be obtained by deletingeμ\+1,…,nνe\_\{\\mu\+1\},\\dots,n\_\{\\nu\}, i\.e\.,π′=n0,…,nμ,eν\+1,nν\+1,…,nk\\pi^\{\\prime\}=n\_\{0\},\\dots,n\_\{\\mu\},e\_\{\\nu\+1\},n\_\{\\nu\+1\},\\dots,n\_\{k\}\. This is a valid walk, sinceeν\+1e\_\{\\nu\+1\}joinsnν=nμn\_\{\\nu\}=n\_\{\\mu\}tonν\+1n\_\{\\nu\+1\}, and its end nodesn0,nkn\_\{0\},n\_\{k\}are unchanged\. The two occurrences ofxxlie in distinct segments ofπ\\pi, since no segment ofπ\\picontains a repeated node\.

Lets′s^\{\\prime\}be the segment ofπ\\picontaining the occurrencenμn\_\{\\mu\}, ands′′s^\{\\prime\\prime\}the segment containingnνn\_\{\\nu\}; both lie in\[x\]\[x\]\. Inπ′\\pi^\{\\prime\}the part ofs′s^\{\\prime\}from its left boundary tonμn\_\{\\mu\}and the part ofs′′s^\{\\prime\\prime\}fromnνn\_\{\\nu\}to its right boundary merge into a single segmentt⊆\[x\]t\\subseteq\[x\]whose left bounding edge is that ofs′s^\{\\prime\}\(if applicable\) and whose right bounding edge is that ofs′′s^\{\\prime\\prime\}\(if applicable\)\. Hencetthas a left exit iffs′s^\{\\prime\}does, and a right exit iffs′′s^\{\\prime\\prime\}does\. Every other segment ofπ′\\pi^\{\\prime\}coincides with a segment ofπ\\pi\(same nodes, bounding edges, exits, and type\), soπ′\\pi^\{\\prime\}can fail to bebb\-open only attt\. We showttdoes notbb\-block\.

*Rule 1 \(collider\)\.*Supposettis a collider, i\.e\.,s′s^\{\\prime\}has no left exit ands′′s^\{\\prime\\prime\}has no right exit\. Ifs′s^\{\\prime\}is itself a collider then, being a segment of thebb\-openπ\\pi, it meetsant\#�G​\(C\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\); ass′⊆\[x\]=\[t\]s^\{\\prime\}\\subseteq\[x\]=\[t\], so doestt\. The same holds ifs′′s^\{\\prime\\prime\}is a collider\. Otherwises′s^\{\\prime\}has a right exitρ′\\rho^\{\\prime\}ands′′s^\{\\prime\\prime\}a left exitλ′′\\lambda^\{\\prime\\prime\}, both variables of\[x\]\[x\]whose exit edges point into strict descendant clusters\. Hence the deleted sub\-walk leaves\[x\]\[x\]downward atρ′\\rho^\{\\prime\}and re\-enters it from below atλ′′\\lambda^\{\\prime\\prime\}, so along it some descending edge \(one traversed from its variable into a child cluster\) is immediately followed—across a single segment—by an ascending edge; the first such segments†s^\{\\dagger\}is a collider, and every directed edge before it descends, so\[x\]\[x\]is anterior tos†s^\{\\dagger\}\. Asπ\\piisbb\-open,s†s^\{\\dagger\}meetsant\#�G​\(C\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\); since\[x\]\[x\]is anterior tos†s^\{\\dagger\}, so does\[x\]\[x\]\.

*Rule 2\.*Every exit ofttis a left exit ofs′s^\{\\prime\}or a right exit ofs′′s^\{\\prime\\prime\}, hence an exit of a segment of thebb\-openπ\\pi, hence not inCC\.

*Rule 3\.*Supposetthas two distinct exits andpa\#�G​\(\[x\]\)⊆C\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[x\]\)\\subseteq C; we derive a contradiction\. Thens′s^\{\\prime\}has a left exitℓ\\ellands′′s^\{\\prime\\prime\}a right exitrrwithℓ≠r\\ell\\neq r\. Consider the right bounding edge ofs′s^\{\\prime\}\.

- •If it points out ofs′s^\{\\prime\}, its endpointρ′\\rho^\{\\prime\}is a right exit ofs′s^\{\\prime\}\. Ifρ′≠ℓ\\rho^\{\\prime\}\\neq\\ell, thens′s^\{\\prime\}has two distinct exits andpa\#�G​\(\[x\]\)⊆C\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[x\]\)\\subseteq C, sos′s^\{\\prime\}alreadybb\-blocksπ\\pi—a contradiction\. Ifρ′=ℓ\\rho^\{\\prime\}=\\ell, thens′=\{ℓ\}s^\{\\prime\}=\\\{\\ell\\\}is a single node, because no segment ofπ\\picontains a repeated node; henceℓ=x\\ell=x\. Nowxxalso lies ons′′s^\{\\prime\\prime\}, whiles′′s^\{\\prime\\prime\}has right exitr≠xr\\neq x\. If the left bounding edge ofs′′s^\{\\prime\\prime\}pointed out ofs′′s^\{\\prime\\prime\}, its left exit would be distinct fromrr\(otherwises′′s^\{\\prime\\prime\}would repeatrr, or would be the single noderr, both impossible since it also containsx≠rx\\neq r\), sos′′s^\{\\prime\\prime\}would have two distinct exits and would alreadybb\-blockπ\\pi\. Thus the left boundary ofs′′s^\{\\prime\\prime\}is an equation entered by an edgeq→⋅q\\to\\cdotwithq∈pa\#�G​\(\[x\]\)⊆Cq\\in\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[x\]\)\\subseteq C; thenqqis a right exit lying inCCof the segment precedings′′s^\{\\prime\\prime\}, which thereforebb\-blocksπ\\pi—a contradiction\.
- •If it points intos′s^\{\\prime\}, the right boundary ofs′s^\{\\prime\}is an equation entered by an edge⋅←q′\\cdot\\mathrel\{\\leftarrow\}q^\{\\prime\}withq′∈pa\#�G​\(\[x\]\)⊆Cq^\{\\prime\}\\in\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[x\]\)\\subseteq C; thenq′q^\{\\prime\}is a left exit lying inCCof the segment followings′s^\{\\prime\}, whichbb\-blocksπ\\pi—a contradiction\.

Hencettdoes notbb\-block, soπ′\\pi^\{\\prime\}isbb\-open\. This contradicts the minimality ofπ\\pi\. Therefore the minimalbb\-open walkπ\\pihas no repeated node, i\.e\., it is abb\-open path fromAAtoBB\. ∎

The following lemma shows that we have correctly designedbb\-separation such that it is equivalent todd\-separation in the acyclification\.

###### Lemma 38\.

For allA,B,C⊆VA,B,C\\subseteq V:

A​⟂\#�GbB\|C⇔A​⟂\#�GacydB\|C\.A\\Perp^\{b\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,C\\iff A\\Perp^\{d\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}B\\,\|\\,C\.

###### Proof\.

“⇒\\Rightarrow”:bb\-separation impliesdd\-separation in the acyclification\. By contrapositive: given a pathπacy=v0,v1,…,vk\\pi^\{\\mathrm\{acy\}\}=v\_\{0\},v\_\{1\},\\ldots,v\_\{k\}in\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}fromv0∈Av\_\{0\}\\in Atovk∈Bv\_\{k\}\\in Bthat isdd\-open givenCC, we construct a walkπ\\piin\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilbetweenv0v\_\{0\}andvkv\_\{k\}that isbb\-open givenCC\. By definition, each edge inπacy\\pi^\{\\mathrm\{acy\}\}connects variables in different clusters of\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil\. It can be lifted to a walk in\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilby lifting each directed edge in the same way as in the proof of Lemma[34](https://arxiv.org/html/2608.19831#Thmtheorem34):

vi→vi\+1v\_\{i\}\\to v\_\{i\+1\}onπacy\\pi^\{\\mathrm\{acy\}\}is lifted tovi→fi=⁣=⋯=⁣=vi\+1v\_\{i\}\\to f\_\{i\}\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}\\cdots\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}v\_\{i\+1\}in\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilvi←vi\+1v\_\{i\}\\mathrel\{\\leftarrow\}v\_\{i\+1\}onπacy\\pi^\{\\mathrm\{acy\}\}is lifted tovi=⁣=⋯=⁣=fi←vi\+1v\_\{i\}\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}\\cdots\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}f\_\{i\}\\mathrel\{\\leftarrow\}v\_\{i\+1\}in\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil\.Concatenating these lifts yields a walkπ\\piin\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilbetweenv0v\_\{0\}andvkv\_\{k\}\.

We show thatπ\\piisbb\-open givenCCby checking each possibility in which it could be blocked \(cf\. Definition[35](https://arxiv.org/html/2608.19831#Thmtheorem35)\)\.

- •A*collider segment*stems from the concatenated liftsvi−1→fi−1=⁣=⋯=⁣=vi=⁣=⋯=⁣=fi←vi\+1v\_\{i\-1\}\\to f\_\{i\-1\}\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}\\cdots\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}v\_\{i\}\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}\\cdots\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}f\_\{i\}\\mathrel\{\\leftarrow\}v\_\{i\+1\}of some collidervi−1→vi←vi\+1v\_\{i\-1\}\\to v\_\{i\}\\mathrel\{\\leftarrow\}v\_\{i\+1\}onπacy\\pi^\{\\mathrm\{acy\}\}\. Sinceπacy\\pi^\{\\mathrm\{acy\}\}isdd\-open givenCC,vi∈anc\#�Gacy​\(C\)v\_\{i\}\\in\\mathrm\{anc\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}\(C\), hencevi∈ant\#�G​\(C\)v\_\{i\}\\in\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\(by Lemma[34](https://arxiv.org/html/2608.19831#Thmtheorem34)\)\. All nodes in the segment are anterior toviv\_\{i\}, and by transitivity, each node in the segment lies inant\#�G​\(C\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\. Thus the segment does notbb\-block\.
- •A*non\-collider segment*stems from a non\-colliderviv\_\{i\}onπacy\\pi^\{\\mathrm\{acy\}\}\(a chain, a fork, or an end node\)\. The lift preserves outgoing edges on variable nodes, and it preserves end points\. Soviv\_\{i\}must be an exit of the segment\. The lifting procedure cannot generate a segment with two distinct exits\. Sinceπacy\\pi^\{\\mathrm\{acy\}\}isdd\-open givenCCandviv\_\{i\}is a non\-collider onπacy\\pi^\{\\mathrm\{acy\}\}, we havevi∉Cv\_\{i\}\\notin C\. Thus the segment does notbb\-block\.

Hence,π\\piisbb\-open givenCC\.

“⇐\\Leftarrow”:dd\-separation in the acyclification impliesbb\-separation\. By contrapositive: given a walkπ=r0,r1,…,rn\\pi=r\_\{0\},r\_\{1\},\\ldots,r\_\{n\}in\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilfromr0∈Ar\_\{0\}\\in Atorn∈Br\_\{n\}\\in Bthat isbb\-open givenCC, we construct a walkπacy\\pi^\{\\mathrm\{acy\}\}in\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}betweenr0r\_\{0\}andrnr\_\{n\}that isdd\-open givenCC\. The walkπ\\piconsists of segmentss1,…,sms\_\{1\},\\dots,s\_\{m\}separated by directed edges; each such directed edge is of the formv→fv\\to forf←vf\\mathrel\{\\leftarrow\}vwithv∈Vv\\in V,f∈Ff\\in F, andv∈pa\#�G​\(\[f\]\)v\\in\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[f\]\)\. We project each segmentsis\_\{i\}to a piece \(nodeviv\_\{i\}or a walkai←pi→bia\_\{i\}\\mathrel\{\\leftarrow\}p\_\{i\}\\to b\_\{i\}\), according to its type:

- •A*collider segment*sis\_\{i\}isbb\-open, so it intersectsant\#�G​\(C\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\); we pick a nodevi∈\[si\]∩V∩anc\#�Gacy​\(C\)v\_\{i\}\\in\[s\_\{i\}\]\\cap V\\cap\\mathrm\{anc\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}\(C\)\(which exists by Lemma[34](https://arxiv.org/html/2608.19831#Thmtheorem34)\) and projectsis\_\{i\}toviv\_\{i\}\. Onπacy\\pi^\{\\mathrm\{acy\}\},viv\_\{i\}will become a collider→vi←\\to v\_\{i\}\\mathrel\{\\leftarrow\}, which isdd\-open givenCCasvi∈anc\#�Gacy​\(C\)v\_\{i\}\\in\\mathrm\{anc\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}\(C\)\.
- •A*non\-collider segment with a single exit*viv\_\{i\}\(a chain, a one\-node fork, or an end noder0r\_\{0\}resp\.rnr\_\{n\}\): sinceπ\\piisbb\-open,vi∉Cv\_\{i\}\\notin C; we projectsis\_\{i\}toviv\_\{i\}\. Asviv\_\{i\}either carries an outgoing edge leaving its cluster or is an end node, it is a non\-collider \(chain, fork, or endpoint\) onπacy\\pi^\{\\mathrm\{acy\}\}, anddd\-open sincevi∉Cv\_\{i\}\\notin C\.
- •A*non\-collider segment with two distinct exits*ai≠bia\_\{i\}\\neq b\_\{i\}: sinceπ\\piisbb\-open,ai,bi∉Ca\_\{i\},b\_\{i\}\\notin Cand there must be a parentpi∈pa\#�G​\(\[si,1\]\)∖Cp\_\{i\}\\in\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[s\_\{i,1\}\]\)\\setminus C; we projectsis\_\{i\}to the forkai←pi→bia\_\{i\}\\mathrel\{\\leftarrow\}p\_\{i\}\\to b\_\{i\}\. As bothai,bia\_\{i\},b\_\{i\}either carry an outgoing edge leaving their cluster or form an end node, they are non\-colliders onπacy\\pi^\{\\mathrm\{acy\}\}, and both aredd\-open givenCCbecauseai,bi∉Ca\_\{i\},b\_\{i\}\\notin C\. Additionally,pip\_\{i\}does notdd\-block givenCConπacy\\pi^\{\\mathrm\{acy\}\}aspi∉Cp\_\{i\}\\notin C\.

The consecutive pieces are joined together to form a walk, preserving the directed edges that separated the segments onπ\\pi\. By construction, two consecutive variable nodes on this sequence must lie in different clusters, and one must be parent of the other in\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\. The resulting walkπacy\\pi^\{\\mathrm\{acy\}\}isdd\-open givenCCby construction\. ∎

We can now prove the global Markov property for partially oriented bipartite graphs via reduction to the Markov property for acyclic SCMs\.

###### Theorem 39\.

Suppose Assumption[10](https://arxiv.org/html/2608.19831#Thmtheorem10)holds\. When assigning independent distributions to all exogenous variables, the resulting joint distributionℙ⁡\(XV\)\\mathbb\{P\}\(X\_\{V\}\)satisfies: for allA,B,C⊆VA,B,C\\subseteq V,

A​⟂\#�GbB\|C⟹XA​⟂⁣⟂ℙ⁡\(XV\)⁡XB\|XC\.A\\Perp^\{b\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,C\\implies X\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\)\}X\_\{B\}\\,\|\\,X\_\{C\}\.

###### Proof\.

The clusters of\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilare partially ordered by the directed edges between them\. For each endogenous cluster\[v\]\[v\]withv∈V∖Uv\\in V\\setminus U, clusterwise unique solvability \(Assumption[10](https://arxiv.org/html/2608.19831#Thmtheorem10)\) provides a cluster solution functionΦ\[v\]\\Phi^\{\[v\]\}that expresses the endogenous variablesX\(V∖U\)∩\[v\]X\_\{\(V\\setminus U\)\\cap\[v\]\}as a function of the parent variablesXpa\#�G​\(\[v\]\)X\_\{\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[v\]\)\}\. WriteΦw\[v\]\\Phi\_\{w\}^\{\[v\]\}for the component ofΦ\[v\]\\Phi^\{\[v\]\}corresponding to variablew∈\(V∖U\)∩\[v\]w\\in\(V\\setminus U\)\\cap\[v\]\.

Replace the original system of equations by the acyclic system: for each endogenous variablev∈\(V∖U\)v\\in\(V\\setminus U\), the structural equation is

Xv=Φv\[v\]​\(Xpa\#�G​\(\[v\]\)\)\.X\_\{v\}=\\Phi\_\{v\}^\{\[v\]\}\(X\_\{\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[v\]\)\}\)\.Each exogenous variableXuX\_\{u\}\(u∈Uu\\in U\) retains its original independent distribution\. By construction, the solutions of the rewritten system coincide with the solutions of the original system \(Proposition[11](https://arxiv.org/html/2608.19831#Thmtheorem11)\): in both cases, variables are determined by recursively substituting cluster solution functions along the partial ordering of the clusters\. The rewritten system is an acyclic SCM: the structural equation for each variablevvdepends only on variables in strictly earlier clusters\.

Now by Lemma[38](https://arxiv.org/html/2608.19831#Thmtheorem38), the assumption

A​⟂\#�Gb​B\|CA\\Perp^\{b\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,Cimplies:

A​⟂\#�GacydB\|C\.A\\Perp^\{d\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}B\\,\|\\,C\.The graph of the rewritten acyclic SCM is a subgraph of the acyclification DAG\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\(it can be a strict subgraph if one or more functional dependences cancel out\)\. By the standard directed global Markov property for acyclic SCMs with independent exogenous variables\[[17](https://arxiv.org/html/2608.19831#bib.bib4), see also[18](https://arxiv.org/html/2608.19831#bib.bib5),[20](https://arxiv.org/html/2608.19831#bib.bib19)\]:

XA​⟂⁣⟂ℙ⁡\(XV\)⁡XB\|XC\.∎X\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\)\}X\_\{B\}\\,\|\\,X\_\{C\}\.\\qed

### C\.2Accounting for Determinism

We now strengthen the Markov property for partially oriented bipartite graphs by taking into account determinism, analogous to howDD\-separation\[[12](https://arxiv.org/html/2608.19831#bib.bib2)\]strengthensdd\-separation in Bayesian networks\.

We will writeS⪯TS\\preceq TiffXSX\_\{S\}is a measurable function ofXTX\_\{T\}\.

###### Lemma 40\.

LetA,B,S,T⊆VA,B,S,T\\subseteq VwithS⪯TS\\preceq T\. Then:

XA​⟂⁣⟂ℙ⁡\(XV\)⁡XB\|XS∪T⇔XA​⟂⁣⟂ℙ⁡\(XV\)⁡XB\|XT\.X\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\)\}X\_\{B\}\\,\|\\,X\_\{S\\cup T\}\\iff X\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\)\}X\_\{B\}\\,\|\\,X\_\{T\}\.

###### Proof\.

This follows from the elementary axioms for conditional independence\[[6](https://arxiv.org/html/2608.19831#bib.bib13)\]\. ∎

AlthoughBB\-blocking is stated withant\#�G​\(C\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)in its collider rule \(Definition[14](https://arxiv.org/html/2608.19831#Thmtheorem14)\),BB\-separation in fact coincides withbb\-separation given the enlarged conditioning setfdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\. This equivalence is the graphical device behind the strengthening\.

###### Lemma 41\.

For allC⊆VC\\subseteq V, a walk isBB\-blocked byCCif and only if it isbb\-blocked byfdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\. Hence, for allA,B,C⊆VA,B,C\\subseteq V:

A​⟂\#�GBB\|C⇔A​⟂\#�GbB\|fdet\#�G​\(C\)\.A\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,C\\iff A\\Perp^\{b\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\.

###### Proof\.

We show the per\-walk equivalence; the separation statement follows by quantifying over all walks \(equivalently, all paths\)\. Running Definition[35](https://arxiv.org/html/2608.19831#Thmtheorem35)atfdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\), rule 3 is subsumed by rule 2: a two\-distinct\-exit segment withpa\#�G​\(\[si,1\]\)⊆fdet\#�G​\(C\)\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[s\_\{i,1\}\]\)\\subseteq\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)has its whole cluster—hence both exits—infdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\. Sobb\-blocking byfdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)reduces to: a collider segment not meetingant\#�G​\(fdet\#�G​\(C\)\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\), or a non\-collider segment with an exit infdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\. This differs fromBB\-blocking byCConly in the collider condition, which usesant\#�G​\(fdet\#�G​\(C\)\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\)rather thanant\#�G​\(C\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\.

Sinceant\#�G​\(C\)⊆ant\#�G​\(fdet\#�G​\(C\)\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\\subseteq\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\), every walkbb\-blocked byfdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)isBB\-blocked byCC\. Conversely, letπ\\pibeBB\-blocked byCC; we show it isbb\-blocked byfdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\. As the two criteria share the non\-collider rule, we may assume the block comes from a collider segmentscs^\{c\}, in a cluster\[c∗\]\[c^\{\\ast\}\], that does not meetant\#�G​\(C\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\. If\[c∗\]\[c^\{\\ast\}\]also fails to meetant\#�G​\(fdet\#�G​\(C\)\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\), thenscs^\{c\}blocksπ\\piunderbb\-blocking byfdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)as well\. Otherwise\[c∗\]\[c^\{\\ast\}\]meetsant\#�G​\(fdet\#�G​\(C\)\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\)but notant\#�G​\(C\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\), and we claimpa\#�G​\(\[c∗\]\)⊆fdet\#�G​\(C\)\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[c^\{\\ast\}\]\)\\subseteq\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\.

By Lemma[34](https://arxiv.org/html/2608.19831#Thmtheorem34), meetingant\#�G​\(S\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(S\)is equivalent to\[c∗\]\[c^\{\\ast\}\]meetinganc\#�Gacy​\(S\)\\mathrm\{anc\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}\(S\); and in the acyclificationanc\#�Gacy​\(fdet\#�G​\(C\)\)=anc\#�Gacy​\(C\)∪fdet\#�G​\(C\)\\mathrm\{anc\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}\(\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\)=\\mathrm\{anc\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}\(C\)\\cup\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\. \(For “⊆\\subseteq”: along a directed path from an ancestorvvto a determined nodeww, either the path meetsCC, sov∈anc\#�Gacy​\(C\)v\\in\\mathrm\{anc\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}\(C\), or, reading it back fromww, each node is a parent of a determined node outsideCCand hence itself determined, sov∈fdet\#�G​\(C\)v\\in\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\.\) Thus\[c∗\]\[c^\{\\ast\}\]meetsfdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\): picky∈\[c∗\]∩fdet\#�G​\(C\)y\\in\[c^\{\\ast\}\]\\cap\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\. As\[c∗\]\[c^\{\\ast\}\]avoidsant\#�G​\(C\)⊇C\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\\supseteq C, we havey∉Cy\\notin C, soyyis endogenous withpa\#�G​\(\[c∗\]\)=pa\#�G​\(\[y\]\)⊆fdet\#�G​\(C\)\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[c^\{\\ast\}\]\)=\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[y\]\)\\subseteq\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\), proving the claim\.

The two walk\-neighbors ofscs^\{c\}are variables ofpa\#�G​\(\[c∗\]\)⊆fdet\#�G​\(C\)\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[c^\{\\ast\}\]\)\\subseteq\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)—they point into the boundary equations ofscs^\{c\}—and each is an exit of the adjacent non\-collider segment\. Hence both neighbors arebb\-blocked byfdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\), soπ\\piisbb\-blocked byfdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\. ∎

Combined with the acyclification, this yields a clean correspondence betweenBB\-separation andDD\-separation that mirrors Lemma[38](https://arxiv.org/html/2608.19831#Thmtheorem38), now accounting for determinism\.

###### Lemma 42\(BB\-separation equalsDD\-separation in the acyclification\)\.

For allA,B,C⊆VA,B,C\\subseteq V:

A​⟂\#�GBB\|C⇔A​⟂\#�GacyDB\|C\.A\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,C\\iff A\\Perp^\{D\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}B\\,\|\\,C\.

###### Proof\.

For everyv∈Vv\\in Vthe acyclification satisfiespa\#�Gacy​\(v\)=pa\#�G​\(\[v\]\)\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}\(v\)=\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[v\]\), so the closure recursions of Definitions[13](https://arxiv.org/html/2608.19831#Thmtheorem13)and[29](https://arxiv.org/html/2608.19831#Thmtheorem29)coincide; hencefdet\#�Gacy​\(C\)=fdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}\(C\)=\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)for allC⊆VC\\subseteq V\. Therefore

A​⟂\#�GB​B\|C\\displaystyle A\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,C⇔A​⟂\#�GbB\|fdet\#�G​\(C\)\\displaystyle\\iff A\\Perp^\{b\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\(Lemma[41](https://arxiv.org/html/2608.19831#Thmtheorem41)\)⇔A​⟂\#�GacydB\|fdet\#�G​\(C\)\\displaystyle\\iff A\\Perp^\{d\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}B\\,\|\\,\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\(Lemma[38](https://arxiv.org/html/2608.19831#Thmtheorem38)\)⇔A​⟂\#�GacydB\|fdet\#�Gacy​\(C\)\\displaystyle\\iff A\\Perp^\{d\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}B\\,\|\\,\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}\(C\)\(fdet\#�Gacy​\(C\)=fdet\#�G​\(C\)\)\\displaystyle\(\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}\(C\)=\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\)⇔A​⟂\#�GacyDB\|C\\displaystyle\\iff A\\Perp^\{D\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}B\\,\|\\,C\(Definition[30](https://arxiv.org/html/2608.19831#Thmtheorem30)\)\.∎

Theorem[16](https://arxiv.org/html/2608.19831#Thmtheorem16)now follows from Theorem[39](https://arxiv.org/html/2608.19831#Thmtheorem39)\. See[16](https://arxiv.org/html/2608.19831#Thmtheorem16)

###### Proof\.

By Lemma[41](https://arxiv.org/html/2608.19831#Thmtheorem41), the hypothesisA​⟂\#�GB​B\|CA\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,Cis equivalent to

A​⟂\#�GbB\|fdet\#�G​\(C\)\.A\\Perp^\{b\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\.By Theorem[39](https://arxiv.org/html/2608.19831#Thmtheorem39),

XA​⟂⁣⟂ℙ⁡\(XV\)⁡XB\|Xfdet\#�G​\(C\)\.X\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\)\}X\_\{B\}\\,\|\\,X\_\{\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\}\.The clusterwise solvability condition gives, for each endogenous cluster\[c\]\[c\]:

\[c\]∩V⪯pa\#�G​\(\[c\]\)\[c\]\\cap V\\preceq\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[c\]\)From the definitions, it follows thatfdet\#�G​\(C\)⪯C\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\\preceq C\. By Lemma[40](https://arxiv.org/html/2608.19831#Thmtheorem40), it then suffices to condition only onCC:

XA​⟂⁣⟂ℙ⁡\(XV\)⁡XB\|XC\.∎X\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\)\}X\_\{B\}\\,\|\\,X\_\{C\}\.\\qed

Similarly tobb\-separation, alsoBB\-separation may equivalently be defined via walks or via paths\.

###### Lemma 43\(BB\-separation via walks or paths\)\.

For allA,B,C⊆VA,B,C\\subseteq V, every walk from a node inAAto a node inBBisBB\-blocked byCCif and only if every path from a node inAAto a node inBBisBB\-blocked byCC\.

###### Proof\.

By Lemma[41](https://arxiv.org/html/2608.19831#Thmtheorem41)\(per\-walk form\), a walk isBB\-blocked byCCiff it isbb\-blocked byfdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\); applied to walks and to paths separately, this gives thatBB\-separation givenCC, defined via walks or via paths, equalsbb\-separation givenfdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)defined via walks resp\. paths\. The latter two coincide by Lemma[37](https://arxiv.org/html/2608.19831#Thmtheorem37)\. ∎

Becausefdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)can be strictly larger thanCC,BB\-separation is genuinely stronger thanbb\-separation:A​⟂\#�Gb​B\|CA\\Perp^\{b\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,CimpliesA​⟂\#�GB​B\|CA\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,C\(comparing the two criteria at the sameCC: the collider rule is common to both, while each non\-collider case ofbb\-blocking—an exit inCC, or two distinct exits whose cluster satisfiespa\#�G​\(\[c\]\)⊆C\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[c\]\)\\subseteq C—puts an exit infdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)and henceBB\-blocks\), but not conversely, as Example[44](https://arxiv.org/html/2608.19831#Thmtheorem44)shows\.

###### Example 44\(BB\-separation is strictly stronger thanbb\-separation\)\.

Consider the observational bathtub graph\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilof Example[6](https://arxiv.org/html/2608.19831#Thmtheorem6)and condition onC=\{XI\}C=\\\{X\_\{I\}\\\}\. Sincepa\#�G​\(\[XO\]\)=\{XI\}\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[X\_\{O\}\]\)=\\\{X\_\{I\}\\\}, the outflow is functionally determined byCC, sofdet\#�G​\(\{XI\}\)=\{XI,XO\}\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\\\{X\_\{I\}\\\}\)=\\\{X\_\{I\},X\_\{O\}\\\}; the variablesXPX\_\{P\}andXDX\_\{D\}are not determined, as their clusters also requireXKX\_\{K\}, resp\.XgX\_\{g\}\.

TakeA=\{XO\}A=\\\{X\_\{O\}\\\}andB=\{XD\}B=\\\{X\_\{D\}\\\}\. The walkXO→f2=⁣=XP→f3=⁣=XDX\_\{O\}\\to f\_\{2\}\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}X\_\{P\}\\to f\_\{3\}\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}X\_\{D\}splits into the segments\{XO\}\\\{X\_\{O\}\\\},\{f2,XP\}\\\{f\_\{2\},X\_\{P\}\\\}, and\{f3,XD\}\\\{f\_\{3\},X\_\{D\}\\\}, with single exitsXOX\_\{O\},XPX\_\{P\}, andXDX\_\{D\}respectively\. None of these exits lies inC=\{XI\}C=\\\{X\_\{I\}\\\}, hence this walk isbb\-open andXO​⟂̸\#�GbXD\|XIX\_\{O\}\\nPerp^\{b\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}X\_\{D\}\\,\|\\,X\_\{I\}\. UnderBB\-separation the outcome differs: every walk out ofXOX\_\{O\}begins with the endpoint segment\{XO\}\\\{X\_\{O\}\\\}whose exitXOX\_\{O\}lies infdet\#�G​\(\{XI\}\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\\\{X\_\{I\}\\\}\), soXO​⟂\#�GBXD\|XIX\_\{O\}\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}X\_\{D\}\\,\|\\,X\_\{I\}\.

TheBB\-separation verdict is the correct one: conditioning onXIX\_\{I\}forcesXO=XIX\_\{O\}=X\_\{I\}\(Example[6](https://arxiv.org/html/2608.19831#Thmtheorem6)\), soXOX\_\{O\}is constant givenXIX\_\{I\}and hence trivially independent ofXDX\_\{D\}\. ThusBB\-separation detects a conditional independence arising from determinism thatbb\-separation misses\.

## Appendix DProof of the Extended Global Markov Property

We prove Theorem[18](https://arxiv.org/html/2608.19831#Thmtheorem18), which extends the Global Markov Property \(Theorem[16](https://arxiv.org/html/2608.19831#Thmtheorem16)\) from joint distributions to Markov kernels with non\-random input variables\. Our strategy will be similar to that of Appendix[C](https://arxiv.org/html/2608.19831#A3), but rather than applying the standard global Markov property for acyclic SCMs, we make use of the global Markov property for causal Bayesian networks with input variables established by[10](https://arxiv.org/html/2608.19831#bib.bib17)\.

[10](https://arxiv.org/html/2608.19831#bib.bib17)proves the global Markov property for CBNs with input variables using*transitional conditional independence*\[[10](https://arxiv.org/html/2608.19831#bib.bib17), Definition 3\.1\]\. This is an asymmetric notion of conditional independence for Markov kernels:XA⟂⁣⟂ℙ\(⋅∥XJ\)XB\|XCX\_\{A\}\\Indep\_\{\\mathbb\{P\}\(\\cdot\\,\\\|\\,X\_\{J\}\)\}X\_\{B\}\\,\|\\,X\_\{C\}means that there*exists*a Markov kernelQ\(XA∥XC\)Q\(X\_\{A\}\\,\\\|\\,X\_\{C\}\)\(not depending onXBX\_\{B\}\) such that

ℙ\(XA,XB,XC∥XJ\)=Q\(XA∥XC\)⊗ℙ\(XB,XC∥XJ\)\.\\mathbb\{P\}\(X\_\{A\},X\_\{B\},X\_\{C\}\\,\\\|\\,X\_\{J\}\)=Q\(X\_\{A\}\\,\\\|\\,X\_\{C\}\)\\otimes\\mathbb\{P\}\(X\_\{B\},X\_\{C\}\\,\\\|\\,X\_\{J\}\)\.The proof of[10](https://arxiv.org/html/2608.19831#bib.bib17)proceeds by induction over the topological ordering of the conditional DAG \(CDAG\)—which marks the variables inJ⊆U⊆VJ\\subseteq U\\subseteq Vas*input*variables—chaining the \(asymmetric\) separoid rules for transitional conditional independence anddd\-separation in CDAGs, an \(asymmetric\) extension ofdd\-separation in DAGs\. A crucial feature of[10](https://arxiv.org/html/2608.19831#bib.bib17)’s approach is that it does*not*rely on symmetry of conditional independence \(which fails for Markov kernels in general\), but instead uses left and right versions of the separoid rules separately\.

###### Theorem 45\.

Suppose Assumption[10](https://arxiv.org/html/2608.19831#Thmtheorem10)holds\. Treat exogenous variablesJ⊆UJ\\subseteq Uas non\-random and assign independent distributions to the remaining exogenous variables inU∖JU\\setminus J, yielding the Markov kernelℙ\(XV∥XJ\)\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\)\. Then for allA,B,C⊆VA,B,C\\subseteq V:

A⟂b\#�GB∪J\|C⟹XA⟂⁣⟂ℙ\(XV∥XJ\)XB\|XC\.A\\Perp^\{b\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\cup J\\,\|\\,C\\implies X\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\)\}X\_\{B\}\\,\|\\,X\_\{C\}\.

###### Proof\.

As in the proof of Theorem[39](https://arxiv.org/html/2608.19831#Thmtheorem39), solutions of the system satisfy the acyclic system of equations

Xv=Φv\[v\]​\(Xpa\#�G​\(\[v\]\)\),v∈V∖U\.X\_\{v\}=\\Phi\_\{v\}^\{\[v\]\}\(X\_\{\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[v\]\)\}\),\\qquad v\\in V\\setminus U\.TreatJ⊆UJ\\subseteq Uas non\-random and put independent distributions on the remaining exogenous variablesW:=U∖JW:=U\\setminus J\. For each endogenous cluster\[v\]\[v\]\(v∈V∖Uv\\in V\\setminus U\), let the deterministic Markov kernel𝒳pa\#�G​\(\[v\]\)→𝒫⁡\(𝒳v\)\\mathcal\{X\}\_\{\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[v\]\)\}\\to\\mathcal\{P\}\(\\mathcal\{X\}\_\{v\}\)be the one corresponding to the cluster solution functionΦv\[v\]\\Phi\_\{v\}^\{\[v\]\}, and for eachw∈Ww\\in Wletℙ⁡\(Xw\)\\mathbb\{P\}\(X\_\{w\}\)be its distribution\. Together with the non\-stochastic inputsXJX\_\{J\}, these Markov kernels define a*causal Bayesian network*ℳ\\mathcal\{M\}in the sense of[10](https://arxiv.org/html/2608.19831#bib.bib17), with:

- •non\-stochastic input variables:XJX\_\{J\},
- •stochastic variables:XV∖JX\_\{V\\setminus J\},
- •graph: the acyclification\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}, viewed as a conditional DAG\#�Gacy​\(V∖J\|do⁡\(J\)\)\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\(V\\setminus J\\,\|\\,\\mathrm\{do\}\(J\)\)\.

This is a valid conditional DAG:\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}is acyclic, and no edge points into anyj∈Jj\\in J, since each suchjjis an exogenous singleton cluster withpa\#�G​\(\[j\]\)=∅\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[j\]\)=\\emptyset\. The joint Markov kernel ofℳ\\mathcal\{M\}is preciselyℙ\(XV∥XJ\)\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\)\.

LetA,B,C⊆VA,B,C\\subseteq Vbe such that

A​⟂\#�GbB∪J\|C\.A\\Perp^\{b\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\cup J\\,\|\\,C\.By Lemma[38](https://arxiv.org/html/2608.19831#Thmtheorem38),

A​⟂\#�GacydB∪J\|C\.A\\Perp^\{d\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\}B\\cup J\\,\|\\,C\.Since\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}is acyclic,dd\-separation coincides withσ\\sigma\-separation\[[10](https://arxiv.org/html/2608.19831#bib.bib17), Definition 5\.9 and Remark 5\.10\]\. Reading the acyclification as the conditional DAG\#�Gacy​\(V∖J\|do⁡\(J\)\)\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\(V\\setminus J\\,\|\\,\\mathrm\{do\}\(J\)\), whoseσ\\sigma\-separation criterion implicitly includes the input nodesJJon the right, this is exactly

A​⟂\#�Gacy​\(V∖J\|do⁡\(J\)\)σB\|C\.A\\Perp^\{\\sigma\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\(V\\setminus J\\,\|\\,\\mathrm\{do\}\(J\)\)\}B\\,\|\\,C\.By the global Markov property of[10](https://arxiv.org/html/2608.19831#bib.bib17)applied toℳ\\mathcal\{M\}, this implies the transitional conditional independence

XA⟂⁣⟂ℙ\(XV∥XJ\)XB\|XC\.∎X\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\)\}X\_\{B\}\\,\|\\,X\_\{C\}\.\\qed

###### Corollary 46\.

Suppose Assumption[10](https://arxiv.org/html/2608.19831#Thmtheorem10)holds\. Treat exogenous variablesJ⊆UJ\\subseteq Uas non\-random, and assign independent distributions to exogenous variables inU∖JU\\setminus J, yielding Markov kernelℙ\(XV∥XJ\)\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\)\. Then for allA,B,C⊆VA,B,C\\subseteq Vsuch thatJ⊆B∪CJ\\subseteq B\\cup C:

A⟂b\#�GB\|C⟹XA⟂⁣⟂ℙ\(XV∥XJ\)XB\|XC\.A\\Perp^\{b\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,C\\implies X\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\)\}X\_\{B\}\\,\|\\,X\_\{C\}\.

###### Proof\.

WhenJ⊆B∪CJ\\subseteq B\\cup Cwe haveJ∖B⊆CJ\\setminus B\\subseteq C, so every walk fromAAto a node ofJ∖BJ\\setminus Bends inCCand is thereforebb\-blocked byCC; henceA​⟂\#�Gb​B\|CA\\Perp^\{b\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,CimpliesA​⟂\#�GbB∪J\|CA\\Perp^\{b\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\cup J\\,\|\\,C, and Theorem[45](https://arxiv.org/html/2608.19831#Thmtheorem45)yieldsXA⟂⁣⟂ℙ\(XV∥XJ\)XB\|XCX\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\)\}X\_\{B\}\\,\|\\,X\_\{C\}\. ∎

### D\.1Accounting for Determinism

Just as we strengthened the global Markov property for partially oriented bipartite graphs by accounting for determinism, we can do the same for the extended version\. Definition[13](https://arxiv.org/html/2608.19831#Thmtheorem13)still applies, and does not need to distinguish exogenous input nodesJJfrom exogenous random nodesU∖JU\\setminus J: exogenous variable nodes are only functionally determined byCCif they are inCC\.

###### Lemma 47\.

LetA,B,S,T⊆VA,B,S,T\\subseteq VwithS⪯TS\\preceq T\. Then:

XA⟂⁣⟂ℙ\(XV∥XJ\)XB\|XS∪T⇔XA⟂⁣⟂ℙ\(XV∥XJ\)XB\|XT\.X\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\)\}X\_\{B\}\\,\|\\,X\_\{S\\cup T\}\\iff X\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\)\}X\_\{B\}\\,\|\\,X\_\{T\}\.

###### Proof\.

This is the transitional\-conditional\-independence analog of Lemma[40](https://arxiv.org/html/2608.19831#Thmtheorem40); because transitional conditional independence is asymmetric, we cannot use the symmetric argument and instead invoke Forré’s Equivalent Exchange rule\. By hypothesisXSX\_\{S\}is a measurable function ofXTX\_\{T\}\. ThenXS∪TX\_\{S\\cup T\}is a measurable function ofXTX\_\{T\}; converselyXTX\_\{T\}is a coordinate projection ofXS∪TX\_\{S\\cup T\}\. Since all model variables are measurable functions of the exogenous variablesXUX\_\{U\}, bothXS∪TX\_\{S\\cup T\}andXTX\_\{T\}are deterministic transitional random variables; as each is a measurable function of the other, they are equivalent\[[10](https://arxiv.org/html/2608.19831#bib.bib17), Notation 2\.19 and Remark 2\.20\]\. The Equivalent Exchange rule for transitional conditional independence\[[10](https://arxiv.org/html/2608.19831#bib.bib17), Corollary 3\.14\], which allows the conditioning variable to be replaced by an equivalent one, then yields both implications:

XA⟂⁣⟂ℙ\(XV∥XJ\)XB\|XS∪T⇔XA⟂⁣⟂ℙ\(XV∥XJ\)XB\|XT\.∎X\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\)\}X\_\{B\}\\,\|\\,X\_\{S\\cup T\}\\iff X\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\)\}X\_\{B\}\\,\|\\,X\_\{T\}\.\\qed

We obtain Theorem[18](https://arxiv.org/html/2608.19831#Thmtheorem18)from Corollary[46](https://arxiv.org/html/2608.19831#Thmtheorem46)\. See[18](https://arxiv.org/html/2608.19831#Thmtheorem18)

###### Proof\.

By Lemma[41](https://arxiv.org/html/2608.19831#Thmtheorem41), the hypothesisA​⟂\#�GB​B\|CA\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,Cis equivalent to

A​⟂\#�GbB\|fdet\#�G​\(C\)\.A\\Perp^\{b\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}B\\,\|\\,\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\.By Corollary[46](https://arxiv.org/html/2608.19831#Thmtheorem46)\(whose hypothesisJ⊆B∪fdet\#�G​\(C\)J\\subseteq B\\cup\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)holds becauseJ⊆B∪C⊆B∪fdet\#�G​\(C\)J\\subseteq B\\cup C\\subseteq B\\cup\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\),

XA⟂⁣⟂ℙ\(XV∥XJ\)XB\|Xfdet\#�G​\(C\)\.X\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\)\}X\_\{B\}\\,\|\\,X\_\{\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\}\.The clusterwise solvability condition gives, for each endogenous cluster\[c\]\[c\]:

\[c\]∩V⪯pa\#�G​\(\[c\]\)\.\[c\]\\cap V\\preceq\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[c\]\)\.From the definitions, it follows thatfdet\#�G​\(C\)⪯C\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\)\\preceq C\. By Lemma[47](https://arxiv.org/html/2608.19831#Thmtheorem47), it then suffices to condition only onCC:

XA⟂⁣⟂ℙ\(XV∥XJ\)XB\|XC\.∎X\_\{A\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\\,\\\|\\,X\_\{J\}\)\}X\_\{B\}\\,\|\\,X\_\{C\}\.\\qed

## Appendix EPhysical Implementations of Hard Interventions

The formal notationdo\(fj:Xv=ξv\)\\mathrm\{do\}\(f\_\{j\}:X\_\{v\}=\\xi\_\{v\}\)has a natural physical interpretation: it specifies*which mechanism*fjf\_\{j\}in the system is replaced in order to enforceXv=ξvX\_\{v\}=\\xi\_\{v\}\. Different choices offjf\_\{j\}correspond to genuinely different physical procedures for achieving the same target value\. We illustrate this for four of the hard interventions on the bathtub model\. Combined with the ones in Example[20](https://arxiv.org/html/2608.19831#Thmtheorem20), this gives a complete “physical implementation” of the causal semantics of the bathtub system under the hard interventions we consider elementary in our framework\.

#### do\(f1:XO=ξO\)\\mathrm\{do\}\(f\_\{1\}:X\_\{O\}=\\xi\_\{O\}\)\.

Equationf1f\_\{1\}\(the equilibrium conditionXI=XOX\_\{I\}=X\_\{O\}\) is replaced byf~1:XO=ξO\\tilde\{f\}\_\{1\}:X\_\{O\}=\\xi\_\{O\}\. The causal ordering is preserved:f~1\\tilde\{f\}\_\{1\}determinesXOX\_\{O\},f2f\_\{2\}determinesXPX\_\{P\}, andf3f\_\{3\}determinesXDX\_\{D\}\. A physical implementation is to divert the original inflow away from the tub and install a new faucet with inflow rateXI2=ξOX\_\{I\_\{2\}\}=\\xi\_\{O\}\.

#### do\(f1:XP=ξP\)\\mathrm\{do\}\(f\_\{1\}:X\_\{P\}=\\xi\_\{P\}\)\.

Equationf1f\_\{1\}is replaced byf~1:XP=ξP\\tilde\{f\}\_\{1\}:X\_\{P\}=\\xi\_\{P\}\. The causal ordering changes:f~1\\tilde\{f\}\_\{1\}now determinesXPX\_\{P\}\(instead ofXOX\_\{O\}\), and consequentlyf2f\_\{2\}must solve forXOX\_\{O\}givenXPX\_\{P\}\. A physical implementation requires diverting the original inflow, installing a sufficiently large new inflow, and connecting a pressure relief valve to the bottom of the tub that activates whenXP\>ξPX\_\{P\}\>\\xi\_\{P\}\.

#### do\(f2:XP=ξP\)\\mathrm\{do\}\(f\_\{2\}:X\_\{P\}=\\xi\_\{P\}\)\.

Here, Torricelli’s lawf2f\_\{2\}is replaced byf~2:XP=ξP\\tilde\{f\}\_\{2\}:X\_\{P\}=\\xi\_\{P\}, while the equilibrium conditionf1f\_\{1\}remains intact\. This requires a more involved physical procedure: clog the drain, reroute the original inflow directly to the outflow through a pipe \(bypassing the tub and drain\), install an additional sufficiently large inflow, and connect a pressure relief valve that activates whenXP\>ξPX\_\{P\}\>\\xi\_\{P\}\. Note that this intervention targets the same variable \(XP=ξPX\_\{P\}=\\xi\_\{P\}\) asdo\(f1:XP=ξP\)\\mathrm\{do\}\(f\_\{1\}:X\_\{P\}=\\xi\_\{P\}\), but through a different mechanism: the equilibrium conditionf1f\_\{1\}is preserved, soXO=XIX\_\{O\}=X\_\{I\}still holds, whereas indo\(f1:XP=ξP\)\\mathrm\{do\}\(f\_\{1\}:X\_\{P\}=\\xi\_\{P\}\)we getXO=XK​ξPX\_\{O\}=X\_\{K\}\\sqrt\{\\xi\_\{P\}\}\.

#### do\(f2:XD=ξD\)\\mathrm\{do\}\(f\_\{2\}:X\_\{D\}=\\xi\_\{D\}\)\.

Torricelli’s lawf2f\_\{2\}is replaced byf~2:XD=ξD\\tilde\{f\}\_\{2\}:X\_\{D\}=\\xi\_\{D\}\. This changes the causal ordering:f3f\_\{3\}now determinesXPX\_\{P\}fromXDX\_\{D\}\(rather thanXDX\_\{D\}fromXPX\_\{P\}\)\. The physical implementation is similar to the previous case—clog the drain, reroute inflow to outflow, install an additional inflow—but instead of a pressure valve, the bathtub is cut at heightξD\\xi\_\{D\}\.

These examples illustrate a key advantage of the BGCM framework: the notiondo\(fj:Xv=ξv\)\\mathrm\{do\}\(f\_\{j\}:X\_\{v\}=\\xi\_\{v\}\)makes the physical implementation explicit by specifying which mechanism is targeted, resolving the ambiguity inherent in the standard notiondo⁡\(Xv=ξv\)\\mathrm\{do\}\(X\_\{v\}=\\xi\_\{v\}\)\. Furthermore, we have shown explicitly that each such hard intervention that leads to a solvable system can indeed be realized as a real\-world intervention\.

## Appendix FPartially Oriented Bipartite Graphs under Interventions

Figure[3](https://arxiv.org/html/2608.19831#A6.F3)shows the partially oriented bipartite graphs for the observational bathtub model, all six well\-defined hard interventions \(cf\. Table[1](https://arxiv.org/html/2608.19831#S5.T1)\), and the three infeasible hard interventions\. Edges that change relative to the observational case are drawn in red; intervened equation nodes are also shown in red\. The systems that \(generically\) do not have solutions are drawn in gray\.

do\(f1:XO=ξO\)\\mathrm\{do\}\(f\_\{1\}:X\_\{O\}=\\xi\_\{O\}\)

XOX\_\{O\}XPX\_\{P\}XDX\_\{D\}f~1\\tilde\{f\}\_\{1\}f2f\_\{2\}f3f\_\{3\}XIX\_\{I\}XKX\_\{K\}XgX\_\{g\}

do\(f2:XO=ξO\)\\mathrm\{do\}\(f\_\{2\}:X\_\{O\}=\\xi\_\{O\}\)

XOX\_\{O\}XPX\_\{P\}XDX\_\{D\}f1f\_\{1\}f~2\\tilde\{f\}\_\{2\}f3f\_\{3\}XIX\_\{I\}XKX\_\{K\}XgX\_\{g\}

do\(f3:XO=ξO\)\\mathrm\{do\}\(f\_\{3\}:X\_\{O\}=\\xi\_\{O\}\)

XOX\_\{O\}XPX\_\{P\}XDX\_\{D\}f1f\_\{1\}f2f\_\{2\}f~3\\tilde\{f\}\_\{3\}XIX\_\{I\}XKX\_\{K\}XgX\_\{g\}

Observational XOX\_\{O\}XPX\_\{P\}XDX\_\{D\}f1f\_\{1\}f2f\_\{2\}f3f\_\{3\}XIX\_\{I\}XKX\_\{K\}XgX\_\{g\}

do\(f1:XP=ξP\)\\mathrm\{do\}\(f\_\{1\}:X\_\{P\}=\\xi\_\{P\}\)

XOX\_\{O\}XPX\_\{P\}XDX\_\{D\}f~1\\tilde\{f\}\_\{1\}f2f\_\{2\}f3f\_\{3\}XIX\_\{I\}XKX\_\{K\}XgX\_\{g\}

do\(f2:XP=ξP\)\\mathrm\{do\}\(f\_\{2\}:X\_\{P\}=\\xi\_\{P\}\)

XOX\_\{O\}XPX\_\{P\}XDX\_\{D\}f1f\_\{1\}f~2\\tilde\{f\}\_\{2\}f3f\_\{3\}XIX\_\{I\}XKX\_\{K\}XgX\_\{g\}

do\(f3:XP=ξP\)\\mathrm\{do\}\(f\_\{3\}:X\_\{P\}=\\xi\_\{P\}\)

XOX\_\{O\}XPX\_\{P\}XDX\_\{D\}f1f\_\{1\}f2f\_\{2\}f~3\\tilde\{f\}\_\{3\}XIX\_\{I\}XKX\_\{K\}XgX\_\{g\}

do\(f1:XD=ξD\)\\mathrm\{do\}\(f\_\{1\}:X\_\{D\}=\\xi\_\{D\}\)

XOX\_\{O\}XPX\_\{P\}XDX\_\{D\}f~1\\tilde\{f\}\_\{1\}f2f\_\{2\}f3f\_\{3\}XIX\_\{I\}XKX\_\{K\}XgX\_\{g\}

do\(f2:XD=ξD\)\\mathrm\{do\}\(f\_\{2\}:X\_\{D\}=\\xi\_\{D\}\)

XOX\_\{O\}XPX\_\{P\}XDX\_\{D\}f1f\_\{1\}f~2\\tilde\{f\}\_\{2\}f3f\_\{3\}XIX\_\{I\}XKX\_\{K\}XgX\_\{g\}

do\(f3:XD=ξD\)\\mathrm\{do\}\(f\_\{3\}:X\_\{D\}=\\xi\_\{D\}\)

XOX\_\{O\}XPX\_\{P\}XDX\_\{D\}f1f\_\{1\}f2f\_\{2\}f~3\\tilde\{f\}\_\{3\}XIX\_\{I\}XKX\_\{K\}XgX\_\{g\}

Figure 3:Partially oriented bipartite graphs for the bathtub model under all hard interventions \(cf\. Table[1](https://arxiv.org/html/2608.19831#S5.T1)\)\. The observational graph is shown top\-right for reference\. Intervened equation nodes are shown in red; edges whose orientation changes relative to the observational case are drawn in red\. Note how replacing different equations can lead to fundamentally different causal orderings: e\.g\.,do\(f1:XD=ξD\)\\mathrm\{do\}\(f\_\{1\}:X\_\{D\}=\\xi\_\{D\}\)reverses the causal flow entirely\. For three hard interventions, the intervened bipartite graph cannot be oriented \(no perfect matching exists\), so the undirected intervened bipartite graph is displayed instead\.
## Appendix GPartially Oriented Bipartite Graphs for the Domain Invariance Examples

Figure[4](https://arxiv.org/html/2608.19831#A7.F4)shows the partially oriented bipartite graphs\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrGR\\hfil\\textstyle G^\{R\}\\hfilof the joint models constructed for each of the four domain invariance examples in Section[6](https://arxiv.org/html/2608.19831#S6)\. In each case, the exogenous domain indicator variableRRis connected to the equation\(s\) that differ between domains\. The graphical structure of\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrGR\\hfil\\textstyle G^\{R\}\\hfildetermines whichBB\-separation statements hold, and hence which domain invariances can be derived from the Markov property\.

Example[21](https://arxiv.org/html/2608.19831#Thmtheorem21):obs\. vs\.do⁡\(Xg=ξg\)\\mathrm\{do\}\(X\_\{g\}=\\xi\_\{g\}\)

XOX\_\{O\}XPX\_\{P\}XDX\_\{D\}XgX\_\{g\}f1f\_\{1\}f2f\_\{2\}f3f\_\{3\}f4f\_\{4\}XIX\_\{I\}XKX\_\{K\}UgU\_\{g\}RR

XP,XO​⟂\#�GRBRX\_\{P\},X\_\{O\}\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G^\{R\}\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G^\{R\}\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G^\{R\}\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G^\{R\}\\hfil$\\crcr\}\}\}\}R

Example[22](https://arxiv.org/html/2608.19831#Thmtheorem22):obs\. vs\.do\(f3:XD=ξD\)\\mathrm\{do\}\(f\_\{3\}:X\_\{D\}=\\xi\_\{D\}\)

XOX\_\{O\}XPX\_\{P\}XDX\_\{D\}f1f\_\{1\}f2f\_\{2\}f3f\_\{3\}XIX\_\{I\}XKX\_\{K\}XgX\_\{g\}RR

XO​⟂\#�GRBR\|XD,XPX\_\{O\}\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G^\{R\}\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G^\{R\}\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G^\{R\}\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G^\{R\}\\hfil$\\crcr\}\}\}\}R\\,\|\\,X\_\{D\},X\_\{P\}

Example[23](https://arxiv.org/html/2608.19831#Thmtheorem23):obs\. vs\.do\(f1:XD=ξD\)\\mathrm\{do\}\(f\_\{1\}:X\_\{D\}=\\xi\_\{D\}\)

XOX\_\{O\}XPX\_\{P\}XDX\_\{D\}f1f\_\{1\}f2f\_\{2\}f3f\_\{3\}XIX\_\{I\}XKX\_\{K\}XgX\_\{g\}RR

No non\-trivialBB\-separation involvingRR

Example[24](https://arxiv.org/html/2608.19831#Thmtheorem24):do\(f1:XD=ξD\)\\mathrm\{do\}\(f\_\{1\}:X\_\{D\}\\\!=\\\!\\xi\_\{D\}\)vs\.do\(f1:XD=ξD′\)\\mathrm\{do\}\(f\_\{1\}:X\_\{D\}\\\!=\\\!\\xi\_\{D\}^\{\\prime\}\)

XOX\_\{O\}XPX\_\{P\}XDX\_\{D\}f1f\_\{1\}f2f\_\{2\}f3f\_\{3\}XIX\_\{I\}XKX\_\{K\}XgX\_\{g\}RR

XO​⟂\#�GRBR\|XPX\_\{O\}\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G^\{R\}\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G^\{R\}\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G^\{R\}\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G^\{R\}\\hfil$\\crcr\}\}\}\}R\\,\|\\,X\_\{P\}

Figure 4:Partially oriented bipartite graphs of the joint models for the four domain invariance examples \(Section[6](https://arxiv.org/html/2608.19831#S6)\)\. In each graph, the exogenous domain indicatorRRis connected to the equation\(s\) that differ between domains\. TheBB\-separation statement below each graph summarizes the graphical criterion from which the corresponding domain invariance is derived via the Markov property\. In Example[23](https://arxiv.org/html/2608.19831#Thmtheorem23), all endogenous nodes and equation nodes form a single cluster, so no non\-trivialBB\-separation involvingRRexists\.
## Appendix HA Worked Example with a Genuine Cycle: Supply and Demand

The bathtub of Example[6](https://arxiv.org/html/2608.19831#Thmtheorem6), although a feedback system at equilibrium, has a partial orientation whose endogenous clusters each consist of one equation and one variable \(\{f1,XO\}\\\{f\_\{1\},X\_\{O\}\\\},\{f2,XP\}\\\{f\_\{2\},X\_\{P\}\\\},\{f3,XD\}\\\{f\_\{3\},X\_\{D\}\\\}\); its causal ordering is therefore acyclic\. To illustrate the part of the framework that deals with genuine cycles—multi\-node clusters, whose segments are treated inBB\-separation as single indivisible units \(Definition[14](https://arxiv.org/html/2608.19831#Thmtheorem14)\)—we work out a classic simultaneous supply–demand system, in which all three mechanisms must be solved jointly\.

Consider a competitive market at equilibrium with endogenous variablesXSX\_\{S\}\(quantity supplied\),XDX\_\{D\}\(quantity demanded\), andXPX\_\{P\}\(price\), and exogenous supply/demand shiftsXUS,XUDX\_\{U\_\{S\}\},X\_\{U\_\{D\}\}\. With supply slopeβ\\betaand demand slopeα\\alphasatisfyingβ\>0\>α\\beta\>0\>\\alpha, the equilibrium is described by three mechanisms:

f1:\\displaystyle f\_\{1\}:0=XS−XD\\displaystyle\\quad 0=X\_\{S\}\-X\_\{D\}\(market clears\)f2:\\displaystyle f\_\{2\}:0=β​XP\+XUS−XS\\displaystyle\\quad 0=\\beta X\_\{P\}\+X\_\{U\_\{S\}\}\-X\_\{S\}\(supply\)f3:\\displaystyle f\_\{3\}:0=α​XP\+XUD−XD\\displaystyle\\quad 0=\\alpha X\_\{P\}\+X\_\{U\_\{D\}\}\-X\_\{D\}\(demand\)
#### A single endogenous cluster\.

The endogenous subgraph \(variables\{XS,XP,XD\}\\\{X\_\{S\},X\_\{P\},X\_\{D\}\\\}, equations\{f1,f2,f3\}\\\{f\_\{1\},f\_\{2\},f\_\{3\}\\\}\) admits the perfect matchingM=\{f1\-⁣\-XS,f2\-⁣\-XP,f3\-⁣\-XD\}M=\\\{f\_\{1\}\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}X\_\{S\},\\ f\_\{2\}\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}X\_\{P\},\\ f\_\{3\}\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}X\_\{D\}\\\}\. The closedMM\-alternating walk

XS\-⁣\-f1\-⁣\-XD\-⁣\-f3\-⁣\-XP\-⁣\-f2\-⁣\-XSX\_\{S\}\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}f\_\{1\}\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}X\_\{D\}\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}f\_\{3\}\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}X\_\{P\}\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}f\_\{2\}\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}X\_\{S\}\(alternating the matched edgesf1\-⁣\-XSf\_\{1\}\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}X\_\{S\},f3\-⁣\-XDf\_\{3\}\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}X\_\{D\},f2\-⁣\-XPf\_\{2\}\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}X\_\{P\}with the unmatched edgesf1\-⁣\-XDf\_\{1\}\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}X\_\{D\},f3\-⁣\-XPf\_\{3\}\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}X\_\{P\},f2\-⁣\-XSf\_\{2\}\\mathrel\{\{\\relbar\\mkern\-9\.0mu\\relbar\}\}X\_\{S\}\) visits all six nodes of the endogenous subgraph\. By Definition[3](https://arxiv.org/html/2608.19831#Thmtheorem3)they therefore form a single cluster\{f1,f2,f3,XS,XP,XD\}\\\{f\_\{1\},f\_\{2\},f\_\{3\},X\_\{S\},X\_\{P\},X\_\{D\}\\\}, and by Lemma[4](https://arxiv.org/html/2608.19831#Thmtheorem4)this is independent of the chosen matching\. The cluster has parentspa\#�G​\(\[f1\]\)=\{XUS,XUD\}\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[f\_\{1\}\]\)=\\\{X\_\{U\_\{S\}\},X\_\{U\_\{D\}\}\\\}\. In the partial orientation\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil, all six intra\-cluster edges are double\-undirected, while the two exogenous shifts point in \(Figure[5](https://arxiv.org/html/2608.19831#A8.F5)a\)\. Unlike the bathtub, the causal ordering here is not acyclic: the entire endogenous system is a single feedback cluster, solved simultaneously\.

f1f\_\{1\}XSX\_\{S\}f2f\_\{2\}XPX\_\{P\}f3f\_\{3\}XDX\_\{D\}XUSX\_\{U\_\{S\}\}XUDX\_\{U\_\{D\}\}

\(a\) partial orientation\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil

XUSX\_\{U\_\{S\}\}XUDX\_\{U\_\{D\}\}XSX\_\{S\}XPX\_\{P\}XDX\_\{D\}

\(b\) acyclification\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}

Figure 5:The supply–demand system and its acyclification\.\(a\)The partial orientation\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil: the three endogenous variables and three equations form a single feedback cluster \(dashed box\) with the exogenous shiftsXUS,XUDX\_\{U\_\{S\}\},X\_\{U\_\{D\}\}as its parents; all intra\-cluster edges are double\-undirected\.\(b\)The acyclification\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}, the DAG on the variable nodes with an edgev→v′v\\to v^\{\\prime\}wheneverv∈pa\#�G​\(\[v′\]\)v\\in\\mathrm\{pa\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\[v^\{\\prime\}\]\); the two shifts become common parents of all three endogenous variables, which are mutually nonadjacent\.BB\-separation in\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilcoincides withDD\-separation in\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\(Lemma[42](https://arxiv.org/html/2608.19831#Thmtheorem42)\), so the two encode the same conditional independences; but only\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfil, which retains the equation nodes, can model interventions\.
#### Unique solvability and solution\.

Becauseβ\>0\>α\\beta\>0\>\\alpha, the cluster is uniquely solvable \(Assumption[10](https://arxiv.org/html/2608.19831#Thmtheorem10)\)\. Substituting the supply and demand relations into the market\-clearing conditionXS=XDX\_\{S\}=X\_\{D\}givesβ​XP\+XUS=α​XP\+XUD\\beta X\_\{P\}\+X\_\{U\_\{S\}\}=\\alpha X\_\{P\}\+X\_\{U\_\{D\}\}, hence

XP=XUD−XUSβ−α,XS=XD=β​XUD−α​XUSβ−α\.X\_\{P\}=\\frac\{X\_\{U\_\{D\}\}\-X\_\{U\_\{S\}\}\}\{\\beta\-\\alpha\},\\qquad X\_\{S\}=X\_\{D\}=\\frac\{\\beta X\_\{U\_\{D\}\}\-\\alpha X\_\{U\_\{S\}\}\}\{\\beta\-\\alpha\}\.

#### Reading off \(in\)dependences\.

Assign independent distributions to the shiftsXUS,XUDX\_\{U\_\{S\}\},X\_\{U\_\{D\}\}, so thatℙ⁡\(XV\)\\mathbb\{P\}\(X\_\{V\}\)is well defined and the Markov property \(Theorem[16](https://arxiv.org/html/2608.19831#Thmtheorem16)\) applies\. Consider the walk

XUS→f2=⁣=XP=⁣=f3←XUD\.X\_\{U\_\{S\}\}\\to f\_\{2\}\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}X\_\{P\}\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}f\_\{3\}\\mathrel\{\\leftarrow\}X\_\{U\_\{D\}\}\.Its middle segmentf2=⁣=XP=⁣=f3f\_\{2\}\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}X\_\{P\}\\mathrel\{\{\\Relbar\\mkern\-9\.0mu\\Relbar\}\}f\_\{3\}lies entirely inside the cluster and is a*collider segment*: both bounding edges point inward\. ForC=∅C=\\emptysetwe haveant\#�G​\(∅\)=∅\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\\emptyset\)=\\emptyset, so this collider segment is disjoint fromant\#�G​\(∅\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\\emptyset\)and blocks \(rule 1 of Definition[14](https://arxiv.org/html/2608.19831#Thmtheorem14)\)\. SinceXUSX\_\{U\_\{S\}\}andXUDX\_\{U\_\{D\}\}attach to the cluster only through the inward edgesXUS→f2X\_\{U\_\{S\}\}\\to f\_\{2\}andXUD→f3X\_\{U\_\{D\}\}\\to f\_\{3\},*every*path between them crosses such a within\-cluster collider segment, so

XUS​⟂\#�GBXUD\|∅⟹XUS​⟂⁣⟂ℙ⁡\(XV\)⁡XUD,X\_\{U\_\{S\}\}\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}X\_\{U\_\{D\}\}\\,\|\\,\\emptyset\\quad\\Longrightarrow\\quad X\_\{U\_\{S\}\}\\Indep\_\{\\mathbb\{P\}\(X\_\{V\}\)\}X\_\{U\_\{D\}\},recovering the assumed independence of the two shifts\.

Conditioning on the equilibrium price reverses the verdict\. AsXPX\_\{P\}lies in the cluster, the whole cluster is anterior toXPX\_\{P\}, so the collider segment now meetsant\#�G​\(\{XP\}\)\\mathrm\{ant\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\\\{X\_\{P\}\\\}\)and no longer blocks\. Moreoverfdet\#�G​\(\{XP\}\)=\{XP\}\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\\\{X\_\{P\}\\\}\)=\\\{X\_\{P\}\\\}: determining the cluster would require*both*exogenous parents, and neither is in\{XP\}\\\{X\_\{P\}\\\}, so no further node is functionally determined\. Hence the endpointsXUS,XUDX\_\{U\_\{S\}\},X\_\{U\_\{D\}\}are not infdet\#�G​\(\{XP\}\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\\\{X\_\{P\}\\\}\), rule 2 does not block either, and the walk isBB\-open:

XUS​⟂̸\#�GBXUD\|XP\.X\_\{U\_\{S\}\}\\nPerp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}X\_\{U\_\{D\}\}\\,\|\\,X\_\{P\}\.This matches the algebra:XP=\(XUD−XUS\)/\(β−α\)X\_\{P\}=\(X\_\{U\_\{D\}\}\-X\_\{U\_\{S\}\}\)/\(\\beta\-\\alpha\), so conditioning on the price imposes the constraintXUD−XUS=\(β−α\)​XPX\_\{U\_\{D\}\}\-X\_\{U\_\{S\}\}=\(\\beta\-\\alpha\)X\_\{P\}and generically renders the two shifts dependent\. It is the familiar phenomenon of conditioning on a common effect—except that here the common effect is an entire feedback cluster rather than a single variable, which is exactly what rule 1 ofBB\-separation is designed to capture\.

#### Determinism \(rule 2\) in a cycle\.

Rule 2 blocks a non\-collider segment as soon as one of its exits is*functionally determined*by the conditioning set\. In a single multi\-variable cluster this is all\-or\-nothing: every endogenous variable has the same parents\{XUS,XUD\}\\\{X\_\{U\_\{S\}\},X\_\{U\_\{D\}\}\\\}, so none is determined until*both*shifts are conditioned on\. Thusfdet\#�G​\(\{XUS\}\)=\{XUS\}\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\\\{X\_\{U\_\{S\}\}\\\}\)=\\\{X\_\{U\_\{S\}\}\\\}, whereas

fdet\#�G​\(\{XUS,XUD\}\)=\{XUS,XUD,XS,XP,XD\}\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(\\\{X\_\{U\_\{S\}\},X\_\{U\_\{D\}\}\\\}\)=\\\{X\_\{U\_\{S\}\},X\_\{U\_\{D\}\},X\_\{S\},X\_\{P\},X\_\{D\}\\\}is the entire set of variable nodes\. In the latter case the endpoint segment of every path already has an exit infdet\#�G​\(C\)\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}\(C\), so rule 2 blocks it; hence, for example,XS​⟂\#�GBXP\|XUS,XUDX\_\{S\}\\Perp^\{B\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}X\_\{P\}\\,\|\\,X\_\{U\_\{S\}\},X\_\{U\_\{D\}\}\. Whereas conditioning on the endogenous price*created*dependence between the shifts \(rule 1\), conditioning on both exogenous shifts*removes*all dependence among the endogenous quantities: fixing both shifts pins down the equilibrium, so every endogenous quantity is constant and hence conditionally independent of the rest\. Needing*all*of a cluster’s parents before any of its variables is determined is not special to cycles; it is a general property offdet\#�G\\mathrm\{fdet\}\_\{\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}\}that already appears at single\-variable clusters whose equation has several parents\. In the bathtub, for instance, the cluster\{f2,XP\}\\\{f\_\{2\},X\_\{P\}\\\}has parents\{XO,XK\}\\\{X\_\{O\},X\_\{K\}\\\}, so pressure is determined only once both outflow and drain area are fixed—conditioning onXOX\_\{O\}alone leavesXPX\_\{P\}undetermined\.

#### The same facts byDD\-separation\.

By Lemma[42](https://arxiv.org/html/2608.19831#Thmtheorem42),BB\-separation in\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilcoincides withDD\-separation in the acyclification\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}\(Figure[5](https://arxiv.org/html/2608.19831#A8.F5)b\), so the three verdicts above can be read off there as well\. In\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}the shiftsXUS,XUDX\_\{U\_\{S\}\},X\_\{U\_\{D\}\}are common parents ofXS,XP,XDX\_\{S\},X\_\{P\},X\_\{D\}, which are otherwise nonadjacent: \(i\) unconditionally the shifts aredd\-separated, their only connections passing through the unconditioned collidersXS,XP,XDX\_\{S\},X\_\{P\},X\_\{D\}; \(ii\) conditioning onXPX\_\{P\}opens that collider, making the shiftsdd\-connected; and \(iii\) conditioning on both shifts blocks each path betweenXSX\_\{S\}andXPX\_\{P\}path at its fork \(XUSX\_\{U\_\{S\}\}orXUDX\_\{U\_\{D\}\}\), givingdd\-separation\. The within\-cluster collider segment of\#�\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldrG\\hfil\\textstyle G\\hfilthus becomes an ordinary collider at a common child in\#�Gacy\\mathchoice\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\displaystyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\displaystyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\displaystyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\textstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\textstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\textstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptstyle G\\hfil$\\crcr\}\}\}\{\\vbox\{\\halign\{\#\\cr\\kern\-0\.7pt\\cr$\\mkern 2\.0mu\\scriptscriptstyle\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraitd\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraitd\}$\}\}\}\\mkern\-1\.5mu\\leaders\{\\hbox\{$\\scriptscriptstyle\\mkern 0\.0mu\\mathrel\{\\mathchoice\{\{\}\\hbox\{$\\displaystyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\textstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptstyle\{\\montraita\}$\}\}\{\{\}\\hbox\{$\\scriptscriptstyle\{\\montraita\}$\}\}\}\\mkern 0\.0mu$\}\}\{\\hfill\}\\mkern\-1\.5mu\\fldr$\\crcr\\kern\-0\.3pt\\cr$\\hfil\\scriptscriptstyle G\\hfil$\\crcr\}\}\}^\{\\mathrm\{acy\}\}, and conditioning on a cluster variable becomes conditioning on that child\. In this exampleDD\-separation reduces to ordinarydd\-separation, since conditioning on the parents that determine a variable already blocks the forks through them\.