Reduction of Probabilistic Chemical Reaction Networks

# Reduction of Probabilistic Chemical Reaction Networks
Source: [https://arxiv.org/html/2606.27737](https://arxiv.org/html/2606.27737)
###### Abstract

Programming adaptive behaviors at the cellular level is a long\-standing goal that raises the question of how probabilistic computation can be implemented in biochemical systems\. Chemical reaction networks \(CRNs\) provide such a substrate and have been shown to realize probabilistic models, including hidden Markov models and factor graphs, with dynamics reproducing Bayesian inference and belief propagation\. However, encoding these algorithms typically requires prohibitively large reaction networks, and classical CRN reduction techniques do not directly apply\. By recovering the factor graph structure encoded in Napp–Adams\-compiled CRNs, we transport recent factor\-graph reduction results to their chemical implementations, obtaining significantly smaller CRNs while preserving the belief\-propagation fixed points on surviving variables\.

Machine Learning, ICML

## 1Introduction

Models of agency in biological organisms, such as the Bayesian brain hypothesis\(Knill and Pouget,[2004](https://arxiv.org/html/2606.27737#bib.bib23)\)and active inference\(Friston,[2010](https://arxiv.org/html/2606.27737#bib.bib21); Parret al\.,[2022](https://arxiv.org/html/2606.27737#bib.bib22)\), posit that decision making relies on maintaining and updating internal beliefs about the environment\. Underlying these frameworks is the assumption that biological agents operate under incomplete information, which they acquire only indirectly through noisy and partial observations\. As a consequence, adaptive behavior requires continuous inference about latent environmental states\.

Modeling agency*in silico*\.Incomplete information is also central to optimal control and, more broadly, to reinforcement learning, where belief updates are formalized through probabilistic inference algorithms such as filtering and belief propagation\. These same inference mechanisms also play a central role in active inference\(Pezzuloet al\.,[2024](https://arxiv.org/html/2606.27737#bib.bib25)\)\. Translating these algorithms into biological substrates therefore appears as a necessary step toward cellular decision making design\. A natural choice for achieving this goal is using chemical reaction networks \(CRNs\), denotedΓ\\Gamma, which are a list of reactions whose mass\-action kinetics define an ordinary differential equation \(ODE\) for concentrations of its species, providing a continuous\-time substrate for computation\. Recent work has shown that CRNs can implement a range of probabilistic inference tasks: Bayesian posterior computation in algal phototaxis\(Colliauxet al\.,[2017](https://arxiv.org/html/2606.27737#bib.bib16)\), filtering\(Nakamura and Kobayashi,[2022](https://arxiv.org/html/2606.27737#bib.bib11),[2021](https://arxiv.org/html/2606.27737#bib.bib24)\), belief propagation\(Napp and Adams,[2013](https://arxiv.org/html/2606.27737#bib.bib5)\), inference\(Pooleet al\.,[2017](https://arxiv.org/html/2606.27737#bib.bib32)\), learning hidden Markov models\(Wiufet al\.,[2023](https://arxiv.org/html/2606.27737#bib.bib17)\), and classification\(Moghimianavvalet al\.,[2024](https://arxiv.org/html/2606.27737#bib.bib26); van der Lindenet al\.,[2022](https://arxiv.org/html/2606.27737#bib.bib27)\)\. Information processing via chemical regulation is imperative to the creation of composable biochemical systems that can generalize to different tasks\(Cardelliet al\.,[2018](https://arxiv.org/html/2606.27737#bib.bib10); Brijder,[2018](https://arxiv.org/html/2606.27737#bib.bib9); Tschantzet al\.,[2019](https://arxiv.org/html/2606.27737#bib.bib12); Vasicet al\.,[2021](https://arxiv.org/html/2606.27737#bib.bib6); Viswa Virinchiet al\.,[2018](https://arxiv.org/html/2606.27737#bib.bib19)\)\. However, implementing these tasks with CRNs scales poorly as the number of distinct chemical species \(and associated reactions/ODEs\) increase, that is, as decision\-making problems become more complex\(Napp and Adams,[2013](https://arxiv.org/html/2606.27737#bib.bib5); Hjelmfeltet al\.,[1991](https://arxiv.org/html/2606.27737#bib.bib31); Viswa Virinchiet al\.,[2018](https://arxiv.org/html/2606.27737#bib.bib19); Wiufet al\.,[2023](https://arxiv.org/html/2606.27737#bib.bib17); Pooleet al\.,[2017](https://arxiv.org/html/2606.27737#bib.bib32)\)\.

#### Inference with CRNs\.

In this article, we focus on inference in factor graphs, denoted byℱ\\mathcal\{F\}, which subsume main ingredients of the methods and models discussed above\(Andersonet al\.,[2021](https://arxiv.org/html/2606.27737#bib.bib2); Napp and Adams,[2013](https://arxiv.org/html/2606.27737#bib.bib5); Andersonet al\.,[2021](https://arxiv.org/html/2606.27737#bib.bib2); Vasicet al\.,[2021](https://arxiv.org/html/2606.27737#bib.bib6); Lakin,[2023](https://arxiv.org/html/2606.27737#bib.bib3)\)\. Engineering biological systems to scale up to large sizes is an essential task at the intersection of these fields\. Just as users do not need detailed knowledge of the intermediate steps of the internal calculations being done by a computer, being able to simplify and interpret designed biochemical systems to the inputs and outputs of interest is necessary to reduce the burden of control in the design of programmable cells\(Tschantzet al\.,[2019](https://arxiv.org/html/2606.27737#bib.bib12)\)\. Limitations to these approaches arise because such implementations typically rely on the control of the stoichiometry of a synthetic CRN\. To that end, we leverage recent factor graph reduction techniques based on deformation retractions of topological spaces associated to factor graphs\(Sergeant\-Perthuis and Boitel,[2025](https://arxiv.org/html/2606.27737#bib.bib4)\)and transport these reductions to their chemical implementation\.

#### Contributions

We develop a theory for recognizing and reducing CRNs that implement belief propagation via the Napp–Adams construction\(Napp and Adams,[2013](https://arxiv.org/html/2606.27737#bib.bib5)\)\. The key insight is that these CRNs contain catalytic dependencies that are not visible from stoichiometry alone: messages are represented by species bundles, factor tables appear as rate parameters, and BP fixed points are encoded by bundle\-normalized steady states\. We make four contributions: \(1\) structural conditions \(W1–W6\) under which a mass\-action CRN encodes a factor graph, together with a constructive reconstruction procedure \(Theorem[4\.7](https://arxiv.org/html/2606.27737#S4.Thmtheorem7)\); \(2\) a proof that positive steady states correspond bijectively to BP fixed points under an additional recycling condition R1 \(Theorem[5\.4](https://arxiv.org/html/2606.27737#S5.Thmtheorem4)\); \(3\) a functorial transport of SP–B factor\-graph retractions to CRN reductions that delete entire message bundles while preserving BP fixed points on surviving variables \(Proposition[6\.2](https://arxiv.org/html/2606.27737#S6.Thmtheorem2), Lemma[6\.3](https://arxiv.org/html/2606.27737#S6.Thmtheorem3)\) — to our knowledge the first semantics\-preserving reduction for BP\-compiled CRNs; and \(4\) empirical validation across several graph families \(Section[7](https://arxiv.org/html/2606.27737#S7)\)\. The scope of \(1–3\) is intentional: W1–W6 characterize the Napp–Adams compiled subclass, not generic biochemical networks, and the pipeline is closed under reduction by Proposition[6\.2](https://arxiv.org/html/2606.27737#S6.Thmtheorem2)\. Table[2](https://arxiv.org/html/2606.27737#A1.T2)\(Appendix[A](https://arxiv.org/html/2606.27737#A1)\) summarizes the correspondence between graphical and chemical concepts that \(W1–W6\) formalize\. We recommend consulting it while reading Sections 4–6\.

The invariant preserved here differs from classical CRN reductions\. Stoichiometric and Laplacian\-based reductions preserve structural or steady\-state properties visible from reaction stoichiometry\. In Napp–Adams compiled CRNs, however, the BP message passing algorithm is carried by catalytic dependencies and bundle\-normalized concentration ratios\. These dependencies are not captured by stoichiometry alone, which is why we reduce at the factor\-graph level and then transport the reduction to the CRN\.

## 2Main Results

We study a class of probabilistic CRNs: mass\-action networks whose positive steady states encode belief propagation \(BP\) fixed points, which give approximate marginals for an associated factor graph, reconstructed from the CRN’s message\-bundle structure\. Our results formalize a pipeline that \(1\) compiles a factor graph into a probabilistic CRN, \(2\) recognizes and reconstructs the underlying factor graph from a given CRN, and \(3\) transports factor\-graph reductions to the chemical setting in a way that preserves BP solutions\.

ℱ\{\\mathcal\{F\}\}ℱ′\{\\mathcal\{F\}^\{\\prime\}\}Γ\{\\Gamma\}Γ′\{\\Gamma^\{\\prime\}\}r\{r\}𝒩\{\\mathcal\{N\}\}𝒩\{\\mathcal\{N\}\}𝒩​\(r\)\{\\mathcal\{N\}\(r\)\}Figure 1:An SP–B retractionrrat the factor\-graph level induces a CRN reduction𝒩​\(r\)\\mathcal\{N\}\(r\)that preserves BP fixed points on surviving variables \(Proposition[6\.2](https://arxiv.org/html/2606.27737#S6.Thmtheorem2), Lemma[6\.3](https://arxiv.org/html/2606.27737#S6.Thmtheorem3)\)\.#### Compilation \(Factor graphs→\\toCRNs\)\.

Napp and Adams\(Napp and Adams,[2013](https://arxiv.org/html/2606.27737#bib.bib5)\)show how to implement message\-passing inference in a CRN by introducing, for each message vector, a bundle of species whose relative concentrations represent that vector\. Reactions are chosen so that the resulting mass\-action ODE performs a continuous\-time analogue of damped asynchronous message updates, converging to the same fixed points\. In our paper we treat this construction as a compilation map from a factor graphℱ\\mathcal\{F\}to a CRNΓ\\Gamma\(formal definitions appear in Section[6](https://arxiv.org/html/2606.27737#S6)and Appendix[D](https://arxiv.org/html/2606.27737#A4)\)\.

#### Recognition and reconstruction \(CRNs→\\toFactor graphs\)\.

We give conditions \(W1\)–\(W6\) under which a CRN encodes a factor graph together with its factor tables, and we provide an explicit reconstruction procedureFG​\(Γ\)\\mathrm\{FG\}\(\\Gamma\)\(Theorem[4\.7](https://arxiv.org/html/2606.27737#S4.Thmtheorem7)\)\.

#### Steady states and BP fixed points\.

With one additional condition \(R1\) ensuring the correct normalization/recycling behavior, we prove that positive steady states ofΓ\\Gammacorrespond to BP fixed points onFG​\(Γ\)\\mathrm\{FG\}\(\\Gamma\)\(Theorem[5\.4](https://arxiv.org/html/2606.27737#S5.Thmtheorem4)\)\.

#### Reduction commutes with compilation\.

Sergeant–Perthuis and Boitel \(SP–B\)\(Sergeant\-Perthuis and Boitel,[2025](https://arxiv.org/html/2606.27737#bib.bib4)\)introduce a notion of*retraction*for factor graphs, obtained by deleting certain variable/factor nodes and updating the neighboring factor tables in a way that preserves the set of BP fixed points on the remaining variables\. We package these SP–B retractions as morphisms in our factor\-graph category \(Section[6](https://arxiv.org/html/2606.27737#S6)\)\. We then show that the Napp–Adams compilation is functorial: an SP–B retractionℱ→ℱ′\\mathcal\{F\}\\to\\mathcal\{F\}^\{\\prime\}induces a corresponding CRN reductionΓ→Γ′\\Gamma\\to\\Gamma^\{\\prime\}that removes entire*message bundles*and updates only the rate parameters in the affected local subnetworks \(Proposition[6\.2](https://arxiv.org/html/2606.27737#S6.Thmtheorem2)\)\. Consequently, compiling after reducing yields the same reduced CRN \(up to relabeling\) as reducing after compiling, and BP fixed points are preserved under this transport \(Lemma[6\.3](https://arxiv.org/html/2606.27737#S6.Thmtheorem3)\)\.

#### Empirical gains\.

Across our benchmark suite, SP–B retractions translate into large*chemical*savings after Napp–Adams compilation, and these savings show up directly as faster ODE integration\. Figure[2](https://arxiv.org/html/2606.27737#S2.F2)illustrates the typical behavior: retracting reducible substructure substantially shrinks the compiled CRN and accelerates convergence of the concentration\-derived marginals\. Table[1](https://arxiv.org/html/2606.27737#S2.T1)summarizes results by graph family\. Tree\- and chain\-structured instances exhibit near\-complete collapse to a small core \(about95%95\\%fewer variables and97%97\\%fewer chemical species at the median\), yielding dramatic simulation speedups \(median270×270\\timeson chains and886×886\\timeson trees\)\. Loopy\-core instances with reducible tendrils see partial but still substantial compression \(median79%79\\%variable and73%73\\%species reduction\) and corresponding speedups \(median22×22\\times\), reflecting that the irreducible loopy core must remain\. By contrast, grid graphs are essentially irreducible under these moves and show no change in size or runtime, serving as a control\. Finally, random planted\-core graphs admit only modest reduction \(median20%20\\%variables and10%10\\%species\) and therefore modest speedups \(median1\.5×1\.5\\times\), consistent with having fewer retractable tendrils\. In all cases, we additionally monitor marginal agreement on the surviving variables \(Section[7](https://arxiv.org/html/2606.27737#S7)\); the observed runtime gains coincide with preserved inferred marginals rather than a change in the underlying beliefs\. Section 7 also reports Erdős–Rényi stress tests over sparse, near\-threshold, and dense regimes, showing that reduction tracks the amount of graph lying outside the irreducible loopy core rather than depending on a planted construction\.

![Refer to caption](https://arxiv.org/html/2606.27737v1/napp_reduction_convergence.png)Figure 2:Simulation ImprovementsSP–B reductions substantially shrink compiled CRNs and yield large compilation and simulation\-time speedups on our benchmark suite while preserving inferred marginals on surviving variables\. Here we have an80%80\\%reduction the size of the CRN \(3\-chain\) and a56%56\\%speed up in convergence to the correct marginal \(Section[7](https://arxiv.org/html/2606.27737#S7)\)\.Table 1:Median compression and simulation speedup of a structured benchmark suite\.

## 3Background

#### Factor graphs\.

We consider discrete probabilistic models whose joint distribution is a product of local functions\. A*factor graph*makes this factorization explicit and provides a convenient interface for message\-passing inference\.

###### Definition 3\.1\(Factor graph\(Yedidiaet al\.,[2000](https://arxiv.org/html/2606.27737#bib.bib29)\)\)\.

A factor graphℱ=\(ℋ,E,f\)\\mathcal\{F\}=\(\\mathcal\{H\},E,f\)consists of a hypergraphℋ=\(I,J\)\\mathcal\{H\}=\(I,J\)with variable indicesIIand factor indicesJJ, a collection of finite state spaces\(Ei\)i∈I\(E\_\{i\}\)\_\{i\\in I\}, and nonnegative factorsψj:∏i∈jEi→ℝ≥0\\psi\_\{j\}:\\prod\_\{i\\in j\}E\_\{i\}\\to\\mathbb\{R\}\_\{\\geq 0\}for eachj∈Jj\\in J\.

The factors define a global probability distribution by

p​\(𝐱\)=1Z​∏j∈Jψj​\(xj\),xj:=\(xi\)i∈j,p\(\\mathbf\{x\}\)=\\frac\{1\}\{Z\}\\prod\_\{j\\in J\}\\psi\_\{j\}\(x\_\{j\}\),\\qquad x\_\{j\}:=\(x\_\{i\}\)\_\{i\\in j\},whereZZis the normalizing constant\. We depictℱ\\mathcal\{F\}as a bipartite graph with variable nodesi∈Ii\\in Iand factor nodesj∈Jj\\in J, with an edge betweenjjandiiwheneveri∈ji\\in j\.

#### Belief propagation\.

Belief propagation \(BP\) is a message\-passing dynamic program that computes exact marginals on tree\-structured factor graphs and is widely used as an approximation on graphs with cycles \(loopy BP\)\(Yedidiaet al\.,[2000](https://arxiv.org/html/2606.27737#bib.bib29)\)\. BP maintains messages along edges: variable\-to\-factor messagesP\(i→j\)∈ℝ≥0EiP^\{\(i\\to j\)\}\\in\\mathbb\{R\}\_\{\\geq 0\}^\{E\_\{i\}\}and factor\-to\-variable messagesS\(j→i\)∈ℝ≥0EiS^\{\(j\\to i\)\}\\in\\mathbb\{R\}\_\{\\geq 0\}^\{E\_\{i\}\}\. Fork∈Eik\\in E\_\{i\},

Sk\(j→i\)\\displaystyle S^\{\(j\\to i\)\}\_\{k\}=∑xj∖ifj​\(xj\)​∏i′∈j∖iPxi′\(i′→j\),\\displaystyle=\\sum\_\{x\_\{j\\setminus i\}\}f\_\{j\}\(x\_\{j\}\)\\prod\_\{i^\{\\prime\}\\in j\\setminus i\}P^\{\(i^\{\\prime\}\\to j\)\}\_\{x\_\{i^\{\\prime\}\}\},Pk\(i→j\)\\displaystyle P^\{\(i\\to j\)\}\_\{k\}=∏j′∈ne​\(i\)∖jSk\(j′→i\)\.\\displaystyle=\\prod\_\{j^\{\\prime\}\\in\\mathrm\{ne\}\(i\)\\setminus j\}S^\{\(j^\{\\prime\}\\to i\)\}\_\{k\}\.An \(unnormalized\) belief at variableiiis p~​\(xi=k\)=∏j∈ne​\(i\)Sk\(j→i\)\\tilde\{p\}\(x\_\{i\}=k\)=\\prod\_\{j\\in\\mathrm\{ne\}\(i\)\}S^\{\(j\\to i\)\}\_\{k\}\.

#### Chemical reaction networks\.

A CRN is a finite set of species and reactions equipped with rate constants\.

###### Definition 3\.2\(CRN\)\.

A CRN is a tripleΓ=\(X,R,κ\)\\Gamma=\(X,R,\\kappa\)whereXXis a finite set of species,RRis a finite set of reactions, andκ:R→ℝ\>0\\kappa:R\\to\\mathbb\{R\}\_\{\>0\}assigns a positive rate to each reaction\.

Under mass\-action kinetics, concentrationsx​\(t\)∈ℝ≥0Xx\(t\)\\in\\mathbb\{R\}\_\{\\geq 0\}^\{X\}evolve according to an ODE determined by\(X,R,κ\)\(X,R,\\kappa\)\. In this work, CRNs serve as a dynamical substrate for implementing and reducing probabilistic inference\.

## 4Recognition Conditions and Reconstruction

We introduce six local conditions on the CRN, \(W1\)–\(W6\), that can be checked directly in terms of stoichiometry and rate polynomials\. Together they guarantee that a factor\-graph structure exists and can be reconstructed\.

Before stating the recognition conditions, we briefly indicate their role\. They abstract the bundle\-and\-rate pattern of the Napp–Adams compilation: messages become species bundles, factor tables become catalyzed production rates, and recycling supplies the normalization needed to recover BP fixed points at steady state\. A fully worked three\-variable chain example appears in Appendix[E\.1](https://arxiv.org/html/2606.27737#A5.Thmexample1)\.

### 4\.1Stoichiometric conditions \(W1–W3\)

LetΓ=\(X,R,κ\)\\Gamma=\(X,R,\\kappa\)be a CRN\. A reactionRk∈RR\_\{k\}\\in Ris expressed asRk:∑i=1Nri​k​Xi⟶∑i=1Npi​k​XiR\_\{k\}:\\ \\sum\_\{i=1\}^\{N\}r\_\{ik\}X\_\{i\}\\longrightarrow\\sum\_\{i=1\}^\{N\}p\_\{ik\}X\_\{i\}with coefficientsri​k,pi​k∈ℕr\_\{ik\},p\_\{ik\}\\in\\mathbb\{N\}\. The*stoichiometric matrix*ofΓ\\Gammais theN×KN\\times KmatrixS=\(xi​k\)S=\(x\_\{ik\}\)with entriesxi​k=pi​k−ri​kx\_\{ik\}\\;=\\;p\_\{ik\}\-r\_\{ik\}, so that columnkkencodes the net change of species in reactionRkR\_\{k\}\.

#### \(W1\) Stoichiometric block decomposition\.

There exists a finite index setℬ\\mathcal\{B\}and a partition of the set of species into disjoint piecesSBS\_\{B\}

X=⨆B∈ℬSBX\\;=\\;\\bigsqcup\_\{B\\in\\mathcal\{B\}\}S\_\{B\}such that, for each reactionRkR\_\{k\}with net stoichiometric column\(xi​k\)i=1N\(x\_\{ik\}\)\_\{i=1\}^\{N\}, the support

\{i:xi​k≠0\}\\\{\\,i:x\_\{ik\}\\neq 0\\,\\\}is contained in a single blockSBS\_\{B\}\. In other words, every reaction has nonzero net stoichiometry in at most one block\.

#### \(W2\) Distinguished zero species in each block\.

For each blockSBS\_\{B\}in the partition of \(W1\) there exists a distinguished speciesXB,0∈SBX\_\{B,0\}\\in S\_\{B\}, called the*zero species*ofSBS\_\{B\}\. We require thatSB∖\{XB,0\}S\_\{B\}\\setminus\\\{X\_\{B,0\}\\\}be nonempty and call its elements the*coordinate species*ofSBS\_\{B\}\.

#### \(W3\) Star–shaped internal stoichiometry\.

For each blockSBS\_\{B\}and each reactionRkR\_\{k\}whose net stoichiometry is supported inSBS\_\{B\}, the restriction of the column\(xi​k\)\(x\_\{ik\}\)toSBS\_\{B\}is either

−XB,0\+XB,korXB,0−XB,k\.\-X\_\{B,0\}\+X\_\{B,k\}\\quad\\text\{or\}\\quad X\_\{B,0\}\-X\_\{B,k\}\.
Under \(W1\)–\(W3\), the species ofΓ\\Gammasplit into disjoint blocksSBS\_\{B\}, each with a unique zero species and several coordinate species, and all reaction columns are intra\-block and of zero↔\\leftrightarrowcoordinate form\.

###### Definition 4\.2\(Catalytic neighbors and assignments\)\.

LetΓ=\(X,R,κ\)\\Gamma=\(X,R,\\kappa\)be a CRN with stoichiometric matrixSSsatisfying \(W1\-W3\)\.

1. 1\.For a bundleBB, the*catalytic neighbor set*Cat​\(B\)\\textrm\{Cat\}\(B\)is the set of bundlesQ≠BQ\\neq Bsuch that there exists a reactionr∈Rr\\in Rwhose net stoichiometry is supported inBBand in which species ofQQappear with zero net stoichiometry\.
2. 2\.For eachBB, writeEQE\_\{Q\}for the coordinate index set of a bundleQ∈Cat​\(B\)Q\\in\\textrm\{Cat\}\(B\), and define the*assignment set* 𝒜B:=∏Q∈Cat​\(B\)EQ\.\\mathcal\{A\}\_\{B\}\\;:=\\;\\prod\_\{Q\\in\\textrm\{Cat\}\(B\)\}E\_\{Q\}\.An elementa∈𝒜Ba\\in\\mathcal\{A\}\_\{B\}is a choicea​\(Q\)∈EQa\(Q\)\\in E\_\{Q\}of one coordinate for each catalytic neighborQ∈Cat​\(B\)Q\\in\\textrm\{Cat\}\(B\)\.

### 4\.2Rate Conditions \(W4–W6\)

LetBBbe a bundle with zero speciesXB,0X\_\{B,0\}and coordinate species\{XB,k\}k∈EB\\\{X\_\{B,k\}\\\}\_\{k\\in E\_\{B\}\}, and letCat​\(B\)\\textrm\{Cat\}\(B\)and𝒜B\\mathcal\{A\}\_\{B\}be as in Definition[4\.2](https://arxiv.org/html/2606.27737#S4.Thmtheorem2)\. For eachk∈EBk\\in E\_\{B\}, letdd​t​\[XB,k\]\+\\frac\{d\}\{dt\}\[X\_\{B,k\}\]^\{\+\}denote the total*positive*contribution todd​t​\[XB,k\]\\frac\{d\}\{dt\}\[X\_\{B,k\}\]coming from reactions whose net stoichiometry restricted toBBisXB,0→XB,kX\_\{B,0\}\\to X\_\{B,k\}\.

#### \(W4\) Separability of production rates\.

We say thatΓ\\Gammasatisfies \(W4\) if for every bundleBBand everyk∈EBk\\in E\_\{B\}, the functiondd​t​\[XB,k\]\+\\frac\{d\}\{dt\}\[X\_\{B,k\}\]^\{\+\}is a polynomial in the concentrations of the catalytic neighbor coordinates, with*at most one*coordinate species from each catalytic neighbor bundle appearing in any monomial\. That is, there exist coefficientsCB​\(k;a\)≥0C\_\{B\}\(k;a\)\\geq 0indexed byk∈EBk\\in E\_\{B\}and assignmentsa∈𝒜Ba\\in\\mathcal\{A\}\_\{B\}such that

dd​t​\[XB,k\]\+=\[XB,0\]​∑a∈𝒜BCB​\(k;a\)​∏Q∈Cat​\(B\)\[XQ,a​\(Q\)\]\.\\frac\{d\}\{dt\}\[X\_\{B,k\}\]^\{\+\}\\;=\\;\[X\_\{B,0\}\]\\sum\_\{a\\in\\mathcal\{A\}\_\{B\}\}C\_\{B\}\(k;a\)\\prod\_\{Q\\in\\textrm\{Cat\}\(B\)\}\[X\_\{Q,a\(Q\)\}\]\.
###### Definition 4\.3\(Sum\-like vs product\-like production\)\.

Assume \(W1\)–\(W4\) and letCB​\(k;a\)C\_\{B\}\(k;a\)be the assignment table of bundleBB\.

1. 1\.We sayBBis*weakly product\-like*if for everyk∈EBk\\in E\_\{B\}there exists at most one assignmenta\(k\)∈𝒜Ba^\{\(k\)\}\\in\\mathcal\{A\}\_\{B\}such thatCB​\(k;a\(k\)\)\>0C\_\{B\}\(k;a^\{\(k\)\}\)\>0, andCB​\(k;a\)=0C\_\{B\}\(k;a\)=0for alla≠a\(k\)a\\neq a^\{\(k\)\}\.
2. 2\.We sayBBis*sum\-like*if there exists somek∈EBk\\in E\_\{B\}and two distinct assignmentsa≠a′a\\neq a^\{\\prime\}in𝒜B\\mathcal\{A\}\_\{B\}such thatCB​\(k;a\)\>0C\_\{B\}\(k;a\)\>0andCB​\(k;a′\)\>0C\_\{B\}\(k;a^\{\\prime\}\)\>0\.

###### Definition 4\.4\(Assignment table\)\.

SupposeΓ\\Gammais a CRN satisfying \(W1\)–\(W4\)\. For each bundleBBandk∈EBk\\in E\_\{B\}, write the positive production partdd​t​\[XB,k\]\+\\frac\{d\}\{dt\}\[X\_\{B,k\}\]^\{\+\}as a polynomial in the catalytic coordinates\{\[XQ,i\]\}Q∈Cat​\(B\),i∈EQ\\\{\[X\_\{Q,i\}\]\\\}\_\{Q\\in\\textrm\{Cat\}\(B\),\\,i\\in E\_\{Q\}\}\. For each assignmenta∈𝒜Ba\\in\\mathcal\{A\}\_\{B\}letCB​\(k;a\)C\_\{B\}\(k;a\)be the coefficient of the monomial\[XB,0\]​∏Q∈Cat​\(B\)\[XQ,a​\(Q\)\]\.\[X\_\{B,0\}\]\\prod\_\{Q\\in\\textrm\{Cat\}\(B\)\}\[X\_\{Q,a\(Q\)\}\]\.

#### \(W5\) Coherent production for product\-like bundles

LetPPbe a bundle withCat​\(P\)≠∅\\textrm\{Cat\}\(P\)\\neq\\varnothingand assignment tableCP​\(k;a\)C\_\{P\}\(k;a\)from \(W4\)\. We say thatPPis*product\-like*if the following conditions hold:

1. 1\.\(Unique supporting assignment\)For eachk∈EPk\\in E\_\{P\}, there existsexactly oneassignmenta\(k\)∈𝒜Pa^\{\(k\)\}\\in\\mathcal\{A\}\_\{P\}such thatCP​\(k;a\(k\)\)\>0C\_\{P\}\(k;a^\{\(k\)\}\)\>0, andCP​\(k;a\)=0C\_\{P\}\(k;a\)=0for alla≠a\(k\)a\\neq a^\{\(k\)\}\.
2. 2\.\(Diagonal structure via bijections\)For each catalytic neighborQ∈Cat​\(P\)Q\\in\\textrm\{Cat\}\(P\), there exists a bijectionθP,Q:EP→EQ\\theta\_\{P,Q\}:E\_\{P\}\\to E\_\{Q\}such that for allk∈EPk\\in E\_\{P\}: a\(k\)​\(Q\)=θP,Q​\(k\)\.a^\{\(k\)\}\(Q\)=\\theta\_\{P,Q\}\(k\)\.

#### \(W6\) Factor structure for sum\-like bundles

Assume there exists a CRN satisfying \(W1\)–\(W5\)\. Condition \(W6\) requires that the sum\-like bundles ofΓ\\Gammacan be partitioned into*factor classes*, each equipped with auxiliary data satisfying the following properties\.

There exists a partition of the set of sum\-like bundles into disjoint classes\{J1,J2,…\}\\\{J\_\{1\},J\_\{2\},\\ldots\\\}\. For each factor classJJ, thefactor\-class dataconsists of:

- •a finite setVJV\_\{J\}of product\-like bundles \(the*incident product bundles*\),
- •for eachQ∈VJQ\\in V\_\{J\}, a bijectionιJ→Q:EQ→≅EBJ→Q\\iota\_\{J\\to Q\}:E\_\{Q\}\\xrightarrow\{\\cong\}E\_\{B\_\{J\\to Q\}\}\(the*state\-space identification*\),
- •a strictly positive functionψJ:∏Q∈VJEQ→\(0,∞\)\\psi\_\{J\}:\\prod\_\{Q\\in V\_\{J\}\}E\_\{Q\}\\to\(0,\\infty\)\(the*factor table*\),
- •positive scalars\{κJ→Q\>0\}Q∈VJ\\\{\\kappa\_\{J\\to Q\}\>0\\\}\_\{Q\\in V\_\{J\}\},

such that the following conditions hold:

\(W6\.1\)\(Missing\-one\-neighbor pattern and targets\)

We have\|J\|=\|VJ\|\|J\|=\|V\_\{J\}\|, and for eachQ∈VJQ\\in V\_\{J\}there exists a unique bundleBJ→Q∈JB\_\{J\\to Q\}\\in Jwith

Cat​\(BJ→Q\)=VJ∖\{Q\}\.\\textrm\{Cat\}\(B\_\{J\\to Q\}\)=V\_\{J\}\\setminus\\\{Q\\\}\.This establishes a bijection betweenJJandVJV\_\{J\}\. We define the*target*ofBJ→QB\_\{J\\to Q\}to be

Tgt​\(BJ→Q\):=Q,\\textrm\{Tgt\}\(B\_\{J\\to Q\}\):=Q,and refer toQQas the*target bundle*ofBJ→QB\_\{J\\to Q\}\.

\(W6\.2\)\(Target\-compatible state spaces\)

For eachQ∈VJQ\\in V\_\{J\}, the bijectionιJ→Q:EQ→EBJ→Q\\iota\_\{J\\to Q\}:E\_\{Q\}\\to E\_\{B\_\{J\\to Q\}\}witnesses that\|EQ\|=\|EBJ→Q\|\|E\_\{Q\}\|=\|E\_\{B\_\{J\\to Q\}\}\|\. Note that differentQ,Q′∈VJQ,Q^\{\\prime\}\\in V\_\{J\}may have\|EQ\|≠\|EQ′\|\|E\_\{Q\}\|\\neq\|E\_\{Q^\{\\prime\}\}\|, corresponding to variables of different cardinalities\.

\(W6\.3\)\(Clamped common table\)

For eachQ∈VJQ\\in V\_\{J\}, eachk∈EQk\\in E\_\{Q\}, and each assignmenta∈∏Q′∈VJ∖\{Q\}EQ′a\\in\\prod\_\{Q^\{\\prime\}\\in V\_\{J\}\\setminus\\\{Q\\\}\}E\_\{Q^\{\\prime\}\}, the assignment table ofB=BJ→QB=B\_\{J\\to Q\}satisfies

CB​\(ιJ→Q​\(k\);a\)=\\displaystyle C\_\{B\}\\bigl\(\\iota\_\{J\\to Q\}\(k\);\\,a\\bigr\)\\;=\\;κJ→Q⋅ψJ​\(xQ=k,xQ′=a​\(Q′\)​for​Q′≠Q\)\.\\displaystyle\\kappa\_\{J\\to Q\}\\cdot\\psi\_\{J\}\\bigl\(x\_\{Q\}=k,\\,x\_\{Q^\{\\prime\}\}=a\(Q^\{\\prime\}\)\\text\{ for \}Q^\{\\prime\}\\neq Q\\bigr\)\.Herek∈EQk\\in E\_\{Q\}is a state of the target product bundle,ιJ→Q​\(k\)∈EB\\iota\_\{J\\to Q\}\(k\)\\in E\_\{B\}is the corresponding state in the sum bundle, andaaassigns states to the remaining product bundles inVJ∖\{Q\}V\_\{J\}\\setminus\\\{Q\\\}\.

###### Definition 4\.5\(Sum bundles and product bundles\)\.

AssumeΓ\\Gammasatisfies \(W1\)–\(W6\)\.

1. 1\.A*sum bundle*is any sum\-like bundle, i\.e\., a bundleBJ→QB\_\{J\\to Q\}appearing in some factor class\(J,VJ\)\(J,V\_\{J\}\)\.
2. 2\.A*product bundle*is any bundlePPthat lies inVJV\_\{J\}for some factor class\(J,VJ\)\(J,V\_\{J\}\)—equivalently,PPis a catalytic neighbor of some sum bundle\.

###### Theorem 4\.7\(CRN→\\toFactor Graph Reconstruction\)\.

LetΓ=\(X,R,κ\)\\Gamma=\(X,R,\\kappa\)be a mass\-action CRN satisfying\(W1\)–\(W6\)\. Then one can reconstruct a factor graphFG​\(Γ\)=\(V,E,H\)\\mathrm\{FG\}\(\\Gamma\)=\(V,E,H\)as follows:

- •The factor nodes are the factor classesJJfrom\(W6\)\.
- •The variable nodes are the variable classes\[P\]\[P\]under∼v\\sim\_\{v\}\(Definition[5\.1](https://arxiv.org/html/2606.27737#S5.Thmtheorem1)\)\.
- •A variable class\[P\]\[P\]is incident to a factor classJJif and only if\[P\]∩VJ≠∅\[P\]\\cap V\_\{J\}\\neq\\varnothing\.
- •The state space of variable class\[P\]\[P\]isE\[P\]:=EPE\_\{\[P\]\}:=E\_\{P\}for any representativeP∈\[P\]P\\in\[P\]\.
- •The factor tableψJ:∏Q∈VJEQ→\(0,∞\)\\psi\_\{J\}:\\prod\_\{Q\\in V\_\{J\}\}E\_\{Q\}\\to\(0,\\infty\)from\(W6\)serves as the potential for factor nodeJJ\.

Moreover, the incidence relation is well\-defined: distinct elements ofVJV\_\{J\}belong to distinct variable classes, establishing a bijection betweenVJV\_\{J\}and the variable classes incident toJJ\. Each factor tableψJ\\psi\_\{J\}is determined byΓ\\Gammaup to multiplication by a positive constant\.

## 5Reconstruction of the Belief Propagation Algorithm

### 5\.1Equivalence relations for factors and variables

###### Definition 5\.1\(Variable equivalence on product bundles\)\.

Assume \(W1\)–\(W6\)\. Let𝒫\\mathcal\{P\}be the set of product bundles and𝒮\\mathcal\{S\}be the set of sum bundles\.

Define a binary relation∼\\simon𝒫\\mathcal\{P\}by: forP,P′∈𝒫P,P^\{\\prime\}\\in\\mathcal\{P\}, P∼P′P\\sim P^\{\\prime\}if and only if:

∃B∈𝒮​such that:\\displaystyle\\exists\\,B\\in\\mathcal\{S\}\\text\{ such that: \}\[Tgt​\(B\)=P​and​B∈Cat​\(P′\)\]\\displaystyle\\Bigl\[\\textrm\{Tgt\}\(B\)=P\\text\{ and \}B\\in\\textrm\{Cat\}\(P^\{\\prime\}\)\\Bigr\]or\[Tgt​\(B\)=P′​and​B∈Cat​\(P\)\]\.\\displaystyle\\Bigl\[\\textrm\{Tgt\}\(B\)=P^\{\\prime\}\\text\{ and \}B\\in\\textrm\{Cat\}\(P\)\\Bigr\]\.
Let∼v\\sim\_\{v\}be the equivalence relation on𝒫\\mathcal\{P\}generated by∼\\sim\(the reflexive, symmetric and transitive closure\)\.

The equivalence classes of𝒫\\mathcal\{P\}under∼v\\sim\_\{v\}are called*variable classes*\.

###### Lemma 5\.2\.

The relation∼v\\sim\_\{v\}of Definition[5\.1](https://arxiv.org/html/2606.27737#S5.Thmtheorem1)is an equivalence relation on𝒫\\mathcal\{P\}\.

###### Lemma 5\.3\.

IfP∼vP′P\\sim\_\{v\}P^\{\\prime\}, then\|EP\|=\|EP′\|\|E\_\{P\}\|=\|E\_\{P^\{\\prime\}\}\|\.

Conditions \(W1\)–\(W6\) characterize when a CRN encodes a factor graph and its factor tables\. To realize the Belief Propagation dynamics on the reconstructed factor graph, we impose one additional condition on the CRN:

#### \(R1\) Bundlewise recycling and uniform product production\.

For each bundleBBwith zero speciesXB,0X\_\{B,0\}and coordinates\{XB,k\}k∈EB\\\{X\_\{B,k\}\\\}\_\{k\\in E\_\{B\}\}there exists a constantρB\>0\\rho\_\{B\}\>0such that, for everyk∈EBk\\in E\_\{B\}, the CRN contains an internal \(non\-catalyzed\) recycling reaction

XB,k→ρBXB,0,X\_\{B,k\}\\xrightarrow\{\\ \\rho\_\{B\}\\ \}X\_\{B,0\},and these are the only internal reactions whose net stoichiometry isXB,k→XB,0X\_\{B,k\}\\to X\_\{B,0\}\.

Moreover, for each product bundlePP, there exists a constantκP\>0\\kappa\_\{P\}\>0such that for everyk∈EPk\\in E\_\{P\},

CP​\(k;a\(k\)\)=κP,C\_\{P\}\(k;a^\{\(k\)\}\)=\\kappa\_\{P\},wherea\(k\)a^\{\(k\)\}is the unique assignment supporting production ofPkP\_\{k\}given by \(W5\)\. Equivalently, every producing reaction with internal net stoichiometryP0→PkP\_\{0\}\\to P\_\{k\}has rate constantκP\\kappa\_\{P\}, independent ofk∈EPk\\in E\_\{P\}\.

###### Theorem 5\.4\(BP Implementation\)\.

LetΓ=\(X,R,κ\)\\Gamma=\(X,R,\\kappa\)be a mass\-action CRN satisfying*\(W1\)–\(W6\)*and*\(R1\)*, and letFG​\(Γ\)\\mathrm\{FG\}\(\\Gamma\)be the factor graph with factor tables\{ψJ\}\\\{\\psi\_\{J\}\\\}reconstructed by Theorem[4\.7](https://arxiv.org/html/2606.27737#S4.Thmtheorem7)\. Then the positive steady states ofΓ\\Gammacorrespond bijectively to BP fixed points onFG​\(Γ\)\\mathrm\{FG\}\(\\Gamma\)\.

More precisely, at any positive steady state\. For each factor classJJand targetQ∈VJQ\\in V\_\{J\}, writeAQ=∏Q′∈VJ∖\{Q\}EQ′A\_\{Q\}=\\prod\_\{Q^\{\\prime\}\\in V\_\{J\}\\setminus\\\{Q\\\}\}E\_\{Q^\{\\prime\}\}for the set of state assignments to the remaining variables, and writeψJ​\(k,a\):=ψJ​\(xQ=k,xQ′=a​\(Q′\)​for all​Q′∈VJ∖\{Q\}\)\\psi\_\{J\}\(k,\\,a\):=\\psi\_\{J\}\\bigl\(x\_\{Q\}=k,\\,x\_\{Q^\{\\prime\}\}=a\(Q^\{\\prime\}\)\\text\{ for all \}Q^\{\\prime\}\\in V\_\{J\}\\setminus\\\{Q\\\}\\bigr\)for the corresponding factor\-table entry\.

1. 1\.For each sum bundleB=BJ→QB=B\_\{J\\to Q\}associated to factor classJJand target product bundleQ∈VJQ\\in V\_\{J\}, the steady\-state concentrations satisfy ρB\[B0\]​\[Bk\]=κJ→Q​∑a∈AQψJ​\(k,a\)​∏Q′∈VJ∖\{Q\}\[Qa​\(Q′\)′\]\\displaystyle\\frac\{\\rho\_\{B\}\}\{\[B\_\{0\}\]\}\[B\_\{k\}\]=\\kappa\_\{J\\to Q\}\\sum\_\{a\\in A\_\{Q\}\}\\psi\_\{J\}\(k,\\,a\)\\prod\_\{Q^\{\\prime\}\\in V\_\{J\}\\setminus\\\{Q\\\}\}\[Q^\{\\prime\}\_\{a\(Q^\{\\prime\}\)\}\]\(1\)for allk∈EQk\\in E\_\{Q\}\.
2. 2\.For each product bundlePP, the steady\-state concentrations satisfy ρPκP​\[P0\]​\[Pk\]=∏B∈Cat​\(P\)\[Bk\]\\frac\{\\rho\_\{P\}\}\{\\kappa\_\{P\}\[P\_\{0\}\]\}\[P\_\{k\}\]=\\prod\_\{B\\in\\textrm\{Cat\}\(P\)\}\[B\_\{k\}\]\(2\)for allk∈EPk\\in E\_\{P\}\.
3. 3\.These equations are precisely the sum\-product BP fixed\-point equations onFG​\(Γ\)\\mathrm\{FG\}\(\\Gamma\), with the factorsρB/\[B0\]\\rho\_\{B\}/\[B\_\{0\}\]andρP/\(κP​\[P0\]\)\\rho\_\{P\}/\(\\kappa\_\{P\}\[P\_\{0\}\]\)serving as the per\-message normalization constants\.

## 6Categories and the Compilation Functor

### 6\.1Source Category𝐅𝐆𝐫𝐚𝐩𝐡𝐑𝐞𝐭\\mathbf\{FGraphRet\}

Objects are pairs\(A,H\)\(A,H\), whereAAis the poset associated to a factor graph andHHis the family of factor Hamiltonians\. Morphisms are finite composites of admissible linear and colinear retractions in the sense of Sergeant–Perthuis–Boitel: these are order\-preserving deformation retractions ofAAwhich act onHHby:

- •Deleting a variableiiand updating its incident factor by marginalizing out the corresponding coordinate inHH\(linear point\);
- •Deleting a factorjjand incorporating its potential into neighboring factors \(colinear point\)\.

For the explicit formulas onHH, see Propositions 5 and 6 of\(Sergeant\-Perthuis and Boitel,[2025](https://arxiv.org/html/2606.27737#bib.bib4)\)\. The explicit factor table updates \(i\.e\. CRN coefficients\) are provided in the Appendix[D\.1](https://arxiv.org/html/2606.27737#A4.SS1)

### 6\.2Target Category𝐂𝐑𝐍\\mathbf\{CRN\}

Objects of𝐂𝐑𝐍\\mathbf\{CRN\}are mass\-action reaction networksΓ=\(X,R,κ\)\\Gamma=\(X,R,\\kappa\)satisfying the recognition conditions \(W1\)–\(W6\), whereXXis the finite species set \(partitioned into bundles\),RRis the finite set of reactions, andκ∈\(0,∞\)R\\kappa\\in\(0,\\infty\)^\{R\}is the vector of rate constants\.

Morphisms in𝐂𝐑𝐍\\mathbf\{CRN\}are finite composites of elementary CRN reductions\. An*elementary reduction*is a triple

\(φ0,φ1,ψ\):\(X,R,κ\)⟶\(X′,R′,κ′\)\(\\varphi\_\{0\},\\varphi\_\{1\},\\psi\):\(X,R,\\kappa\)\\longrightarrow\(X^\{\\prime\},R^\{\\prime\},\\kappa^\{\\prime\}\)with:

- •φ0:X→X′\\varphi\_\{0\}:X\\to X^\{\\prime\}obtained by deleting a union of whole bundles, so thatX′=X∖⋃B∈ℬdelSBX^\{\\prime\}=X\\setminus\\bigcup\_\{B\\in\\mathcal\{B\}\_\{\\mathrm\{del\}\}\}S\_\{B\};
- •φ1:R→R′\\varphi\_\{1\}:R\\to R^\{\\prime\}the induced map on reactions, removing exactly those reactions that involve deleted species;
- •ψ:\(0,∞\)R→\(0,∞\)R′\\psi:\(0,\\infty\)^\{R\}\\to\(0,\\infty\)^\{R^\{\\prime\}\}a map on rate vectors, producingκ′=ψ​\(κ\)\\kappa^\{\\prime\}=\\psi\(\\kappa\)for the reduced network\(X′,R′\)\(X^\{\\prime\},R^\{\\prime\}\)\.

A general morphism is a finite composite of such reductions, with composition defined componentwise:

\(φ0′,φ1′,ψ′\)∘\(φ0,φ1,ψ\):=\(φ0′∘φ0,φ1′∘φ1,ψ′∘ψ\),\(\\varphi^\{\\prime\}\_\{0\},\\varphi^\{\\prime\}\_\{1\},\\psi^\{\\prime\}\)\\circ\(\\varphi\_\{0\},\\varphi\_\{1\},\\psi\):=\(\\varphi^\{\\prime\}\_\{0\}\\circ\\varphi\_\{0\},\\varphi^\{\\prime\}\_\{1\}\\circ\\varphi\_\{1\},\\psi^\{\\prime\}\\circ\\psi\),and identity morphisms given by\(idX,idR,id\)\(\\mathrm\{id\}\_\{X\},\\mathrm\{id\}\_\{R\},\\mathrm\{id\}\)\.

### 6\.3The Compilation Functor𝒩\\mathcal\{N\}

We now package the Napp compilation into a functor

𝒩:𝐅𝐆𝐫𝐚𝐩𝐡𝐑𝐞𝐭⟶𝐂𝐑𝐍\.\\mathcal\{N\}:\\mathbf\{FGraphRet\}\\longrightarrow\\mathbf\{CRN\}\.
#### Action on objects\.

Let\(A,H\)\(A,H\)be a factor graph in the sense of Sergeant–Perthuis–Boitel, with HamiltoniansHcH\_\{c\}and factor tablesψc∝e−Hc\\psi\_\{c\}\\propto e^\{\-H\_\{c\}\}\. The compiled CRN is

𝒩​\(A,H\)=\(X,R,κ\),\\mathcal\{N\}\(A,H\)=\(X,R,\\kappa\),constructed exactly as in Definition[D\.1](https://arxiv.org/html/2606.27737#A4.Thmtheorem1)\.

#### Action on morphisms\.

Let

r:\(A,H\)⟶\(A′,H′\)r:\(A,H\)\\longrightarrow\(A^\{\\prime\},H^\{\\prime\}\)be a morphism in𝐅𝐆𝐫𝐚𝐩𝐡𝐑𝐞𝐭\\mathbf\{FGraphRet\}, i\.e\., a finite composite of linear and colinear retractions in the sense of Sergeant–Perthuis–Boitel\. Writeψc∝e−Hc\\psi\_\{c\}\\propto e^\{\-H\_\{c\}\}andψc′∝e−Hc′\\psi^\{\\prime\}\_\{c\}\\propto e^\{\-H^\{\\prime\}\_\{c\}\}for the initial and updated factor tables\.

On CRNs, we define

𝒩​\(r\)=\(φ0,φ1,ψ\):𝒩​\(A,H\)→𝒩​\(A′,H′\),\\mathcal\{N\}\(r\)=\(\\varphi\_\{0\},\\varphi\_\{1\},\\psi\):\\mathcal\{N\}\(A,H\)\\to\\mathcal\{N\}\(A^\{\\prime\},H^\{\\prime\}\),as follows:

- •φ0:X→X′\\varphi\_\{0\}:X\\to X^\{\\prime\}deletes exactly the bundles corresponding to edges and variables removed byrr\(linear or colinear point\), and is the identity on all remaining species\. At the level of bundles,φ0\\varphi\_\{0\}is just restriction of the Napp compilation to the surviving part of the factor graph\.
- •φ1:R→R′\\varphi\_\{1\}:R\\to R^\{\\prime\}deletes all reactions whose reactants or products contain a deleted bundle; it is the identity on all other reactions \(recycling, product, marginal, and surviving sum reactions\)\.
- •ψ:κ↦κ′\\psi:\\kappa\\mapsto\\kappa^\{\\prime\}acts only on the rate constants of surviving sum\-bundle reactions\. For each surviving factor nodejjand edgej→nj\\to n, we keep the same scalarcj→n\>0c\_\{j\\to n\}\>0and set λj→n′​\(𝐤j\)=cj→n⋅ψj′​\(𝐤j\),\\lambda^\{\\prime\}\_\{j\\to n\}\(\\mathbf\{k\}^\{j\}\)=c\_\{j\\to n\}\\cdot\\psi^\{\\prime\}\_\{j\}\(\\mathbf\{k\}^\{j\}\),whereψj′\\psi^\{\\prime\}\_\{j\}is the updated factor from the SP–B retraction\. Recycling and product rates are left unchanged: κr′=κr,κprod′=κprod\.\\kappa^\{\\prime\}\_\{r\}=\\kappa\_\{r\},\\quad\\kappa^\{\\prime\}\_\{\\mathrm\{prod\}\}=\\kappa\_\{\\mathrm\{prod\}\}\.

By construction,\(φ0,φ1,ψ\)\(\\varphi\_\{0\},\\varphi\_\{1\},\\psi\)is a morphism in𝐂𝐑𝐍\\mathbf\{CRN\}: it removes entire bundles and their reactions, and only rescales rate parameters on the surviving sum bundles, leaving the long\-time behavior of the surviving species compatible with the factor\-table update\.

###### Lemma 6\.1\(Recycling structure of𝒩​\(A,H\)\\mathcal\{N\}\(A,H\)\)\.

For any factor graph\(A,H\)\(A,H\), the compiled CRN𝒩​\(𝒜,H\)=\(X,R,κ\)\\mathcal\{N\}\(\\mathcal\{A\},H\)=\(X,R,\\kappa\)satisfies\(R1\):

- •each bundleBB\(sum, product, or marginal\) contains a zero speciesB0B\_\{0\}and coordinate species\{Bk\}k∈EB\\\{B\_\{k\}\\\}\_\{k\\in E\_\{B\}\};
- •there is a single constantρB\>0\\rho\_\{B\}\>0such that, for everyk∈EBk\\in E\_\{B\}, the only internal reactions with net flowBk→B0B\_\{k\}\\to B\_\{0\}are the recycling reactions Bk→ρBB0,B\_\{k\}\\xrightarrow\{\\ \\rho\_\{B\}\\ \}B\_\{0\},which involve no catalysts from other bundles\.

In particular, the image of𝒩\\mathcal\{N\}lies in the full subcategory of CRNs satisfying\(W1\)–\(W6\)and\(R1\)\.

###### Proof of Lemma[6\.1](https://arxiv.org/html/2606.27737#S6.Thmtheorem1)\.

By construction of𝒩​\(A,H\)\\mathcal\{N\}\(A,H\), each bundleBBis equipped with recycling reactions

Bk→κrB0orBk→κprodB0,B\_\{k\}\\xrightarrow\{\\ \\kappa\_\{r\}\\ \}B\_\{0\}\\quad\\text\{or\}\\quad B\_\{k\}\\xrightarrow\{\\ \\kappa\_\{\\mathrm\{prod\}\}\\ \}B\_\{0\},depending on whetherBBis a message, belief, or marginal bundle, and no other reactions inRRhave net stoichiometry supported in\{B0,Bk\}\\\{B\_\{0\},B\_\{k\}\\\}\. SettingρB\\rho\_\{B\}equal to the appropriate global recycling constant yields exactly the structure required by \(R1\)\. ∎

###### Proposition 6\.2\(Functoriality\)\.

The assignment

\(A,H\)⟼𝒩​\(A,H\),r⟼𝒩​\(r\)\(A,H\)\\longmapsto\\mathcal\{N\}\(A,H\),\\quad r\\longmapsto\\mathcal\{N\}\(r\)defines a functor𝒩:𝐅𝐆𝐫𝐚𝐩𝐡𝐑𝐞𝐭→𝐂𝐑𝐍\\mathcal\{N\}:\\mathbf\{FGraphRet\}\\to\\mathbf\{CRN\}\. Moreover, by Lemma[6\.1](https://arxiv.org/html/2606.27737#S6.Thmtheorem1)and the description of𝒩​\(r\)\\mathcal\{N\}\(r\), the image of𝒩\\mathcal\{N\}lies in the full subcategory of CRNs satisfying*\(W1\)–\(W6\)*and*\(R1\)*\.

###### Lemma 6\.3\(Functoriality at BP Fixed Points\)\.

Let

r:\(A,H\)⟶\(A′,H′\)r:\(A,H\)\\longrightarrow\(A^\{\\prime\},H^\{\\prime\}\)be a morphism in𝐅𝐆𝐫𝐚𝐩𝐡𝐑𝐞𝐭\\mathbf\{FGraphRet\}, and let

Γ=𝒩​\(A,H\),Γ′=𝒩​\(A′,H′\)\\Gamma=\\mathcal\{N\}\(A,H\),\\quad\\Gamma^\{\\prime\}=\\mathcal\{N\}\(A^\{\\prime\},H^\{\\prime\}\)be the associated CRNs\. Then:

1. 1\.Γ\\GammaandΓ′\\Gamma^\{\\prime\}satisfy*\(W1\)–\(W6\)*and*\(R1\)*, and hence their positive steady states implement BP on\(A,H\)\(A,H\)and\(A′,H′\)\(A^\{\\prime\},H^\{\\prime\}\), respectively, by Theorem[5\.4](https://arxiv.org/html/2606.27737#S5.Thmtheorem4)\.
2. 2\.The CRN morphism 𝒩​\(r\)=\(φ0,φ1,ψ\):Γ→Γ′\\mathcal\{N\}\(r\)=\(\\varphi\_\{0\},\\varphi\_\{1\},\\psi\):\\Gamma\\to\\Gamma^\{\\prime\}deletes exactly the bundles corresponding to sites/regions removed byrr, leaves all recycling ratesρB\\rho\_\{B\}for surviving bundles unchanged, and updates sum\-bundle production coefficients by λj→n​\(𝐤j\)=cj→n​ψj​\(𝐤j\)⟼λj→n′​\(𝐤j\)=cj→n​ψj′​\(𝐤j\)\\lambda\_\{j\\to n\}\(\\mathbf\{k\}^\{j\}\)=c\_\{j\\to n\}\\psi\_\{j\}\(\\mathbf\{k\}^\{j\}\)\\longmapsto\\lambda^\{\\prime\}\_\{j\\to n\}\(\\mathbf\{k\}^\{j\}\)=c\_\{j\\to n\}\\psi^\{\\prime\}\_\{j\}\(\\mathbf\{k\}^\{j\}\)whereψj↦ψj′\\psi\_\{j\}\\mapsto\\psi^\{\\prime\}\_\{j\}is the factor\-table update induced byrrin the sense of Sergeant–Perthuis–Boitel\.
3. 3\.Ifπ\\piis a BP fixed point on\(A,H\)\(A,H\), then its restrictionπ′\\pi^\{\\prime\}to the surviving variables of\(A′,H′\)\(A^\{\\prime\},H^\{\\prime\}\)is a BP fixed point there, and any positive steady statesx∗x^\{\*\}ofΓ\\Gammaandx′⁣∗x^\{\\prime\*\}ofΓ′\\Gamma^\{\\prime\}whose bundle\-normalized concentrations correspond toπ\\piandπ′\\pi^\{\\prime\}are related by𝒩​\(r\)\\mathcal\{N\}\(r\)on the surviving bundles\.

## 7Simulations and Experiments

We benchmark the computational impact of SP–B retractions on CRNs obtained via the Napp–Adams compilation\. For each factor\-graph instanceℱ\\mathcal\{F\}, we compare*compile\-first*\(compileℱ\\mathcal\{F\}into𝒩​\(ℱ\)\\mathcal\{N\}\(\\mathcal\{F\}\)\) against*reduce\-then\-compile*\(apply the SP–B retraction to obtainr​\(ℱ\)r\(\\mathcal\{F\}\), then compile𝒩​\(r​\(ℱ\)\)\\mathcal\{N\}\(r\(\\mathcal\{F\}\)\)\)\. We report reductions in factor\-graph size, compiled CRN size, wall\-clock simulation time for integrating the induced mass\-action ODEs, and marginal agreement on the surviving variables\.

Unless otherwise stated, the experiments in the main text use binary state spaces,\|Ei\|=2\|E\_\{i\}\|=2, in order to isolate the effect of graph topology on reduction\. This is the relevant structural variable for SP–B retractions: reducibility depends on the poset structure of the factor graph, not on the cardinalities of the variable state spaces\. Larger state spaces increase the absolute number of compiled species and reactions, but do not change which variables or factors are removed by the retraction\. Appendix[B](https://arxiv.org/html/2606.27737#A2)reports cardinality sweeps and mixed\-cardinality examples confirming this behavior\.

#### Tree\-structured factor graphs \(exact BP regime\)\.

We generate binary\-tree factor graphs with depthsd∈\{3,4,5,6\}d\\in\\\{3,4,5,6\\\}, giving\|I\|∈\{7,15,31,63\}\|I\|\\in\\\{7,15,31,63\\\}variables\. On trees, BP computes exact marginals and SP–B reductions retract away essentially all non\-core acyclic structure, collapsing each instance to a trivial core\. Median variable reduction is≈95%\\approx 95\\%and median species reduction is≈97\.5%\\approx 97\.5\\%\(Table[1](https://arxiv.org/html/2606.27737#S2.T1)\), yielding dramatic simulation speedups \(median∼885×\\sim 885\\times\)\. Trees are faster than chains in our benchmarks \(chains:∼270×\\sim 270\\times\) not because they reduce more, but because unreduced tree instances are substantially more expensive: compiled CRN size grows more aggressively with depth for trees than for chains, so the speedup ratio is larger\.

#### Loopy BP: loopy cores with reducible tendrils\.

We generate loopy\-core instances with core sizec∈\{3,4,5\}c\\in\\\{3,4,5\\\}and attached tendrils of lengtht∈\{1,3,5,10\}t\\in\\\{1,3,5,10\\\}\. SP–B reductions remove the acyclic tendrils while preserving the irreducible loopy core, yielding partial but substantial compression \(median≈79%\\approx 79\\%variable and≈73%\\approx 73\\%species reduction\) and corresponding speedups \(median∼22×\\sim 22\\times\) \(Table[1](https://arxiv.org/html/2606.27737#S2.T1)\)\.

![Refer to caption](https://arxiv.org/html/2606.27737v1/er_phase_transition.png)Figure 3:Variable reduction onG​\(n,p\)G\(n,p\)as a function of edge probabilitypp\. Dashed verticals markp∗=ln⁡\(n\)/np^\{\*\}=\\ln\(n\)/n\. Reduction peaks below the connectivity threshold, where graphs contain small loopy cores with retractable tendrils, and falls to zero once the irreducible core dominates\.
#### Erdős–Rényi random graphs\.

To test whether the method depends on planted reducible structure, we also benchmark Erdős–Rényi graphsG​\(n,p\)G\(n,p\)withn∈\[20,100\]n\\in\[20,100\]and edge probabilities spanning sparse, near\-threshold, and dense regimes\. The relevant structural scale is the connectivity thresholdp∗=ln⁡\(n\)/np^\{\*\}=\\ln\(n\)/n\. Below and near this threshold, graphs contain many tree\-like components and tendrils attached to small loopy cores; above it, the irreducible core occupies an increasing fraction of the graph\.

Figure[3](https://arxiv.org/html/2606.27737#S7.F3)shows that variable reduction follows this core/tendril transition\. Reduction is small nearp=0p=0, rises in the sparse regime, peaks belowp∗p^\{\*\}, and then falls to zero as the graph becomes dense\. Species reduction shows the same pattern \(Figure[4](https://arxiv.org/html/2606.27737#S7.F4)\), exceeding60%60\\%in representative subcritical regimes and reaching about69%69\\%in the best median cells\. Median simulation speedups peak around16×16\\times\(Figure[5](https://arxiv.org/html/2606.27737#S7.F5)\), with larger individual gains, but runtime is more variable because it also depends on ODE stiffness and solver behavior\.

![Refer to caption](https://arxiv.org/html/2606.27737v1/er_species_heatmap.png)Figure 4:Median CRN species reduction onG​\(n,p\)G\(n,p\)instances\. Species reduction is concentrated in the sparse/subcritical wedge and vanishes in the dense regime, where the irreducible loopy core dominates\.The ER benchmark therefore supports the same interpretation as the structured families: SP–B reductions remove graph regions outside the irreducible loopy core\. In dense ER graphs, where the core dominates, the method correctly halts with little or no reduction rather than altering the inferential structure\.

All wall\-clock ER timings were measured on Auburn University’s Easley high\-performance computing cluster\. Each job used one node, one task, one CPU per task, 8GB memory, a 12\-hour wall\-time limit; each array task evaluated one\(n,p\)\(n,p\)cell with 10 random seeds\.

![Refer to caption](https://arxiv.org/html/2606.27737v1/er_speedup_heatmap.png)Figure 5:Median simulation speedup onG​\(n,p\)G\(n,p\)instances\. Speedup is largest in the sparse/subcritical regime where reductions remove many species\.
#### Correctness diagnostics\.

BP converged in all440440ER runs on both the original and the reduced networks\. We additionally track three marginal\-agreement diagnostics: BP\(original\) vs\. BP\(reduced\) on surviving variables \(the SP–B preservation test\), CRN\-derived marginals vs\. BP marginals on each network \(end\-to\-end validation of the compiled chemical implementation\), and CRN\(original\) vs\. CRN\(reduced\) on surviving species\.

For tree instances, BP is exact and the BP\(orig\) vs\. BP\(reduced\) comparison is a stringent correctness test; observed differences are dominated by numerical precision \. For loopy and ER instances, we interpret the comparison as a core fixed\-point preservation diagnostic\. The median BP\(original\) vs\. BP\(reduced\) discrepancy on ER instances is4×10−84\\times 10^\{\-8\}\. A small fraction of near\-critical runs \(∼1%\\sim 1\\%above0\.10\.1\) show larger discrepancies; these correspond to graphs near the connectivity threshold where loopy BP can have multiple fixed points and the original and reduced systems converge to different basins\. These gains reflect computational savings from retraction rather than changes in the inferred beliefs, with the caveat that loopy BP non\-uniqueness can select different fixed points across the two systems in the near\-critical regime\.

## Future Work

A natural next step is physical realization via DNA strand displacement, which by the universality result of\(Soloveichiket al\.,[2010](https://arxiv.org/html/2606.27737#bib.bib33)\)can implement any mass\-action CRN\. Our reduction pipeline therefore provides a direct path to smaller, more experimentally feasible DNA implementations of BP inference: reduce the factor graph, compile via Napp\-Adams, then implement the reduced CRN in DNA\. The species counts reported here suggest this could bring moderately complex inference tasks within reach of current DNA nanotechnology\. A longer\-horizon question is whether gene regulatory networks that perform approximate inference can be cast in Napp\-Adams form, which would make them amenable to the same reduction pipeline; this depends on whether their ODE structure can be shown to satisfy conditions \(W1\)–\(W6\), and we leave this as an open problem\.

## Impact Statement

We are not aware of any immediate negative societal impacts arising from this work\. Methods that make molecular computation more physically realizable could contribute to the design of programmable cells and synthetic biological systems\. The ethical implications of such downstream applications would need to be assessed as the technology matures\.

## Acknowledgments

This material is based upon research supported by the Chateaubriand Fellowship of the Office for Science & Technology of the Embassy of France in the United States as well as by NSF Grant DMS\-2246127\-PSC and DOE Grant DE\-SC0025649

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## Reduction of CRNs

A common class of exact reduction methods exploits stoichiometric structure to eliminate subnetworks while preserving steady\-state behavior of the unreduced portion\. In particular, Hirono et al\.\(Hironoet al\.,[2021](https://arxiv.org/html/2606.27737#bib.bib8)\)identify output\-complete subnetworksγ\\gammaand compute an*influence index*λ​\(γ\)\\lambda\(\\gamma\), an integer determined by its topology,χ​\(γ\)=\|Vγ\|−\|Eγ\|\\chi\(\\gamma\)=\|V\_\{\\gamma\}\|\-\|E\_\{\\gamma\}\|together with stoichiometric cycle and conservation subspaces\. Whenλ​\(γ\)=0\\lambda\(\\gamma\)=0, they prove a localization principle: steady\-state sensitivities outsideγ\\gammaare unaffected by perturbations internal toγ\\gamma, which justifies eliminatingγ\\gammavia a Schur complement reduction on the stoichiometry\. Related reduction strategies apply Schur complements to weighted graph Laplacians derived from CRNs\. For complex\-balanced mass\-action systems, Rao et al\.\(Raoet al\.,[2013](https://arxiv.org/html/2606.27737#bib.bib13)\)use the weighted Laplacian on the graph of complexes to delete complexes while retaining equilibrium structure, and later work\(Gasparyanet al\.,[2025](https://arxiv.org/html/2606.27737#bib.bib14)\)proposes a Schur complement on*species–reaction graphs*with an emphasis on approximation error in dynamical trajectories\. Broader families of model simplification and reduction such as conservation analysis, decomposition, timescale separation, and sensitivity\-based methods are surveyed in\(Snowdenet al\.,[2017](https://arxiv.org/html/2606.27737#bib.bib15)\)\.

However, these methods do not apply directly to CRNs arising from hidden Markov models, graphical models, and factor graphs\(Wiufet al\.,[2023](https://arxiv.org/html/2606.27737#bib.bib17); Napp and Adams,[2013](https://arxiv.org/html/2606.27737#bib.bib5); Colliauxet al\.,[2017](https://arxiv.org/html/2606.27737#bib.bib16)\)\. In these networks, key reactions are catalyzed, and the stoichiometric matrix alone fails to capture the underlying message\-passing structure\. We therefore exploit reduction techniques developed for graphical models that preserve collections of distributions corresponding to fixed points of the belief propagation algorithm\(Sergeant\-Perthuis and Boitel,[2025](https://arxiv.org/html/2606.27737#bib.bib4); Yedidiaet al\.,[2000](https://arxiv.org/html/2606.27737#bib.bib29)\)\. To this end, we propose a dictionary that maps CRNs to graphical models in a structure\-preserving manner\.

## Appendix AConceptual Overview and Scope

This section makes explicit the dictionary used throughout Sections 4–6\. Table[2](https://arxiv.org/html/2606.27737#A1.T2)summarizes the correspondence between factor\-graph concepts and their chemical realizations in a Napp–Adams compiled CRN; conditions \(W1\)–\(W6\) formalize the right column, and \(R1\) ensures the resulting steady states are precisely the BP fixed\-point equations after bundle normalization\. Table[3](https://arxiv.org/html/2606.27737#A1.T3)clarifies which classes of CRNs fall within and outside this framework\.

Table 2:Factor\-graph/CRN correspondence in a Napp–Adams compiled CRN\.Table 3:Scope of the recognition and reduction framework\.Entries marked “outside” indicate only that the relevant semantics differ from the BP message\-bundle structure captured by \(W1\)–\(W6\) and \(R1\), not that such CRNs are unreducible by other means\.

## Appendix BState\-Cardinality Sweeps

The main experiments use binary variables in order to isolate the role of graph topology in SP–B reduction\. Here we report additional experiments showing that the same reduction behavior persists for larger state spaces\. The reason is structural: SP–B reducibility is determined by the factor\-graph poset, not by the cardinality of the variable state spaces\. Increasing the alphabet size changes the absolute number of compiled CRN species and reactions, but it does not change which variables or factors are linear or colinear points\.

We tested cardinalitiesK∈\{2,3,4,5\}K\\in\\\{2,3,4,5\\\}on chains, depth\-4 trees, and loopy\-core graphs with tendrils\. We also tested a mixed\-cardinality chain\. For each instance, we apply the same reduce\-then\-compile pipeline as in Section[7](https://arxiv.org/html/2606.27737#S7), and we report variable reduction, species reduction, and maximum BP marginal discrepancy on the surviving variables\.

Table 4:State\-cardinality sweeps\. Reduction percentages remain stable across cardinalities because SP–B reducibility depends on graph topology rather than alphabet size\. HereKKdenotes the common state\-space cardinality of each variable, except in the mixed\-cardinality row\.For the 10\-variable chains, the compiled CRN sizes grow linearly withKKin this family: the unreduced networks have150,200,250,150,200,250,and300300species forK=2,3,4,5K=2,3,4,5, respectively, while the reduced networks have9,12,15,9,12,15,and1818species\. Thus the variable reduction is90\.0%90\.0\\%and the species reduction is94\.0%94\.0\\%in every case\. The BP discrepancy on the surviving variable remains negligible, decreasing from7\.93×10−107\.93\\times 10^\{\-10\}forK=2K=2to5\.55×10−175\.55\\times 10^\{\-17\}forK=5K=5\.

The tree and loopy\-core families show the same qualitative behavior\. Depth\-4 trees reduce from1515variables to11variable, with96\.6%96\.6\\%species reduction across all tested cardinalities\. Loopy\-core graphs with tendrils reduce from1616variables to44variables, with68\.2%68\.2\\%species reduction across all tested cardinalities\. In the loopy case, the reduction removes the tendrils while preserving the irreducible core, so increasing the state\-space cardinality changes the size of the compiled CRN but not the graph\-theoretic core\.

Finally, the mixed\-cardinality chain\[2,3,4,5,4,3,2\]\[2,3,4,5,4,3,2\]reduces from77variables to11variable, with94\.0%94\.0\\%species reduction and BP discrepancy5\.79×10−135\.79\\times 10^\{\-13\}\. This confirms that the reduction does not require all variables to have the same alphabet size\. The state spaces affect the sizes of the species bundles and factor tables, while reducibility is controlled by the underlying factor\-graph topology\.

## Appendix CTopological Background

###### Definition C\.1\(Alexandrov topology\)\.

Let\(X,≤\)\(X,\\leq\)be a poset\. The*Alexandrov topology*onXXis the topology whose open sets are the*up\-sets*:

U⊆X​is open⟺\(∀x∈U\)​\(∀y∈X\)​\(x≤y⇒y∈U\)\.U\\subseteq X\\text\{ is open \}\\quad\\Longleftrightarrow\\quad\(\\forall x\\in U\)\(\\forall y\\in X\)\\ \(x\\leq y\\Rightarrow y\\in U\)\.Equivalently, the closed sets are exactly the*down\-sets*:

C⊆X​is closed⟺\(∀x∈C\)​\(∀y∈X\)​\(y≤x⇒y∈C\)\.C\\subseteq X\\text\{ is closed \}\\quad\\Longleftrightarrow\\quad\(\\forall x\\in C\)\(\\forall y\\in X\)\\ \(y\\leq x\\Rightarrow y\\in C\)\.We writeX𝒜X\_\{\\mathcal\{A\}\}for the topological space obtained from a poset𝒜\\mathcal\{A\}equipped with its Alexandrov topology\.

### Homotopy for finite posets

We adopt the notion of homotopy for finite posets used in\(Sergeant\-Perthuis and Boitel,[2025](https://arxiv.org/html/2606.27737#bib.bib4)\)\. Informally, one views a finite poset𝒜\\mathcal\{A\}as an Alexandrov spaceX𝒜X\_\{\\mathcal\{A\}\}and uses continuous \(equivalently, order\-preserving\) maps to the target posetX𝒜×\[0,1\]→XℬX\_\{\\mathcal\{A\}\}\\times\[0,1\]\\to X\_\{\\mathcal\{B\}\}\.

###### Definition C\.3\(Deformation retracts of posets\)\.

Letℬ⊆𝒜\\mathcal\{B\}\\subseteq\\mathcal\{A\}be a subposet and leti:ℬ↪𝒜i:\\mathcal\{B\}\\hookrightarrow\\mathcal\{A\}be the inclusion\. A*deformation retract*of𝒜\\mathcal\{A\}ontoℬ\\mathcal\{B\}consists of an order\-preserving mapr:𝒜→ℬr:\\mathcal\{A\}\\to\\mathcal\{B\}such that

r∘i=idℬandi∘r≃id𝒜,r\\circ i=\\textrm\{id\}\_\{\\mathcal\{B\}\}\\qquad\\text\{and\}\\qquad i\\circ r\\simeq\\textrm\{id\}\_\{\\mathcal\{A\}\},where≃\\simeqdenotes homotopy in the sense of\(Sergeant\-Perthuis and Boitel,[2025](https://arxiv.org/html/2606.27737#bib.bib4)\)\. It is a*strong deformation retract*if, moreover, the homotopy fromi∘ri\\circ rtoid𝒜\\textrm\{id\}\_\{\\mathcal\{A\}\}fixesℬ\\mathcal\{B\}pointwise\.

### Linear/colinear points and cores

The reductions we use are based on deleting elements of a finite poset without changing its homotopy type\. The relevant notions are the following\.

###### Definition C\.5\(Linear and colinear points\)\.

Let𝒜\\mathcal\{A\}be a finite poset and leta∈𝒜a\\in\\mathcal\{A\}\.

\(Linear\)\.We sayaais*linear*if there exists an elementa↑∈𝒜a^\{\\uparrow\}\\in\\mathcal\{A\}witha<a↑a<a^\{\\uparrow\}such that every element aboveaais also abovea↑a^\{\\uparrow\}, i\.e\.

\(∀b∈𝒜\)​\(b≥a⇒b≥a↑\)\.\(\\forall b\\in\\mathcal\{A\}\)\\ \\bigl\(b\\geq a\\Rightarrow b\\geq a^\{\\uparrow\}\\bigr\)\.\(L\)
\(Colinear\)\.Dually, we sayaais*colinear*if there existsa↓∈𝒜a^\{\\downarrow\}\\in\\mathcal\{A\}witha↓<aa^\{\\downarrow\}<asuch that every element belowaais also belowa↓a^\{\\downarrow\}, i\.e\.

\(∀b∈𝒜\)​\(b≤a⇒b≤a↓\)\.\(\\forall b\\in\\mathcal\{A\}\)\\ \\bigl\(b\\leq a\\Rightarrow b\\leq a^\{\\downarrow\}\\bigr\)\.\(coL\)

###### Definition C\.7\(Core of a finite poset\(Stong,[1966](https://arxiv.org/html/2606.27737#bib.bib28)\)\)\.

The*core*of a finite poset𝒜\\mathcal\{A\}is a subposetℬ⊆𝒜\\mathcal\{B\}\\subseteq\\mathcal\{A\}such that:

1. 1\.ℬ\\mathcal\{B\}has no linear or colinear points \(computed withinℬ\\mathcal\{B\}\), and
2. 2\.ℬ\\mathcal\{B\}is a strong deformation retract of𝒜\\mathcal\{A\}\(Definition[C\.3](https://arxiv.org/html/2606.27737#A3.Thmtheorem3)\)\.

We denote a core of𝒜\\mathcal\{A\}byc​o​𝒜co\\mathcal\{A\}\.

###### Proposition C\.8\.

Every finite poset𝒜\\mathcal\{A\}admits a corec​o​𝒜co\\mathcal\{A\}\. Moreover, one can obtain a strong deformation retract𝒜→c​o​𝒜\\mathcal\{A\}\\to co\\mathcal\{A\}by iteratively applying up\- and down\-retractions that remove linear and colinear points\. Finally, two finite posets are homotopy equivalent if and only if their cores are isomorphic\.

###### Proof\.

This is Theorem 2 of\(Stong,[1966](https://arxiv.org/html/2606.27737#bib.bib28)\)\. ∎

## Appendix DProofs of propositions, lemmas, and theorems

###### Proof of Theorem[4\.7](https://arxiv.org/html/2606.27737#S4.Thmtheorem7)\.

The factor classesJJfrom \(W6\) and variable classes\[P\]\[P\]from Definition[5\.1](https://arxiv.org/html/2606.27737#S5.Thmtheorem1)immediately provide the node sets\. By Lemma[5\.3](https://arxiv.org/html/2606.27737#S5.Thmtheorem3), ifP∼vP′P\\sim\_\{v\}P^\{\\prime\}then\|EP\|=\|EP′\|\|E\_\{P\}\|=\|E\_\{P^\{\\prime\}\}\|, so the state spaceE\[P\]E\_\{\[P\]\}is well\-defined up to the canonical bijections induced by \(W5\) and \(W6\)\.

We wish to show that the incidence relation is well defined, which amounts to showing that for each factor classJJand distinctQ,Q′∈VJQ,Q^\{\\prime\}\\in V\_\{J\}, the variable classes\[Q\]\[Q\]and\[Q′\]\[Q^\{\\prime\}\]are distinct\.

Suppose for contradiction thatQ∼vQ′Q\\sim\_\{v\}Q^\{\\prime\}for someQ≠Q′Q\\neq Q^\{\\prime\}inVJV\_\{J\}\. Then there exists a chainQ=P0∼P1∼⋯∼Pm=Q′Q=P\_\{0\}\\sim P\_\{1\}\\sim\\cdots\\sim P\_\{m\}=Q^\{\\prime\}of direct cross\-links\. Consider the first link:Q∼P1Q\\sim P\_\{1\}means there exists a sum bundleBBwith eitherTgt​\(B\)=Q\\textrm\{Tgt\}\(B\)=QandB∈Cat​\(P1\)B\\in\\textrm\{Cat\}\(P\_\{1\}\), orTgt​\(B\)=P1\\textrm\{Tgt\}\(B\)=P\_\{1\}andB∈Cat​\(Q\)B\\in\\textrm\{Cat\}\(Q\)\.

In the first case, letJ′J^\{\\prime\}be the factor class containingBB, soB=BJ′→QB=B\_\{J^\{\\prime\}\\to Q\}and henceQ∈VJ′Q\\in V\_\{J^\{\\prime\}\}\. SinceB∈Cat​\(P1\)B\\in\\textrm\{Cat\}\(P\_\{1\}\)andP1P\_\{1\}is product\-like, condition \(W5\) tells us thatBBcatalyzes production ofP1P\_\{1\}\. By the structure of \(W6\), the sum bundleBBtargetsQQand hasCat​\(B\)=VJ′∖\{Q\}\\textrm\{Cat\}\(B\)=V\_\{J^\{\\prime\}\}\\setminus\\\{Q\\\}\. ForBBto appear inCat​\(P1\)\\textrm\{Cat\}\(P\_\{1\}\), the product bundleP1P\_\{1\}must be receiving messages from the same variable asQQbut through a different factor thanJ′J^\{\\prime\}\. In particular,P1∉VJ′P\_\{1\}\\notin V\_\{J^\{\\prime\}\}\.

The second case is symmetric: ifTgt​\(B\)=P1\\textrm\{Tgt\}\(B\)=P\_\{1\}andB∈Cat​\(Q\)B\\in\\textrm\{Cat\}\(Q\), thenP1P\_\{1\}is the target of some sum bundle whileQQis catalyzed by that bundle, again placingQQandP1P\_\{1\}as product bundles for the same variable going to different factors\.

In either case, each cross\-linkPi∼Pi\+1P\_\{i\}\\sim P\_\{i\+1\}in the chain connects product bundles that represent the same variable but are incident to different factor classes\. The key observation is that the relation∼\\sim*cannot*connect two product bundles that are both in the sameVJV\_\{J\}: ifQ,Q′∈VJQ,Q^\{\\prime\}\\in V\_\{J\}withQ≠Q′Q\\neq Q^\{\\prime\}, thenQQandQ′Q^\{\\prime\}are targets of distinct sum bundles inJJ, namelyBJ→QB\_\{J\\to Q\}andBJ→Q′B\_\{J\\to Q^\{\\prime\}\}\. By \(W6\.1\),Cat​\(BJ→Q\)=VJ∖\{Q\}\\textrm\{Cat\}\(B\_\{J\\to Q\}\)=V\_\{J\}\\setminus\\\{Q\\\}andCat​\(BJ→Q′\)=VJ∖\{Q′\}\\textrm\{Cat\}\(B\_\{J\\to Q^\{\\prime\}\}\)=V\_\{J\}\\setminus\\\{Q^\{\\prime\}\\\}\.

ForQ∼Q′Q\\sim Q^\{\\prime\}directly, we would need a sum bundleBBwithTgt​\(B\)=Q\\textrm\{Tgt\}\(B\)=QandB∈Cat​\(Q′\)B\\in\\textrm\{Cat\}\(Q^\{\\prime\}\)\(or vice versa\)\. IfB∈JB\\in J, thenB=BJ→QB=B\_\{J\\to Q\}, andB∈Cat​\(Q′\)B\\in\\textrm\{Cat\}\(Q^\{\\prime\}\)would requireQ′Q^\{\\prime\}to be a product bundle catalyzed by this sum bundle\. ButCat​\(BJ→Q\)=VJ∖\{Q\}\\textrm\{Cat\}\(B\_\{J\\to Q\}\)=V\_\{J\}\\setminus\\\{Q\\\}consists of product bundles inVJV\_\{J\}, not the catalytic neighbors of those bundles\. The setCat​\(Q′\)\\textrm\{Cat\}\(Q^\{\\prime\}\)consists of sum bundles that catalyzeQ′Q^\{\\prime\}, which by the Napp–Adams structure are sum bundles targeting the same variable asQ′Q^\{\\prime\}from other factors\. SinceBJ→QB\_\{J\\to Q\}targetsQQ\(a different element ofVJV\_\{J\}, hence a different variable\), we haveBJ→Q∉Cat​\(Q′\)B\_\{J\\to Q\}\\notin\\textrm\{Cat\}\(Q^\{\\prime\}\)\.

If insteadB∉JB\\notin J, sayB∈J′B\\in J^\{\\prime\}for some other factor class, thenTgt​\(B\)=Q\\textrm\{Tgt\}\(B\)=QimpliesQ∈VJ′Q\\in V\_\{J^\{\\prime\}\}\. ForB∈Cat​\(Q′\)B\\in\\textrm\{Cat\}\(Q^\{\\prime\}\), the sum bundleBBmust be one of the catalytic inputs toQ′Q^\{\\prime\}\. These are sum bundles for the variable corresponding toQ′Q^\{\\prime\}, coming from factors other than the oneQ′Q^\{\\prime\}points to\. SinceBBtargetsQQandQ∈VJQ\\in V\_\{J\}, the variable ofBBis the variable ofQQ\. ForB∈Cat​\(Q′\)B\\in\\textrm\{Cat\}\(Q^\{\\prime\}\), the variable ofQ′Q^\{\\prime\}must equal the variable ofQQ\. ButQQandQ′Q^\{\\prime\}are distinct elements ofVJV\_\{J\}, which by \(W6\.1\) correspond to distinct targets of sum bundles inJJ—hence to distinct variables incident to factorJJ\. This is a contradiction\.

ThereforeQ≁Q′Q\\not\\sim Q^\{\\prime\}for distinctQ,Q′∈VJQ,Q^\{\\prime\}\\in V\_\{J\}, and since∼v\\sim\_\{v\}is generated by∼\\sim, we haveQ≁vQ′Q\\not\\sim\_\{v\}Q^\{\\prime\}\. This shows\[Q\]≠\[Q′\]\[Q\]\\neq\[Q^\{\\prime\}\], and moreover\[Q\]∩VJ=\{Q\}\[Q\]\\cap V\_\{J\}=\\\{Q\\\}\(any other element of\[Q\]\[Q\]inVJV\_\{J\}would have to equalQQby the above\)\.

Consequently, the mapQ↦\[Q\]Q\\mapsto\[Q\]is a bijection fromVJV\_\{J\}to the set of variable classes incident toJJ\. The incidence relation is well\-defined, and the factor tableψJ:∏Q∈VJEQ→\(0,∞\)\\psi\_\{J\}:\\prod\_\{Q\\in V\_\{J\}\}E\_\{Q\}\\to\(0,\\infty\)from \(W6\.3\) can be viewed as a function on∏\[Q\]:Q∈VJE\[Q\]\\prod\_\{\[Q\]:Q\\in V\_\{J\}\}E\_\{\[Q\]\}via the canonical identificationsEQ≅E\[Q\]E\_\{Q\}\\cong E\_\{\[Q\]\}\.

Finally, uniqueness ofψJ\\psi\_\{J\}up to positive scalar follows from \(W6\.3\): the coefficient arraysCBJ→QC\_\{B\_\{J\\to Q\}\}are determined byΓ\\Gamma, and any two tablesψJ,ψJ′\\psi\_\{J\},\\psi^\{\\prime\}\_\{J\}satisfying \(W6\.3\) must have constant ratioψJ′/ψJ=λ\\psi^\{\\prime\}\_\{J\}/\\psi\_\{J\}=\\lambdafor someλ\>0\\lambda\>0, with the scalarsκJ→Q\\kappa\_\{J\\to Q\}adjusting accordingly\. ∎

###### Proof of Lemma[5\.2](https://arxiv.org/html/2606.27737#S5.Thmtheorem2)\.

The reflexive\-symmetric\-transitive closure of any binary relation is an equivalence relation\. ∎

###### Proof of Lemma[5\.3](https://arxiv.org/html/2606.27737#S5.Thmtheorem3)\.

It suffices to showP∼P′P\\sim P^\{\\prime\}implies\|EP\|=\|EP′\|\|E\_\{P\}\|=\|E\_\{P^\{\\prime\}\}\|; the general case follows by induction along∼\\sim\-chains\.

SupposeP∼P′P\\sim P^\{\\prime\}via sum bundleBBwithTgt​\(B\)=P\\textrm\{Tgt\}\(B\)=PandB∈Cat​\(P′\)B\\in\\textrm\{Cat\}\(P^\{\\prime\}\)\. LetJJbe the factor class containingBB, soB=BJ→PB=B\_\{J\\to P\}andP∈VJP\\in V\_\{J\}\.

- •By \(W6\.2\),ιJ→P:EP→≅EB\\iota\_\{J\\to P\}:E\_\{P\}\\xrightarrow\{\\cong\}E\_\{B\}, so\|EP\|=\|EB\|\|E\_\{P\}\|=\|E\_\{B\}\|\.
- •SinceB∈Cat​\(P′\)B\\in\\textrm\{Cat\}\(P^\{\\prime\}\)andP′P^\{\\prime\}is product\-like, \(W5\.2\) gives a bijectionθP′,B:EP′→EB\\theta\_\{P^\{\\prime\},B\}:E\_\{P^\{\\prime\}\}\\to E\_\{B\}, so\|EP′\|=\|EB\|\|E\_\{P^\{\\prime\}\}\|=\|E\_\{B\}\|\.

Hence\|EP\|=\|EB\|=\|EP′\|\|E\_\{P\}\|=\|E\_\{B\}\|=\|E\_\{P^\{\\prime\}\}\|\. ∎

###### Proof of Theorem[5\.4](https://arxiv.org/html/2606.27737#S5.Thmtheorem4)\.

Fix a factor class\(J,VJ\)\(J,V\_\{J\}\)and a product bundleQ∈VJQ\\in V\_\{J\}\. Consider the associated sum bundleB:=BJ→Q∈JB:=B\_\{J\\to Q\}\\in Jfrom \(W6\)\. By conditions \(W1\)–\(W3\), every reaction with nonzero net stoichiometry inBBhas internal net stoichiometry of the formB0↔BkB\_\{0\}\\leftrightarrow B\_\{k\}for a uniquek∈EBk\\in E\_\{B\}, and no other bundle has nonzero net stoichiometry in that reaction\.

For eachk∈EBk\\in E\_\{B\}, letdd​t​\[Bk\]\+\\frac\{d\}\{dt\}\[B\_\{k\}\]^\{\+\}denote the total positive contribution todd​t​\[Bk\]\\frac\{d\}\{dt\}\[B\_\{k\}\]coming from reactions whose internal net stoichiometry isB0→BkB\_\{0\}\\to B\_\{k\}\. By \(W4\), we may expand:

dd​t​\[Bk\]\+=\[B0\]​∑a∈𝒜BCB​\(k;a\)​∏Q′∈Cat​\(B\)\[XQ′,a​\(Q′\)\]\.\\frac\{d\}\{dt\}\[B\_\{k\}\]^\{\+\}=\[B\_\{0\}\]\\sum\_\{a\\in\\mathcal\{A\}\_\{B\}\}C\_\{B\}\(k;a\)\\prod\_\{Q^\{\\prime\}\\in\\textrm\{Cat\}\(B\)\}\[X\_\{Q^\{\\prime\},a\(Q^\{\\prime\}\)\}\]\.
By the missing\-one\-neighbor property of \(W6\.1\), we haveCat​\(B\)=VJ∖\{Q\}\\textrm\{Cat\}\(B\)=V\_\{J\}\\setminus\\\{Q\\\}\. Hence each assignmentj∈JB=∏Q′∈VJ∖\{Q\}EQ′j\\in J\_\{B\}=\\prod\_\{Q^\{\\prime\}\\in V\_\{J\}\\setminus\\\{Q\\\}\}E\_\{Q^\{\\prime\}\}is equivalently a choice of states\(xQ′\)Q′≠Q\(x\_\{Q^\{\\prime\}\}\)\_\{Q^\{\\prime\}\\neq Q\}for all other product bundles inVJV\_\{J\}\.

Now apply the clamped\-table property \(W6\.3\): there exists a strictly positive tableψJ:∏Q′∈VJEQ′→\(0,∞\)\\psi\_\{J\}:\\prod\_\{Q^\{\\prime\}\\in V\_\{J\}\}E\_\{Q^\{\\prime\}\}\\to\(0,\\infty\)and a scalarκJ→Q\>0\\kappa\_\{J\\to Q\}\>0such that for everyk∈EQk\\in E\_\{Q\}\(identified withEBE\_\{B\}via the bijectionιJ→Q\\iota\_\{J\\to Q\}\) and every assignmenta∈∏Q′∈VJ∖\{Q\}EQ′a\\in\\prod\_\{Q^\{\\prime\}\\in V\_\{J\}\\setminus\\\{Q\\\}\}E\_\{Q^\{\\prime\}\}:

CB​\(ιJ→Q​\(k\);a\)=κJ→Q⋅ψJ​\(xQ=k,xQ′=a​\(Q′\)​for​Q′≠Q\)\.C\_\{B\}\(\\iota\_\{J\\to Q\}\(k\);a\)=\\kappa\_\{J\\to Q\}\\cdot\\psi\_\{J\}\\bigl\(x\_\{Q\}=k,\\,x\_\{Q^\{\\prime\}\}=a\(Q^\{\\prime\}\)\\text\{ for \}Q^\{\\prime\}\\neq Q\\bigr\)\.
Substituting this identity into the expansion fordd​t​\[Bk\]\+\\frac\{d\}\{dt\}\[B\_\{k\}\]^\{\+\}:

dd​t​\[Bk\]\+=\[B0\]​κJ→Q​∑a∈∏Q′∈VJ∖\{Q\}EQ′ψJ​\(xQ=k,xQ′=a​\(Q′\)\)​∏Q′∈VJ∖\{Q\}\[Qa​\(Q′\)′\]\.\\frac\{d\}\{dt\}\[B\_\{k\}\]^\{\+\}=\[B\_\{0\}\]\\kappa\_\{J\\to Q\}\\sum\_\{a\\in\\prod\_\{Q^\{\\prime\}\\in V\_\{J\}\\setminus\\\{Q\\\}\}E\_\{Q^\{\\prime\}\}\}\\psi\_\{J\}\\bigl\(x\_\{Q\}=k,\\,x\_\{Q^\{\\prime\}\}=a\(Q^\{\\prime\}\)\\bigr\)\\prod\_\{Q^\{\\prime\}\\in V\_\{J\}\\setminus\\\{Q\\\}\}\[Q^\{\\prime\}\_\{a\(Q^\{\\prime\}\)\}\]\.
Next, we identify the negative contribution to the ODE\. By \(R1\), the only internal reactions with net stoichiometryBk→B0B\_\{k\}\\to B\_\{0\}are the recycling reactionsBk→ρBB0B\_\{k\}\\xrightarrow\{\\rho\_\{B\}\}B\_\{0\}, with no catalysts and a bundlewise constant rateρB=ρBJ→Q\\rho\_\{B\}=\\rho\_\{B\_\{J\\to Q\}\}\. Hence the mass\-action dynamics for coordinateBkB\_\{k\}are:

dd​t​\[Bk\]=dd​t​\[Bk\]\+−ρB​\[Bk\]\.\\frac\{d\}\{dt\}\[B\_\{k\}\]=\\frac\{d\}\{dt\}\[B\_\{k\}\]^\{\+\}\-\\rho\_\{B\}\[B\_\{k\}\]\.
At a positive steady state,dd​t​\[Bk\]=0\\frac\{d\}\{dt\}\[B\_\{k\}\]=0\. Rearranging:

ρB\[B0\]​\[Bk\]=κJ→Q​∑a∈∏Q′∈VJ∖\{Q\}EQ′ψJ​\(xQ=k,xQ′=a​\(Q′\)\)​∏Q′∈VJ∖\{Q\}\[Qa​\(Q′\)′\]\.\\frac\{\\rho\_\{B\}\}\{\[B\_\{0\}\]\}\[B\_\{k\}\]=\\kappa\_\{J\\to Q\}\\sum\_\{a\\in\\prod\_\{Q^\{\\prime\}\\in V\_\{J\}\\setminus\\\{Q\\\}\}E\_\{Q^\{\\prime\}\}\}\\psi\_\{J\}\\bigl\(x\_\{Q\}=k,\\,x\_\{Q^\{\\prime\}\}=a\(Q^\{\\prime\}\)\\bigr\)\\prod\_\{Q^\{\\prime\}\\in V\_\{J\}\\setminus\\\{Q\\\}\}\[Q^\{\\prime\}\_\{a\(Q^\{\\prime\}\)\}\]\.\(3\)
This establishes equation \([1](https://arxiv.org/html/2606.27737#S5.E1)\)\.

LetPPbe a product bundle\. By \(W5\),PPis product\-like: for eachk∈EPk\\in E\_\{P\}there is a unique assignmenta\(k\)∈𝒜Pa^\{\(k\)\}\\in\\mathcal\{A\}\_\{P\}supporting production ofPkP\_\{k\}, and along each catalytic neighbor the induced map on state labels is a bijection\.

By the structure of the Napp–Adams construction \(which our recognition conditions capture\), the producing reactions forPkP\_\{k\}have the schematic form:

P0\+∑B∈Cat​\(P\)Bk⟶Pk\+∑B∈Cat​\(P\)Bk,P\_\{0\}\+\\sum\_\{B\\in\\textrm\{Cat\}\(P\)\}B\_\{k\}\\longrightarrow P\_\{k\}\+\\sum\_\{B\\in\\textrm\{Cat\}\(P\)\}B\_\{k\},where one coordinate labeledkkfrom each incident sum bundle catalyzes production of the coordinate labeledkkinPP\. \(Here we use the bijections from \(W5\) to identify state labels across bundles\.\)

Under \(R1\), every such producing reaction has rate constantκP\\kappa\_\{P\}independent ofkk\. Thus the total production rate intoPkP\_\{k\}is:

dd​t​\[Pk\]\+=κP​\[P0\]​∏B∈Cat​\(P\)\[Bk\]\.\\frac\{d\}\{dt\}\[P\_\{k\}\]^\{\+\}=\\kappa\_\{P\}\[P\_\{0\}\]\\prod\_\{B\\in\\textrm\{Cat\}\(P\)\}\[B\_\{k\}\]\.
HereCat​\(P\)\\textrm\{Cat\}\(P\)denotes the set of sum bundles incident toPP—equivalently, for each factor classJJcontainingPPin its variable setVJV\_\{J\}, we include the sum bundleBJ→PB\_\{J\\to P\}\.

Again by \(R1\), the only internal reactions with net stoichiometryPk→P0P\_\{k\}\\to P\_\{0\}are the recycling reactionsPk→ρPP0P\_\{k\}\\xrightarrow\{\\rho\_\{P\}\}P\_\{0\}\. Hence:

dd​t​\[Pk\]=κP​\[P0\]​∏B∈Cat​\(P\)\[Bk\]−ρP​\[Pk\]\.\\frac\{d\}\{dt\}\[P\_\{k\}\]=\\kappa\_\{P\}\[P\_\{0\}\]\\prod\_\{B\\in\\textrm\{Cat\}\(P\)\}\[B\_\{k\}\]\-\\rho\_\{P\}\[P\_\{k\}\]\.
At a positive steady state:

ρPκP​\[P0\]​\[Pk\]=∏B∈Cat​\(P\)\[Bk\]\.\\frac\{\\rho\_\{P\}\}\{\\kappa\_\{P\}\[P\_\{0\}\]\}\[P\_\{k\}\]=\\prod\_\{B\\in\\textrm\{Cat\}\(P\)\}\[B\_\{k\}\]\.\(4\)
This establishes equation \([2](https://arxiv.org/html/2606.27737#S5.E2)\)\.

We now show that equations \([3](https://arxiv.org/html/2606.27737#A4.E3)\) and \([4](https://arxiv.org/html/2606.27737#A4.E4)\) are precisely the sum\-product BP fixed\-point equations on the reconstructed factor graphFG​\(Γ\)\\mathrm\{FG\}\(\\Gamma\)\.

Recall from Theorem[4\.7](https://arxiv.org/html/2606.27737#S4.Thmtheorem7)thatFG​\(Γ\)\\mathrm\{FG\}\(\\Gamma\)has:

- •Factor nodes corresponding to factor classesJJ;
- •Variable nodes corresponding to variable classes\[P\]\[P\]\(equivalence classes of product bundles under∼v\\sim\_\{v\}\);
- •An edge between factorJJand variable\[P\]\[P\]whenever\[P\]∩VJ≠∅\[P\]\\cap V\_\{J\}\\neq\\emptyset;
- •Factor tableψJ:∏Q∈VJEQ→\(0,∞\)\\psi\_\{J\}:\\prod\_\{Q\\in V\_\{J\}\}E\_\{Q\}\\to\(0,\\infty\)for each factorJJ\.

In the standard BP notation on this factor graph:

- •Sk\(j→n\)S^\{\(j\\to n\)\}\_\{k\}denotes thekk\-th component of the sum message from factorjjto variablenn;
- •Pk\(n→j\)P^\{\(n\\to j\)\}\_\{k\}denotes thekk\-th component of the product message from variablennto factorjj\.

The BP sum\-product fixed\-point equations are \(cf\.\(Napp and Adams,[2013](https://arxiv.org/html/2606.27737#bib.bib5)\)\):

Sk\(j→n\)\\displaystyle S^\{\(j\\to n\)\}\_\{k\}∝∑𝐤j:knj=kψj​\(𝐤j\)​∏n′∈ne​\(j\)∖nPkn′j\(n′→j\),\\displaystyle\\propto\\sum\_\{\\mathbf\{k\}^\{j\}:k^\{j\}\_\{n\}=k\}\\psi\_\{j\}\(\\mathbf\{k\}^\{j\}\)\\prod\_\{n^\{\\prime\}\\in\\mathrm\{ne\}\(j\)\\setminus n\}P^\{\(n^\{\\prime\}\\to j\)\}\_\{k^\{j\}\_\{n^\{\\prime\}\}\},\(5\)Pk\(n→j\)\\displaystyle P^\{\(n\\to j\)\}\_\{k\}∝∏j′∈ne​\(n\)∖jSk\(j′→n\)\.\\displaystyle\\propto\\prod\_\{j^\{\\prime\}\\in\\mathrm\{ne\}\(n\)\\setminus j\}S^\{\(j^\{\\prime\}\\to n\)\}\_\{k\}\.\(6\)
We establish the correspondence by setting:

Sk\(J→\[Q\]\)\\displaystyle S^\{\(J\\to\[Q\]\)\}\_\{k\}:=\[BJ→Q,k\],\\displaystyle:=\[B\_\{J\\to Q,k\}\],Pk\(\[Q\]→J\)\\displaystyle P^\{\(\[Q\]\\to J\)\}\_\{k\}:=\[PQ,k\],\\displaystyle:=\[P\_\{Q,k\}\],whereQQis the unique representative of\[Q\]\[Q\]inVJV\_\{J\}\.

Sum messages:In our CRN, the sum bundleBJ→QB\_\{J\\to Q\}corresponds to the message from factorJJto variable\[Q\]\[Q\]\. Equation \([3](https://arxiv.org/html/2606.27737#A4.E3)\) states:

ρB\[B0\]​\[Bk\]=κJ→Q​∑aψJ​\(xQ=k,xQ′=a​\(Q′\)\)​∏Q′∈VJ∖\{Q\}\[Qa​\(Q′\)′\]\.\\frac\{\\rho\_\{B\}\}\{\[B\_\{0\}\]\}\[B\_\{k\}\]=\\kappa\_\{J\\to Q\}\\sum\_\{a\}\\psi\_\{J\}\(x\_\{Q\}=k,x\_\{Q^\{\\prime\}\}=a\(Q^\{\\prime\}\)\)\\prod\_\{Q^\{\\prime\}\\in V\_\{J\}\\setminus\\\{Q\\\}\}\[Q^\{\\prime\}\_\{a\(Q^\{\\prime\}\)\}\]\.
Since product bundlesQ′∈VJ∖\{Q\}Q^\{\\prime\}\\in V\_\{J\}\\setminus\\\{Q\\\}correspond to the other variables incident to factorJJ, and\[Qa​\(Q′\)′\]\[Q^\{\\prime\}\_\{a\(Q^\{\\prime\}\)\}\]is the concentration of the product\-message species, we have:

\[Bk\]∝∑𝐤J:kQJ=kψJ​\(𝐤J\)​∏Q′∈VJ∖\{Q\}\[QkQ′J′\],\[B\_\{k\}\]\\propto\\sum\_\{\\mathbf\{k\}^\{J\}:k^\{J\}\_\{Q\}=k\}\\psi\_\{J\}\(\\mathbf\{k\}^\{J\}\)\\prod\_\{Q^\{\\prime\}\\in V\_\{J\}\\setminus\\\{Q\\\}\}\[Q^\{\\prime\}\_\{k^\{J\}\_\{Q^\{\\prime\}\}\}\],which matches equation \([5](https://arxiv.org/html/2606.27737#A4.E5)\) under our identification\.

Product messages:For a product bundlePPcorresponding to variable\[P\]\[P\]sending to factorJJ, equation \([4](https://arxiv.org/html/2606.27737#A4.E4)\) states:

ρPκP​\[P0\]​\[Pk\]=∏B∈Cat​\(P\)\[Bk\]\.\\frac\{\\rho\_\{P\}\}\{\\kappa\_\{P\}\[P\_\{0\}\]\}\[P\_\{k\}\]=\\prod\_\{B\\in\\textrm\{Cat\}\(P\)\}\[B\_\{k\}\]\.
The catalytic neighborsCat​\(P\)\\textrm\{Cat\}\(P\)are exactly the sum bundles from factorsJ′≠JJ^\{\\prime\}\\neq Jincident to variable\[P\]\[P\]\. Hence:

\[Pk\]∝∏J′∈ne​\(\[P\]\)∖J\[BJ′→P,k\],\[P\_\{k\}\]\\propto\\prod\_\{J^\{\\prime\}\\in\\mathrm\{ne\}\(\[P\]\)\\setminus J\}\[B\_\{J^\{\\prime\}\\to P,k\}\],which matches equation \([6](https://arxiv.org/html/2606.27737#A4.E6)\) under our identification\.

Normalization:The proportionality constantsρB/\[B0\]\\rho\_\{B\}/\[B\_\{0\}\]andρP/\(κP​\[P0\]\)\\rho\_\{P\}/\(\\kappa\_\{P\}\[P\_\{0\}\]\)serve as the per\-message normalization factors\. In standard BP, messages are defined only up to positive rescaling, and these normalization factors are precisely what allow the concentrations to be interpreted as \(unnormalized\) message components\.

Since the sum of concentrations within each bundle is conserved \(a consequence of the stoichiometric structure\), normalization is automatically maintained by the reaction dynamics\.

This completes the identification of CRN steady states with BP fixed points\. ∎

###### Definition D\.1\(Napp Compilation\)\.

Given a factor graph\(A,H\)\(A,H\)with factor tablesψj∝e−Hj\\psi\_\{j\}\\propto e^\{\-H\_\{j\}\}, the*Napp compilation*is the mass\-action CRN

constructed as follows:

1. 1\.For every directed edgej→nj\\to nin the factor graph, introduce a sum bundle𝖲j→n\\mathsf\{S\}^\{j\\to n\}and a product bundle𝖯n→j\\mathsf\{P\}^\{n\\to j\}\. Each bundleBBconsists of a zero speciesB0B\_\{0\}and one coordinate speciesBkB\_\{k\}for every symbolkkof the underlying variable\.
2. 2\.For each bundleBB, add recycling reactions: Bk→κrB0,k≥1,B\_\{k\}\\xrightarrow\{\\kappa\_\{r\}\}B\_\{0\},\\quad k\\geq 1,with a single global recycling rateκr\>0\\kappa\_\{r\}\>0\.
3. 3\.For each edgej→nj\\to nand each assignment𝐤j∈Kj\\mathbf\{k\}^\{j\}\\in K^\{j\}with\(𝐤j\)n=k\(\\mathbf\{k\}^\{j\}\)\_\{n\}=k, add a sum\-bundle production reaction: 𝖲0j→n\+∑n′∈ne​\(j\)∖n𝖯\(𝐤j\)n′n′→j→ψj​\(𝐤j\)𝖲kj→n\+∑n′∈ne​\(j\)∖n𝖯\(𝐤j\)n′n′→j\.\\mathsf\{S\}^\{j\\to n\}\_\{0\}\+\\sum\_\{n^\{\\prime\}\\in\\mathrm\{ne\}\(j\)\\setminus n\}\\mathsf\{P\}^\{n^\{\\prime\}\\to j\}\_\{\(\\mathbf\{k\}^\{j\}\)\_\{n^\{\\prime\}\}\}\\xrightarrow\{\\psi\_\{j\}\(\\mathbf\{k\}^\{j\}\)\}\\mathsf\{S\}^\{j\\to n\}\_\{k\}\+\\sum\_\{n^\{\\prime\}\\in\\mathrm\{ne\}\(j\)\\setminus n\}\\mathsf\{P\}^\{n^\{\\prime\}\\to j\}\_\{\(\\mathbf\{k\}^\{j\}\)\_\{n^\{\\prime\}\}\}\.
4. 4\.For each variablennand each statekk, add product\-bundle reactions: 𝖯0n→j\+∑j′∈ne​\(n\)∖j𝖲kj′→n→κprod𝖯kn→j\+∑j′∈ne​\(n\)∖j𝖲kj′→n,\\mathsf\{P\}^\{n\\to j\}\_\{0\}\+\\sum\_\{j^\{\\prime\}\\in\\mathrm\{ne\}\(n\)\\setminus j\}\\mathsf\{S\}^\{j^\{\\prime\}\\to n\}\_\{k\}\\xrightarrow\{\\kappa\_\{\\mathrm\{prod\}\}\}\\mathsf\{P\}^\{n\\to j\}\_\{k\}\+\\sum\_\{j^\{\\prime\}\\in\\mathrm\{ne\}\(n\)\\setminus j\}\\mathsf\{S\}^\{j^\{\\prime\}\\to n\}\_\{k\},with a single global production rateκprod\>0\\kappa\_\{\\mathrm\{prod\}\}\>0\.

We write𝒩​\(A,H\)\\mathcal\{N\}\(A,H\)\(or simply𝒩​\(FG\)\\mathcal\{N\}\(\\mathrm\{FG\}\)\) for this CRN and refer to the map𝒩:𝐅𝐆𝐫𝐚𝐩𝐡→𝐂𝐑𝐍\\mathcal\{N\}:\\mathbf\{FGraph\}\\to\\mathbf\{CRN\}as the Napp compilation\.

###### Corollary D\.2\(BP↔\\leftrightarrowCRN\)\.

LetΓ=\(X,R,κ\)\\Gamma=\(X,R,\\kappa\)be a mass–action CRN satisfying\(W1\)–\(W6\)and\(R1\), and letFG​\(Γ\)\\mathrm\{FG\}\(\\Gamma\)be the factor graph with factor tables\{ψJ\}\\\{\\psi\_\{J\}\\\}reconstructed by Theorem[4\.7](https://arxiv.org/html/2606.27737#S4.Thmtheorem7)\. Let

ΓN:=𝒩​\(FG​\(Γ\)\)\\Gamma^\{\\mathrm\{N\}\}\\;:=\\;\\mathcal\{N\}\\bigl\(\\mathrm\{FG\}\(\\Gamma\)\\bigr\)be the BP–CRN obtained by applying the Napp–Adams compilation toFG​\(Γ\)\\mathrm\{FG\}\(\\Gamma\), with any fixed choice of recycling and production constants\.

Then there exist positive scalars\{sB\>0\}\\\{s\_\{B\}\>0\\\}, one for each bundleBB, and a choice of rate constants forΓN\\Gamma^\{\\mathrm\{N\}\}such that the following hold\.

1. 1\.The species ofΓ\\GammaandΓN\\Gamma^\{\\mathrm\{N\}\}can be put into bijection bundlewise, preserving zero versus coordinate species, the distinction between sum and product bundles, and catalytic neighbor sets, and under this bijection the change of variables \[XB,k\]N=sB​\[XB,k\]\[X\_\{B,k\}\]^\{\\mathrm\{N\}\}\\;=\\;s\_\{B\}\\,\[X\_\{B,k\}\]conjugates the mass–action vector field ofΓ\\Gammato that ofΓN\\Gamma^\{\\mathrm\{N\}\}\.
2. 2\.In particular, positive steady states ofΓ\\Gammacorrespond bijectively, via the same bundlewise rescaling, to positive steady states ofΓN\\Gamma^\{\\mathrm\{N\}\}\.
3. 3\.From the main result of Napp–Adams applied toΓN\\Gamma^\{\\mathrm\{N\}\}, any positive steady state ofΓ\\Gammatherefore determines a collection of BP messages \(defined up to the usual per–message positive rescaling\) onFG​\(Γ\)\\mathrm\{FG\}\(\\Gamma\)that satisfy the sum–product fixed–point equations for the recovered factor tables\{ψJ\}\\\{\\psi\_\{J\}\\\}\.

###### Proof of[D\.2](https://arxiv.org/html/2606.27737#A4.Thmtheorem2)\.

LetFG​\(Γ\)\\mathrm\{FG\}\(\\Gamma\)be reconstructed fromΓ\\Gammaunder \(W1\)–\(W6\), with factor classes\(J,VJ\)\(J,V\_\{J\}\)and factor tablesψJ\\psi\_\{J\}from \(W6\)\. Consider the Napp compilationΓN:=𝒩​\(FG​\(Γ\)\)\\Gamma^\{N\}:=\\mathcal\{N\}\(\\mathrm\{FG\}\(\\Gamma\)\), in which each directed incidence\(J,Q\)\(J,Q\)withQ∈VJQ\\in V\_\{J\}produces a sum bundle𝖲J→Q\\mathsf\{S\}^\{J\\to Q\}and each directed incidence\(Q,J\)\(Q,J\)produces a product bundle𝖯Q→J\\mathsf\{P\}^\{Q\\to J\}\.

By the construction ofFG​\(Γ\)\\mathrm\{FG\}\(\\Gamma\)in Theorem[4\.7](https://arxiv.org/html/2606.27737#S4.Thmtheorem7), the bundles ofΓ\\Gammadecompose into:

1. 1\.Sum bundlesBJ→QB\_\{J\\to Q\}indexed by pairs\(J,Q\)\(J,Q\)withQ∈VJQ\\in V\_\{J\};
2. 2\.Product bundlesPPindexed by directed incidences\(Q,J\)\(Q,J\)\.

The Napp compilation produces exactly one bundle of each type for each such directed incidence\. Hence there is a canonical bundlewise bijection between the species sets ofΓ\\GammaandΓN\\Gamma^\{N\}, preserving bundle type \(sum vs\. product\), zero vs\. coordinate species, and state labels\.

Moreover, by \(W4\) and \(W5\), the reaction monomials ofΓ\\Gammamatch the Napp monomials:

- •Sum\-bundle production terms are multilinear monomials with one coordinate from each catalytic neighbor;
- •Product\-bundle production terms are diagonal in the state labelkk\.

Fix a factor class\(J,VJ\)\(J,V\_\{J\}\)andQ∈VJQ\\in V\_\{J\}\. By \(W6\.3\), the production coefficients of the corresponding sum bundleBJ→QB\_\{J\\to Q\}satisfy:

CBJ→Q​\(k;a\)=κJ→Q⋅ψJ​\(xQ=k,xQ′=a​\(Q′\)​for​Q′≠Q\)\.C\_\{B\_\{J\\to Q\}\}\(k;a\)=\\kappa\_\{J\\to Q\}\\cdot\\psi\_\{J\}\(x\_\{Q\}=k,x\_\{Q^\{\\prime\}\}=a\(Q^\{\\prime\}\)\\text\{ for \}Q^\{\\prime\}\\neq Q\)\.
In the Napp compilation, the analogous producing reactions for𝖲kJ→Q\\mathsf\{S\}^\{J\\to Q\}\_\{k\}have rate constants proportional to the same clamped table entries ofψJ\\psi\_\{J\}, with a user\-chosen proportionality constant\. Choose these Napp proportionality constants so that, after a bundlewise rescaling

\[XB,k\]N=sB​\[XB,k\],\[X\_\{B,k\}\]^\{N\}=s\_\{B\}\[X\_\{B,k\}\],the coefficients of every monomial in the sum\-bundle production polynomials agree term\-by\-term betweenΓ\\GammaandΓN\\Gamma^\{N\}\.

Similarly, by \(R1\) we may choose Napp recycling and product\-production constants to match the internalkk\-uniform recycling andkk\-uniform product production inΓ\\Gamma\(again after the same bundlewise rescaling\)\.

Because mass\-action vector fields are polynomial and determined by the reaction monomials and their coefficients, the term\-by\-term agreement implies that the change of variables\[XB,k\]N=sB​\[XB,k\]\[X\_\{B,k\}\]^\{N\}=s\_\{B\}\[X\_\{B,k\}\]conjugates the ODE ofΓ\\Gammato that ofΓN\\Gamma^\{N\}\. Since the vector fields are conjugate under a linear \(diagonal, positive\) change of coordinates, positive steady states correspond bijectively under this rescaling\. A pointx∗∈ℝ\>0Mx^\{\*\}\\in\\mathbb\{R\}^\{M\}\_\{\>0\}is a steady state ofΓ\\Gammaif and only if the rescaled point\(sB​xB,k∗\)\(s\_\{B\}x^\{\*\}\_\{B,k\}\)is a steady state ofΓN\\Gamma^\{N\}\. Theorem[5\.4](https://arxiv.org/html/2606.27737#S5.Thmtheorem4)establishes that positive steady states of any CRN satisfying \(W1\)–\(W6\) and \(R1\) correspond to BP fixed points on the reconstructed factor graph\. Applying this to bothΓ\\GammaandΓN\\Gamma^\{N\}:

- •Any positive steady statex∗x^\{\*\}ofΓ\\Gammacorresponds to a BP fixed point onFG​\(Γ\)\\mathrm\{FG\}\(\\Gamma\);
- •The rescaled steady state\(sB​xB,k∗\)\(s\_\{B\}x^\{\*\}\_\{B,k\}\)ofΓN\\Gamma^\{N\}corresponds to the same BP fixed point \(since rescaling messages by positive constants does not change BP fixed\-point status\)\.

Thus steady states ofΓ\\Gammadetermine BP fixed points onFG​\(Γ\)\\mathrm\{FG\}\(\\Gamma\), with factor tables\{ψJ\}\\\{\\psi\_\{J\}\\\}recovered from the reaction coefficients\. ∎

### D\.1SP–B retractions in the space of factor tables

We briefly recall the SP–B update rules in the space of factor tables\. Writeψc∝e−Hc\\psi\_\{c\}\\propto e^\{\-H\_\{c\}\}\.

- •Linear retraction\.The Hamiltonians on survivors are unchanged \(restriction\), henceψc′=ψc\\psi^\{\\prime\}\_\{c\}=\\psi\_\{c\}for all survivingcc\.
- •Colinear retraction\.Letaabe maximal colinear with lower neighbora↓a^\{\\downarrow\}\. LetB=𝒜∖\{a\}B=\\mathcal\{A\}\\setminus\\\{a\\\}\. Letb↑b^\{\\uparrow\}be the unique upper cover ofa↓a^\{\\downarrow\}inBB\. Define H^a↓​\(x\)=ln​∑z∈Ea:za↓=xexp⁡\{−Ha​\(z\)\+Ha↓​\(x\)\}\.\\widehat\{H\}\_\{a^\{\\downarrow\}\}\(x\)\\;=\\;\\ln\\sum\_\{z\\in E\_\{a\}:\\ z\_\{a^\{\\downarrow\}\}=x\}\\exp\\\{\-H\_\{a\}\(z\)\+H\_\{a^\{\\downarrow\}\}\(x\)\\\}\.ThenH~c=Hc\\widetilde\{H\}\_\{c\}=H\_\{c\}forc≠b↑c\\neq b^\{\\uparrow\}andH~b↑=Hb↑−H^a↓∘πb↑​a↓\\widetilde\{H\}\_\{b^\{\\uparrow\}\}=H\_\{b^\{\\uparrow\}\}\-\\widehat\{H\}\_\{a^\{\\downarrow\}\}\\circ\\pi\_\{b^\{\\uparrow\}a^\{\\downarrow\}\}\. Exponentiating gives the two standard factor–table forms: \(Unary→higher parity\)ψb↑′​\(𝐤\)=ψb↑​\(𝐤\)​ψa​\(ka↓\),\\text\{\(Unary \}\\to\\text\{ higher parity\)\}\\quad\\psi^\{\\prime\}\_\{b^\{\\uparrow\}\}\(\\mathbf\{k\}\)=\\psi\_\{b^\{\\uparrow\}\}\(\\mathbf\{k\}\)\\,\\psi\_\{a\}\(k\_\{a^\{\\downarrow\}\}\),\(Higher parity→unary ona↓\)ψ′a↓\(x\)=∑z:za↓=xψa\(z\)\.\\text\{\(Higher parity \}\\to\\text\{ unary on \}a^\{\\downarrow\}\)\\quad\\psi^\{\\prime\}\_\{a^\{\\downarrow\}\}\(x\)=\\sum\_\{z:\\,z\_\{a^\{\\downarrow\}\}=x\}\\psi\_\{a\}\(z\)\.

###### Lemma D\.3\(Factor version of S–P Eq\. 4\.29: colinear, survivor linear\)\.

Leta∈𝒜a\\in\\mathcal\{A\}be colinear and maximal, letB=𝒜∖\{a\}B=\\mathcal\{A\}\\setminus\\\{a\\\}, and letb=a↓∈Bb=a^\{\\downarrow\}\\in Bbe*linear*inBBwith unique upper coverb↑∈Bb^\{\\uparrow\}\\in B\. Write factor tablesψc∝e−Hc\\psi\_\{c\}\\propto e^\{\-H\_\{c\}\}onBB\. Forx∈Ebx\\in E\_\{b\}and any𝐤∈Eb↑\\mathbf\{k\}\\in E\_\{b^\{\\uparrow\}\}with𝐤\|b=x\\mathbf\{k\}\|\_\{b\}=x,

ψb↑′​\(𝐤\)∝ψb↑​\(𝐤\)⋅∑z∈Ea:zb=xψa​\(z\)ψb​\(x\)\\quad\\psi^\{\\prime\}\_\{b^\{\\uparrow\}\}\(\\mathbf\{k\}\)\\ \\propto\\ \\psi\_\{b^\{\\uparrow\}\}\(\\mathbf\{k\}\)\\;\\cdot\\;\\frac\{\\displaystyle\\sum\_\{z\\in E\_\{a\}:\\ z\_\{b\}=x\}\\ \\psi\_\{a\}\(z\)\}\{\\psi\_\{b\}\(x\)\}\\quadandψc′=ψc\\psi^\{\\prime\}\_\{c\}=\\psi\_\{c\}for allc≠b↑c\\neq b^\{\\uparrow\}\.

###### Lemma D\.4\(Factor version of S–P Eq\. 4\.30: colinear, survivor not linear\)\.

Leta∈𝒜a\\in\\mathcal\{A\}be colinear and maximal,B=𝒜∖\{a\}B=\\mathcal\{A\}\\setminus\\\{a\\\},b=a↓∈Bb=a^\{\\downarrow\}\\in B*not*linear inBB, and letcB​\(b\)c\_\{B\}\(b\)be the Möbius coefficient ofbbin the posetBB\. Then

ψb′​\(x\)∝\(ψb​\(x\)\)1−1cB​\(b\)​\(∑z∈Ea:zb=xψa​\(z\)\)1cB​\(b\)\(x∈Eb\),\\quad\\psi^\{\\prime\}\_\{b\}\(x\)\\ \\propto\\ \\big\(\\psi\_\{b\}\(x\)\\big\)^\{\\,1\-\\frac\{1\}\{c\_\{B\}\(b\)\}\}\\;\\Big\(\\ \\sum\_\{z\\in E\_\{a\}:\\ z\_\{b\}=x\}\\ \\psi\_\{a\}\(z\)\\ \\Big\)^\{\\frac\{1\}\{c\_\{B\}\(b\)\}\}\\quad\\qquad\(x\\in E\_\{b\}\),andψc′=ψc\\psi^\{\\prime\}\_\{c\}=\\psi\_\{c\}for allc≠bc\\neq b\.

###### Both lemmas\.

Start from theHH\-space statements in S–P and exponentiate\. For Eq\. 4\.29:H~b↑=Hb↑−H^b∘πb↑​b\\widetilde\{H\}\_\{b^\{\\uparrow\}\}=H\_\{b^\{\\uparrow\}\}\-\\widehat\{H\}\_\{b\}\\circ\\pi\_\{b^\{\\uparrow\}b\}withH^b​\(x\)=ln​∑z:zb=xe−Ha​\(z\)\+Hb​\(x\)\\widehat\{H\}\_\{b\}\(x\)=\\ln\\sum\_\{z:\\ z\_\{b\}=x\}e^\{\-H\_\{a\}\(z\)\+H\_\{b\}\(x\)\}gives

ψb↑′​\(𝐤\)∝e−H~b↑​\(𝐤\)∝e−Hb↑​\(𝐤\)​eH^b​\(x\)=ψb↑​\(𝐤\)​∑z:zb=xe−Ha​\(z\)e−Hb​\(x\)=ψb↑​\(𝐤\)​∑z:zb=xψa​\(z\)ψb​\(x\)\.\\psi^\{\\prime\}\_\{b^\{\\uparrow\}\}\(\\mathbf\{k\}\)\\ \\propto\\ e^\{\-\\widetilde\{H\}\_\{b^\{\\uparrow\}\}\(\\mathbf\{k\}\)\}\\ \\propto\\ e^\{\-H\_\{b^\{\\uparrow\}\}\(\\mathbf\{k\}\)\}\\ e^\{\\widehat\{H\}\_\{b\}\(x\)\}=\\psi\_\{b^\{\\uparrow\}\}\(\\mathbf\{k\}\)\\ \\frac\{\\sum\_\{z:\\ z\_\{b\}=x\}e^\{\-H\_\{a\}\(z\)\}\}\{e^\{\-H\_\{b\}\(x\)\}\}=\\psi\_\{b^\{\\uparrow\}\}\(\\mathbf\{k\}\)\\ \\frac\{\\sum\_\{z:\\ z\_\{b\}=x\}\\psi\_\{a\}\(z\)\}\{\\psi\_\{b\}\(x\)\}\.For Eq\. 4\.30:H~b=Hb−1cB​\(b\)​ln​∑z:zb=xe−Ha​\(z\)\+Hb​\(x\)\\widetilde\{H\}\_\{b\}=H\_\{b\}\-\\frac\{1\}\{c\_\{B\}\(b\)\}\\ln\\sum\_\{z:\\ z\_\{b\}=x\}e^\{\-H\_\{a\}\(z\)\+H\_\{b\}\(x\)\}yields

ψb′​\(x\)∝e−H~b​\(x\)=ψb​\(x\)​\(∑z:zb=xe−Ha​\(z\)\+Hb​\(x\)\)1/cB​\(b\)=ψb​\(x\)​\(1ψb​\(x\)​∑z:zb=xψa​\(z\)\)1/cB​\(b\),\\psi^\{\\prime\}\_\{b\}\(x\)\\ \\propto\\ e^\{\-\\widetilde\{H\}\_\{b\}\(x\)\}=\\psi\_\{b\}\(x\)\\,\\Big\(\\sum\_\{z:\\ z\_\{b\}=x\}e^\{\-H\_\{a\}\(z\)\+H\_\{b\}\(x\)\}\\Big\)^\{\\\!1/c\_\{B\}\(b\)\}=\\psi\_\{b\}\(x\)\\,\\Big\(\\tfrac\{1\}\{\\psi\_\{b\}\(x\)\}\\sum\_\{z:\\ z\_\{b\}=x\}\\psi\_\{a\}\(z\)\\Big\)^\{\\\!1/c\_\{B\}\(b\)\},we can verify this is correct by taking logs\. ∎

###### Proof of Prop[6\.2](https://arxiv.org/html/2606.27737#S6.Thmtheorem2)\.

We verify the two functor axioms: preservation of identities and preservation of composition\.

\(Identities\)\.Ifr=id\(A,H\)r=\\mathrm\{id\}\_\{\(A,H\)\}, then no poset point is retracted and the HamiltoniansHcH\_\{c\}\(hence the tablesψc\\psi\_\{c\}\) are unchanged\. The compilation on objects produces the same CRN\(X,R,κ\)\(X,R,\\kappa\), and on morphisms we have

φ0=idX,φ1=idR,ψ=idκ,\\varphi\_\{0\}=\\mathrm\{id\}\_\{X\},\\quad\\varphi\_\{1\}=\\mathrm\{id\}\_\{R\},\\quad\\psi=\\mathrm\{id\}\_\{\\kappa\},so𝒩​\(id\(A,H\)\)=id𝒩​\(A,H\)\\mathcal\{N\}\(\\mathrm\{id\}\_\{\(A,H\)\}\)=\\mathrm\{id\}\_\{\\mathcal\{N\}\(A,H\)\}\.

\(Composition\)\.Let

r1:\(A,H\)→\(B,G\),r2:\(B,G\)→\(C,K\)r\_\{1\}:\(A,H\)\\to\(B,G\),\\quad r\_\{2\}:\(B,G\)\\to\(C,K\)be two morphisms in𝐅𝐆𝐫𝐚𝐩𝐡𝐑𝐞𝐭\\mathbf\{FGraphRet\}\. Their compositer2∘r1r\_\{2\}\\circ r\_\{1\}is again a finite composite of linear/colinear retractions\. On factor tables, the SP–B update rules are defined by local formulas inψ\\psi\-space; applyingr1r\_\{1\}and thenr2r\_\{2\}gives the same final tablesψc′′\\psi^\{\\prime\\prime\}\_\{c\}as applying the compositer2∘r1r\_\{2\}\\circ r\_\{1\}once\.

On CRNs we have:

- •φ0r2∘r1\\varphi\_\{0\}^\{r\_\{2\}\\circ r\_\{1\}\}deletes exactly the union of bundles deleted byr1r\_\{1\}andr2r\_\{2\}; this is the same asφ0r2∘φ0r1\\varphi\_\{0\}^\{r\_\{2\}\}\\circ\\varphi\_\{0\}^\{r\_\{1\}\}bundlewise\.
- •Likewise,φ1r2∘r1=φ1r2∘φ1r1\\varphi\_\{1\}^\{r\_\{2\}\\circ r\_\{1\}\}=\\varphi\_\{1\}^\{r\_\{2\}\}\\circ\\varphi\_\{1\}^\{r\_\{1\}\}because a reaction is present after the composite if and only if all its bundles survive both stages\.
- •For the rate part, each surviving edgej→nj\\to ncarries a scalarcj→nc\_\{j\\to n\}that we keep fixed along the entire chain\. Ifψj↦ψj′↦ψj′′\\psi\_\{j\}\\mapsto\\psi^\{\\prime\}\_\{j\}\\mapsto\\psi^\{\\prime\\prime\}\_\{j\}are the factor tables underr1r\_\{1\}andr2r\_\{2\}, then λj→n=cj→n​ψj↦ψr1cj→n​ψj′↦ψr2cj→n​ψj′′,\\lambda\_\{j\\to n\}=c\_\{j\\to n\}\\psi\_\{j\}\\xmapsto\{\\psi^\{r\_\{1\}\}\}c\_\{j\\to n\}\\psi^\{\\prime\}\_\{j\}\\xmapsto\{\\psi^\{r\_\{2\}\}\}c\_\{j\\to n\}\\psi^\{\\prime\\prime\}\_\{j\},which is exactly the update prescribed byψr2∘r1\\psi^\{r\_\{2\}\\circ r\_\{1\}\}\. Henceψr2∘r1=ψr2∘ψr1\\psi^\{r\_\{2\}\\circ r\_\{1\}\}=\\psi^\{r\_\{2\}\}\\circ\\psi^\{r\_\{1\}\}\.

Therefore

𝒩​\(r2∘r1\)=\(φ0r2∘r1,φ1r2∘r1,ψr2∘r1\)=\(φ0r2,φ1r2,ψr2\)∘\(φ0r1,φ1r1,ψr1\)=𝒩​\(r2\)∘𝒩​\(r1\)\.\\mathcal\{N\}\(r\_\{2\}\\circ r\_\{1\}\)=\(\\varphi\_\{0\}^\{r\_\{2\}\\circ r\_\{1\}\},\\varphi\_\{1\}^\{r\_\{2\}\\circ r\_\{1\}\},\\psi^\{r\_\{2\}\\circ r\_\{1\}\}\)=\(\\varphi\_\{0\}^\{r\_\{2\}\},\\varphi\_\{1\}^\{r\_\{2\}\},\\psi^\{r\_\{2\}\}\)\\circ\(\\varphi\_\{0\}^\{r\_\{1\}\},\\varphi\_\{1\}^\{r\_\{1\}\},\\psi^\{r\_\{1\}\}\)=\\mathcal\{N\}\(r\_\{2\}\)\\circ\\mathcal\{N\}\(r\_\{1\}\)\.
\(Closure under \(W1\)–\(W6\)\)\.For any factor graph\(A,H\)\(A,H\), the construction of𝒩​\(A,H\)\\mathcal\{N\}\(A,H\)forces the bundle decomposition and zero↔\\leftrightarrowcoordinate structure, so \(W1\)–\(W3\) always hold\.

We verify \(W4\)–\(W6\) directly:

- •\(W4\):By construction, sum\-bundle production reactions have the form S0j→n\+∑n′≠nPkn′jn′→j→Skj→n\+∑n′≠nPkn′jn′→j,S^\{j\\to n\}\_\{0\}\+\\sum\_\{n^\{\\prime\}\\neq n\}P^\{n^\{\\prime\}\\to j\}\_\{k^\{j\}\_\{n^\{\\prime\}\}\}\\to S^\{j\\to n\}\_\{k\}\+\\sum\_\{n^\{\\prime\}\\neq n\}P^\{n^\{\\prime\}\\to j\}\_\{k^\{j\}\_\{n^\{\\prime\}\}\},with rateψj​\(𝐤j\)\\psi\_\{j\}\(\\mathbf\{k\}^\{j\}\)\. The catalytic neighbors ofSj→nS^\{j\\to n\}are the product bundles\{Pn′→j\}n′∈ne​\(j\)∖n\\\{P^\{n^\{\\prime\}\\to j\}\\\}\_\{n^\{\\prime\}\\in\\mathrm\{ne\}\(j\)\\setminus n\}, and each monomial uses exactly one coordinate from each such bundle\. This is precisely the assignment\-form expansion required by \(W4\)\.
- •\(W5\):Product bundles have production reactions of the form P0n→j\+∑j′≠jSkj′→n→Pkn→j\+∑j′≠jSkj′→n\.P^\{n\\to j\}\_\{0\}\+\\sum\_\{j^\{\\prime\}\\neq j\}S^\{j^\{\\prime\}\\to n\}\_\{k\}\\to P^\{n\\to j\}\_\{k\}\+\\sum\_\{j^\{\\prime\}\\neq j\}S^\{j^\{\\prime\}\\to n\}\_\{k\}\.For eachkk, there is exactly one assignment \(all catalysts contribute statekk\), and the bijectionsθP,B\\theta\_\{P,B\}are simply the identity on the state setEnE\_\{n\}\. This satisfies \(W5\)\.
- •\(W6\):Sum bundles for the same factorjjform a factor classJjJ\_\{j\}\. For each variablen∈ne​\(j\)n\\in\\mathrm\{ne\}\(j\), the sum bundleSj→nS^\{j\\to n\}hasCat​\(Sj→n\)=\{Pn′→j:n′≠n\}=VJj∖\{Pn→j\}\\textrm\{Cat\}\(S^\{j\\to n\}\)=\\\{P^\{n^\{\\prime\}\\to j\}:n^\{\\prime\}\\neq n\\\}=V\_\{J\_\{j\}\}\\setminus\\\{P^\{n\\to j\}\\\}, establishing the missing\-one\-neighbor pattern\. The coefficient arrays areCSj→n​\(k;a\)=ψj​\(xn=k,xn′=a​\(n′\)\)C\_\{S^\{j\\to n\}\}\(k;a\)=\\psi\_\{j\}\(x\_\{n\}=k,x\_\{n^\{\\prime\}\}=a\(n^\{\\prime\}\)\), which is the clamped\-table property withκj→n=1\\kappa\_\{j\\to n\}=1\.

SP–B retractions only change the tablesψJ\\psi\_\{J\}within a factor class and do not permute or merge catalytic bundles\. Thus \(W4\)–\(W6\) continue to hold after any sequence of retractions, and the target CRN of𝒩​\(r\)\\mathcal\{N\}\(r\)still lies in the subcategory of CRNs satisfying \(W1\)–\(W6\)\.

Altogether,𝒩\\mathcal\{N\}preserves identities and composition, and by Lemma[6\.1](https://arxiv.org/html/2606.27737#S6.Thmtheorem1)its image is contained in the CRNs satisfying \(W1\)–\(W6\) and \(R1\), as claimed\. ∎

###### Proof of Theorem[6\.3](https://arxiv.org/html/2606.27737#S6.Thmtheorem3)\.

By Lemma[6\.1](https://arxiv.org/html/2606.27737#S6.Thmtheorem1), bothΓ=𝒩​\(A,H\)\\Gamma=\\mathcal\{N\}\(A,H\)andΓ′=𝒩​\(A′,H′\)\\Gamma^\{\\prime\}=\\mathcal\{N\}\(A^\{\\prime\},H^\{\\prime\}\)satisfy \(W1\)–\(W6\) and \(R1\)\. By Theorem[5\.4](https://arxiv.org/html/2606.27737#S5.Thmtheorem4), their positive steady states correspond to BP fixed points on the respective factor graphs\.

This is exactly the definition of𝒩​\(r\)\\mathcal\{N\}\(r\)given above\. The key points are:

- •φ0\\varphi\_\{0\}deletes bundles corresponding to removed edges/variables;
- •φ1\\varphi\_\{1\}deletes reactions involving those bundles;
- •ψ\\psiupdates production rates according to the SP–B table update, while leaving recycling rates unchanged\.

The main theorem of Sergeant–Perthuis–Boitel\(Sergeant\-Perthuis and Boitel,[2025](https://arxiv.org/html/2606.27737#bib.bib4)\)shows that SP–B retractionsr:\(A,H\)→\(A′,H′\)r:\(A,H\)\\to\(A^\{\\prime\},H^\{\\prime\}\)map BP fixed points on\(A,H\)\(A,H\)to BP fixed points on\(A′,H′\)\(A^\{\\prime\},H^\{\\prime\}\)by restricting to the surviving sites and updating factor tables\. We verify that this correspondence is reflected at the CRN level\.

Letπ\\pibe a BP fixed point on\(A,H\)\(A,H\)\. By Theorem[5\.4](https://arxiv.org/html/2606.27737#S5.Thmtheorem4), there exists a positive steady statex∗x^\{\*\}ofΓ=𝒩​\(A,H\)\\Gamma=\\mathcal\{N\}\(A,H\)such that the bundle\-normalized concentrations

\[Bk\]∗∑k′\[Bk′\]∗\\frac\{\[B\_\{k\}\]^\{\*\}\}\{\\sum\_\{k^\{\\prime\}\}\[B\_\{k^\{\\prime\}\}\]^\{\*\}\}equal the corresponding message components ofπ\\pi\.

Letπ′\\pi^\{\\prime\}be the restriction ofπ\\pito\(A′,H′\)\(A^\{\\prime\},H^\{\\prime\}\)as defined by the SP–B retraction\. The Sergeant–Perthuis–Boitel theorem guarantees thatπ′\\pi^\{\\prime\}is a BP fixed point on\(A′,H′\)\(A^\{\\prime\},H^\{\\prime\}\)\.

Now consider the CRN morphism𝒩​\(r\)=\(φ0,φ1,ψ\)\\mathcal\{N\}\(r\)=\(\\varphi\_\{0\},\\varphi\_\{1\},\\psi\)\. For surviving bundlesBB, the morphism:

- •Preserves the bundle structure \(same zero and coordinate species\);
- •Preserves recycling rates \(ρB\\rho\_\{B\}unchanged\);
- •Updates production rates according toψj↦ψj′\\psi\_\{j\}\\mapsto\\psi^\{\\prime\}\_\{j\}\.

The steady\-state equations for surviving bundles inΓ′\\Gamma^\{\\prime\}are:

ρB\[B0\]′​\[Bk\]′=κj→n​∑aψj′​\(𝐤j\)​∏Q′∈VJ∖\{Q\}\[Qa​\(Q′\)′\]′\.\\frac\{\\rho\_\{B\}\}\{\[B\_\{0\}\]^\{\\prime\}\}\[B\_\{k\}\]^\{\\prime\}=\\kappa\_\{j\\to n\}\\sum\_\{a\}\\psi^\{\\prime\}\_\{j\}\(\\mathbf\{k\}^\{j\}\)\\prod\_\{Q^\{\\prime\}\\in V\_\{J\}\\setminus\\\{Q\\\}\}\[Q^\{\\prime\}\_\{a\(Q^\{\\prime\}\)\}\]^\{\\prime\}\.
Since the factor\-table updateψj↦ψj′\\psi\_\{j\}\\mapsto\\psi^\{\\prime\}\_\{j\}is exactly what appears in the SP–B prescription, and sinceπ′\\pi^\{\\prime\}satisfies the BP equations with tablesψj′\\psi^\{\\prime\}\_\{j\}, there exists a positive steady statex′⁣∗x^\{\\prime\*\}ofΓ′\\Gamma^\{\\prime\}whose bundle\-normalized concentrations equal the message components ofπ′\\pi^\{\\prime\}\.

The relationship betweenx∗x^\{\*\}andx′⁣∗x^\{\\prime\*\}is that their restrictions to surviving bundles are related by the concentration rescaling induced by𝒩​\(r\)\\mathcal\{N\}\(r\)\. Specifically, ifBBis a surviving bundle, then

\[Bk\]′⁣∗=sB⋅\[Bk\]∗\[B\_\{k\}\]^\{\\prime\*\}=s\_\{B\}\\cdot\[B\_\{k\}\]^\{\*\}for some positive scalarsBs\_\{B\}that depends on the normalization conventions but preserves the bundle\-normalized ratios\.

This completes the proof that𝒩​\(r\)\\mathcal\{N\}\(r\)transports BP fixed points fromΓ\\GammatoΓ′\\Gamma^\{\\prime\}\. ∎

## Appendix EExamples

These examples illustrate the two main concrete mechanisms behind the paper\. Example[E\.1](https://arxiv.org/html/2606.27737#A5.Thmexample1)shows that the steady\-state equations of the compiled CRN reproduce the BP fixed\-point equations on a small mixed\-cardinality chain\. Example[E\.2](https://arxiv.org/html/2606.27737#A5.Thmexample2)then shows how an SP–B retraction acts on the same instance and how the induced CRN morphism removes the corresponding bundles while preserving the BP semantics on the surviving variables\.

###### Example E\.1\(Chain graph with mixed cardinalities\)\.

Consider variables\{1,2,3\}\\\{1,2,3\\\}with state spacesE1=\{1,2\}E\_\{1\}=\\\{1,2\\\},E2=\{1,2,3\}E\_\{2\}=\\\{1,2,3\\\},E3=\{1,2\}E\_\{3\}=\\\{1,2\\\}, and factorsa,ba,bwithne​\(a\)=\{1,2\}\\mathrm\{ne\}\(a\)=\\\{1,2\\\},ne​\(b\)=\{2,3\}\\mathrm\{ne\}\(b\)=\\\{2,3\\\}\. The factor tables are

ψa:E1×E2→\(0,∞\),ψb:E2×E3→\(0,∞\),\\psi\_\{a\}:E\_\{1\}\\times E\_\{2\}\\to\(0,\\infty\),\\qquad\\psi\_\{b\}:E\_\{2\}\\times E\_\{3\}\\to\(0,\\infty\),soψa\\psi\_\{a\}is2×32\\times 3andψb\\psi\_\{b\}is3×23\\times 2\.

X1X\_\{1\}X2X\_\{2\}X3X\_\{3\}aabbSa→1S^\{a\\to 1\}01122Sa→2S^\{a\\to 2\}0112233Sb→2S^\{b\\to 2\}0112233Sb→3S^\{b\\to 3\}01122P1→aP^\{1\\to a\}01122P2→aP^\{2\\to a\}0112233P2→bP^\{2\\to b\}0112233P3→bP^\{3\\to b\}01122\(a\) Full CRNΓ\\Gamma: 28 species𝒩​\(r\)\\mathcal\{N\}\(r\)SP–BX1X\_\{1\}aaX2X\_\{2\}bbX3X\_\{3\}Sa→1S^\{a\\to 1\}01122Sa→2S^\{a\\to 2\}0112233Sb→2S^\{b\\to 2\}0112233Sb→3S^\{b\\to 3\}01122P1→aP^\{1\\to a\}01122P2→aP^\{2\\to a\}0112233P2→bP^\{2\\to b\}0112233P3→bP^\{3\\to b\}01122\(b\) Reduced CRNΓ′\\Gamma^\{\\prime\}: 14 speciesFigure 6:Bundle structure of the compiled CRNΓ=𝒩​\(ℱ\)\\Gamma=\\mathcal\{N\}\(\\mathcal\{F\}\)\(left\) and the reduced CRNΓ′=𝒩​\(ℱ′\)\\Gamma^\{\\prime\}=\\mathcal\{N\}\(\\mathcal\{F\}^\{\\prime\}\)\(right\) for the mixed\-cardinality chain of Example[E\.1](https://arxiv.org/html/2606.27737#A5.Thmexample1)and its reduction in Example[E\.2](https://arxiv.org/html/2606.27737#A5.Thmexample2)\. Each directed edge of the factor graph produces one sum bundle \(above\) and one product bundle \(below\); dots represent individual chemical species \(zero species indexed0, coordinate species indexed1,…,\|E\|1,\\dots,\|E\|\)\. The SP–B retractionrrremoves variableX1X\_\{1\}and factoraa, deleting the four faded bundles and their associated reactions\. The surviving bundles carry the same BP beliefs onX2X\_\{2\}andX3X\_\{3\}as the full network, with the influence ofX1X\_\{1\}absorbed into the updated rate parameters ofSb→2S^\{b\\to 2\}andSb→3S^\{b\\to 3\}\.
#### Bundles\.

Each directed edge of the factor graph produces one bundle\. The four sum bundles and four product bundles are:

Here subscript0is the zero \(unassigned\) speciesXB,0X\_\{B,0\}, and subscripts1,…,\|EB\|1,\\ldots,\|E\_\{B\}\|are the coordinate species\. Note that bundles associated with variable22have\|EB\|=3\|E\_\{B\}\|=3coordinate species, while those for variables11or33have\|EB\|=2\|E\_\{B\}\|=2\.

#### Catalytic neighbor sets\.

For each sum bundleBB,Cat​\(B\)\\mathrm\{Cat\}\(B\)consists of the product bundles in the same factor class minus the target \(the missing\-one\-neighbor pattern of \(W6\.1\)\):

Cat​\(Sa→1\)\\displaystyle\\mathrm\{Cat\}\(S^\{a\\to 1\}\)=VJa∖\{P1→a\}=\{P2→a\},\\displaystyle=V\_\{J\_\{a\}\}\\setminus\\\{P^\{1\\to a\}\\\}=\\\{P^\{2\\to a\}\\\},Cat​\(Sa→2\)\\displaystyle\\mathrm\{Cat\}\(S^\{a\\to 2\}\)=VJa∖\{P2→a\}=\{P1→a\},\\displaystyle=V\_\{J\_\{a\}\}\\setminus\\\{P^\{2\\to a\}\\\}=\\\{P^\{1\\to a\}\\\},Cat​\(Sb→2\)\\displaystyle\\mathrm\{Cat\}\(S^\{b\\to 2\}\)=VJb∖\{P2→b\}=\{P3→b\},\\displaystyle=V\_\{J\_\{b\}\}\\setminus\\\{P^\{2\\to b\}\\\}=\\\{P^\{3\\to b\}\\\},Cat​\(Sb→3\)\\displaystyle\\mathrm\{Cat\}\(S^\{b\\to 3\}\)=VJb∖\{P3→b\}=\{P2→b\}\.\\displaystyle=V\_\{J\_\{b\}\}\\setminus\\\{P^\{3\\to b\}\\\}=\\\{P^\{2\\to b\}\\\}\.For each product bundlePP,Cat​\(P\)\\mathrm\{Cat\}\(P\)consists of the sum bundles targeting the same variable from*other*factors:

Cat​\(P1→a\)\\displaystyle\\mathrm\{Cat\}\(P^\{1\\to a\}\)=∅\\displaystyle=\\varnothing\(variable1touches only factora\),\\displaystyle\\text\{\(variable $1$ touches only factor $a$\)\},Cat​\(P2→a\)\\displaystyle\\mathrm\{Cat\}\(P^\{2\\to a\}\)=\{Sb→2\}\\displaystyle=\\\{S^\{b\\to 2\}\\\}\(sum bundle from factorbtargeting var2\),\\displaystyle\\text\{\(sum bundle from factor $b$ targeting var $2$\)\},Cat​\(P2→b\)\\displaystyle\\mathrm\{Cat\}\(P^\{2\\to b\}\)=\{Sa→2\}\\displaystyle=\\\{S^\{a\\to 2\}\\\}\(sum bundle from factoratargeting var2\),\\displaystyle\\text\{\(sum bundle from factor $a$ targeting var $2$\)\},Cat​\(P3→b\)\\displaystyle\\mathrm\{Cat\}\(P^\{3\\to b\}\)=∅\\displaystyle=\\varnothing\(variable3touches only factorb\)\.\\displaystyle\\text\{\(variable $3$ touches only factor $b$\)\}\.

#### Factor classes and variable classes\.

- •Ja=\{Sa→1,Sa→2\}J\_\{a\}=\\\{S^\{a\\to 1\},\\,S^\{a\\to 2\}\\\}withVJa=\{P1→a,P2→a\}V\_\{J\_\{a\}\}=\\\{P^\{1\\to a\},\\,P^\{2\\to a\}\\\}; factor tableψa:E1×E2→\(0,∞\)\\psi\_\{a\}:E\_\{1\}\\times E\_\{2\}\\to\(0,\\infty\), a2×32\\times 3matrix\.
- •Jb=\{Sb→2,Sb→3\}J\_\{b\}=\\\{S^\{b\\to 2\},\\,S^\{b\\to 3\}\\\}withVJb=\{P2→b,P3→b\}V\_\{J\_\{b\}\}=\\\{P^\{2\\to b\},\\,P^\{3\\to b\}\\\}; factor tableψb:E2×E3→\(0,∞\)\\psi\_\{b\}:E\_\{2\}\\times E\_\{3\}\\to\(0,\\infty\), a3×23\\times 2matrix\.

The cross\-link relation∼\\simon product bundles \(Definition 5\.1\) gives:P2→a∼P2→bP^\{2\\to a\}\\sim P^\{2\\to b\}viaSb→2S^\{b\\to 2\}\(targetP2→bP^\{2\\to b\}, catalyzesP2→aP^\{2\\to a\}\) and symmetrically viaSa→2S^\{a\\to 2\}\. No other pairs are cross\-linked\. The variable classes are therefore

\{P1→a\},\{P2→a,P2→b\},\{P3→b\},\\\{P^\{1\\to a\}\\\},\\quad\\\{P^\{2\\to a\},\\,P^\{2\\to b\}\\\},\\quad\\\{P^\{3\\to b\}\\\},with cardinalities2,3,22,3,2, matching variables1,2,31,2,3\.

#### Napp compilation𝓝​\(𝓕\)\\boldsymbol\{\\mathcal\{N\}\(\\mathcal\{F\}\)\}: species\.

Each bundleBBwith\|EB\|\|E\_\{B\}\|coordinate species contributes\|EB\|\+1\|E\_\{B\}\|\+1species \(one zero species \+\|EB\|\|E\_\{B\}\|coordinate species\)\. The total species count is

\(2\+1\)⏟Sa→1\+\(3\+1\)⏟Sa→2\+\(3\+1\)⏟Sb→2\+\(2\+1\)⏟Sb→3\+\(2\+1\)⏟P1→a\+\(3\+1\)⏟P2→a\+\(3\+1\)⏟P2→b\+\(2\+1\)⏟P3→b=𝟐𝟖​species\.\\underbrace\{\(2\{\+\}1\)\}\_\{S^\{a\\to 1\}\}\+\\underbrace\{\(3\{\+\}1\)\}\_\{S^\{a\\to 2\}\}\+\\underbrace\{\(3\{\+\}1\)\}\_\{S^\{b\\to 2\}\}\+\\underbrace\{\(2\{\+\}1\)\}\_\{S^\{b\\to 3\}\}\+\\underbrace\{\(2\{\+\}1\)\}\_\{P^\{1\\to a\}\}\+\\underbrace\{\(3\{\+\}1\)\}\_\{P^\{2\\to a\}\}\+\\underbrace\{\(3\{\+\}1\)\}\_\{P^\{2\\to b\}\}\+\\underbrace\{\(2\{\+\}1\)\}\_\{P^\{3\\to b\}\}=\\mathbf\{28\\textbf\{ species\}\}\.

#### Recycling reactions\.

For each bundleBBand each coordinate speciesBkB\_\{k\},k≥1k\\geq 1\(Definition[D\.1](https://arxiv.org/html/2606.27737#A4.SS1), step 2\):

Bk→κrB0\.B\_\{k\}\\xrightarrow\{\\;\\kappa\_\{r\}\\;\}B\_\{0\}\.The number of recycling reactions per bundle equals\|EB\|\|E\_\{B\}\|, giving2\+3\+3\+2\+2\+3\+3\+2=𝟐𝟎2\+3\+3\+2\+2\+3\+3\+2=\\mathbf\{20\}recycling reactions\. We list them explicitly:

*Sum\-bundle recycling \(10 reactions\):*

S1a→1→κrS0a→1,\\displaystyle S^\{a\\to 1\}\_\{1\}\\xrightarrow\{\\kappa\_\{r\}\}S^\{a\\to 1\}\_\{0\},\\qquadS2a→1→κrS0a→1,\\displaystyle S^\{a\\to 1\}\_\{2\}\\xrightarrow\{\\kappa\_\{r\}\}S^\{a\\to 1\}\_\{0\},S1a→2→κrS0a→2,\\displaystyle S^\{a\\to 2\}\_\{1\}\\xrightarrow\{\\kappa\_\{r\}\}S^\{a\\to 2\}\_\{0\},\\quadS2a→2→κrS0a→2,\\displaystyle S^\{a\\to 2\}\_\{2\}\\xrightarrow\{\\kappa\_\{r\}\}S^\{a\\to 2\}\_\{0\},\\quadS3a→2→κrS0a→2,\\displaystyle S^\{a\\to 2\}\_\{3\}\\xrightarrow\{\\kappa\_\{r\}\}S^\{a\\to 2\}\_\{0\},S1b→2→κrS0b→2,\\displaystyle S^\{b\\to 2\}\_\{1\}\\xrightarrow\{\\kappa\_\{r\}\}S^\{b\\to 2\}\_\{0\},\\quadS2b→2→κrS0b→2,\\displaystyle S^\{b\\to 2\}\_\{2\}\\xrightarrow\{\\kappa\_\{r\}\}S^\{b\\to 2\}\_\{0\},\\quadS3b→2→κrS0b→2,\\displaystyle S^\{b\\to 2\}\_\{3\}\\xrightarrow\{\\kappa\_\{r\}\}S^\{b\\to 2\}\_\{0\},S1b→3→κrS0b→3,\\displaystyle S^\{b\\to 3\}\_\{1\}\\xrightarrow\{\\kappa\_\{r\}\}S^\{b\\to 3\}\_\{0\},\\qquadS2b→3→κrS0b→3\.\\displaystyle S^\{b\\to 3\}\_\{2\}\\xrightarrow\{\\kappa\_\{r\}\}S^\{b\\to 3\}\_\{0\}\.
*Product\-bundle recycling \(10 reactions\):*

P11→a→κrP01→a,\\displaystyle P^\{1\\to a\}\_\{1\}\\xrightarrow\{\\kappa\_\{r\}\}P^\{1\\to a\}\_\{0\},\\qquadP21→a→κrP01→a,\\displaystyle P^\{1\\to a\}\_\{2\}\\xrightarrow\{\\kappa\_\{r\}\}P^\{1\\to a\}\_\{0\},P12→a→κrP02→a,\\displaystyle P^\{2\\to a\}\_\{1\}\\xrightarrow\{\\kappa\_\{r\}\}P^\{2\\to a\}\_\{0\},\\quadP22→a→κrP02→a,\\displaystyle P^\{2\\to a\}\_\{2\}\\xrightarrow\{\\kappa\_\{r\}\}P^\{2\\to a\}\_\{0\},\\quadP32→a→κrP02→a,\\displaystyle P^\{2\\to a\}\_\{3\}\\xrightarrow\{\\kappa\_\{r\}\}P^\{2\\to a\}\_\{0\},P12→b→κrP02→b,\\displaystyle P^\{2\\to b\}\_\{1\}\\xrightarrow\{\\kappa\_\{r\}\}P^\{2\\to b\}\_\{0\},\\quadP22→b→κrP02→b,\\displaystyle P^\{2\\to b\}\_\{2\}\\xrightarrow\{\\kappa\_\{r\}\}P^\{2\\to b\}\_\{0\},\\quadP32→b→κrP02→b,\\displaystyle P^\{2\\to b\}\_\{3\}\\xrightarrow\{\\kappa\_\{r\}\}P^\{2\\to b\}\_\{0\},P13→b→κrP03→b,\\displaystyle P^\{3\\to b\}\_\{1\}\\xrightarrow\{\\kappa\_\{r\}\}P^\{3\\to b\}\_\{0\},\\qquadP23→b→κrP03→b\.\\displaystyle P^\{3\\to b\}\_\{2\}\\xrightarrow\{\\kappa\_\{r\}\}P^\{3\\to b\}\_\{0\}\.

#### Sum\-bundle production reactions\.

For each directed edgej→nj\\to nand each joint assignmentkj∈∏n′∈ne​\(j\)En′k^\{j\}\\in\\prod\_\{n^\{\\prime\}\\in\\mathrm\{ne\}\(j\)\}E\_\{n^\{\\prime\}\}with\(kj\)n=k\(k^\{j\}\)\_\{n\}=k\(Definition[D\.1](https://arxiv.org/html/2606.27737#A4.SS1), step 3\):

S0j→n\+∑n′∈ne​\(j\)∖nP\(kj\)n′n′→j→ψj​\(kj\)Skj→n\+∑n′∈ne​\(j\)∖nP\(kj\)n′n′→j\.S^\{j\\to n\}\_\{0\}\\;\+\\\!\\\!\\sum\_\{n^\{\\prime\}\\in\\mathrm\{ne\}\(j\)\\setminus n\}\\\!\\\!P^\{n^\{\\prime\}\\to j\}\_\{\(k^\{j\}\)\_\{n^\{\\prime\}\}\}\\;\\xrightarrow\{\\;\\psi\_\{j\}\(k^\{j\}\)\\;\}S^\{j\\to n\}\_\{k\}\\;\+\\\!\\\!\\sum\_\{n^\{\\prime\}\\in\\mathrm\{ne\}\(j\)\\setminus n\}\\\!\\\!P^\{n^\{\\prime\}\\to j\}\_\{\(k^\{j\}\)\_\{n^\{\\prime\}\}\}\.
Sum bundleSa→1S^\{a\\to 1\}\(target: variable11; catalyst:P2→aP^\{2\\to a\}\)\. The assignment iska=\(k1,k2\)∈E1×E2k^\{a\}=\(k\_\{1\},k\_\{2\}\)\\in E\_\{1\}\\times E\_\{2\}with output statek=k1k=k\_\{1\}\. There are\|E1\|×\|E2\|=2×3=6\|E\_\{1\}\|\\times\|E\_\{2\}\|=2\\times 3=6reactions, one per entry ofψa\\psi\_\{a\}:

Observe thatP2→aP^\{2\\to a\}species appear on both sides \(zero net stoichiometry\)—they are catalysts\. The net stoichiometry of each reaction isS0a→1→Ska→1S^\{a\\to 1\}\_\{0\}\\to S^\{a\\to 1\}\_\{k\}, supported entirely in the bundleSa→1S^\{a\\to 1\}, verifying \(W1\) and \(W3\)\.

Sum bundleSa→2S^\{a\\to 2\}\(target: variable22; catalyst:P1→aP^\{1\\to a\}\)\. Assignmentka=\(k1,k2\)k^\{a\}=\(k\_\{1\},k\_\{2\}\)with outputk=k2k=k\_\{2\}\. There are2×3=62\\times 3=6reactions:

Here the catalystP1→aP^\{1\\to a\}has only\|E1\|=2\|E\_\{1\}\|=2coordinate species, while the target bundleSa→2S^\{a\\to 2\}has\|ESa→2\|=3\|E\_\{S^\{a\\to 2\}\}\|=3coordinate species\. This is where mixed cardinalities become visible in the reaction stoichiometry: the number of reactions producing a given coordinateSka→2S^\{a\\to 2\}\_\{k\}equals\|E1\|=2\|E\_\{1\}\|=2\(one per state of the catalyst\), whereas each reaction producingSka→1S^\{a\\to 1\}\_\{k\}\(above\) had\|E2\|=3\|E\_\{2\}\|=3such variants\.

Sum bundleSb→2S^\{b\\to 2\}\(target: variable22; catalyst:P3→bP^\{3\\to b\}\)\. Assignmentkb=\(k2,k3\)k^\{b\}=\(k\_\{2\},k\_\{3\}\)with outputk=k2k=k\_\{2\}\. There are3×2=63\\times 2=6reactions:

Sum bundleSb→3S^\{b\\to 3\}\(target: variable33; catalyst:P2→bP^\{2\\to b\}\)\. Assignmentkb=\(k2,k3\)k^\{b\}=\(k\_\{2\},k\_\{3\}\)with outputk=k3k=k\_\{3\}\. There are3×2=63\\times 2=6reactions:

Sum\-bundle production count:6\+6\+6\+6=𝟐𝟒6\+6\+6\+6=\\mathbf\{24\}reactions\.

#### Product\-bundle production reactions\.

For each variablenn, each factorj∈ne​\(n\)j\\in\\mathrm\{ne\}\(n\), and each statek∈Enk\\in E\_\{n\}\(Definition[D\.1](https://arxiv.org/html/2606.27737#A4.SS1), step 4\):

P0n→j\+∑j′∈ne​\(n\)∖jSkj′→n→κprodPkn→j\+∑j′∈ne​\(n\)∖jSkj′→n\.P^\{n\\to j\}\_\{0\}\\;\+\\\!\\\!\\sum\_\{j^\{\\prime\}\\in\\mathrm\{ne\}\(n\)\\setminus j\}\\\!\\\!S^\{j^\{\\prime\}\\to n\}\_\{k\}\\;\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{n\\to j\}\_\{k\}\\;\+\\\!\\\!\\sum\_\{j^\{\\prime\}\\in\\mathrm\{ne\}\(n\)\\setminus j\}\\\!\\\!S^\{j^\{\\prime\}\\to n\}\_\{k\}\.When\|ne​\(n\)∖j\|=0\|\\mathrm\{ne\}\(n\)\\setminus j\|=0\(i\.e\. the variable touches only one factor\), there are no catalysts and the reaction is simplyP0n→j→κprodPkn→jP^\{n\\to j\}\_\{0\}\\xrightarrow\{\\kappa\_\{\\mathrm\{prod\}\}\}P^\{n\\to j\}\_\{k\}\.

Product bundleP1→aP^\{1\\to a\}\(ne​\(1\)=\{a\}\\mathrm\{ne\}\(1\)=\\\{a\\\}, sone​\(1\)∖a=∅\\mathrm\{ne\}\(1\)\\setminus a=\\varnothing; no catalysts\)\. Two reactions \(\|E1\|=2\|E\_\{1\}\|=2\):

P01→a→κprodP11→a,P01→a→κprodP21→a\.P^\{1\\to a\}\_\{0\}\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{1\\to a\}\_\{1\},\\qquad P^\{1\\to a\}\_\{0\}\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{1\\to a\}\_\{2\}\.
Product bundleP2→aP^\{2\\to a\}\(ne​\(2\)=\{a,b\}\\mathrm\{ne\}\(2\)=\\\{a,b\\\}, sone​\(2\)∖a=\{b\}\\mathrm\{ne\}\(2\)\\setminus a=\\\{b\\\}; catalyst:Sb→2S^\{b\\to 2\}\)\. Three reactions \(\|E2\|=3\|E\_\{2\}\|=3\):

P02→a\+S1b→2\\displaystyle P^\{2\\to a\}\_\{0\}\+S^\{b\\to 2\}\_\{1\}→κprodP12→a\+S1b→2,\\displaystyle\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{2\\to a\}\_\{1\}\+S^\{b\\to 2\}\_\{1\},P02→a\+S2b→2\\displaystyle P^\{2\\to a\}\_\{0\}\+S^\{b\\to 2\}\_\{2\}→κprodP22→a\+S2b→2,\\displaystyle\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{2\\to a\}\_\{2\}\+S^\{b\\to 2\}\_\{2\},P02→a\+S3b→2\\displaystyle P^\{2\\to a\}\_\{0\}\+S^\{b\\to 2\}\_\{3\}→κprodP32→a\+S3b→2\.\\displaystyle\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{2\\to a\}\_\{3\}\+S^\{b\\to 2\}\_\{3\}\.
Here the bijectionθP2→a,Sb→2\\theta\_\{P^\{2\\to a\},\\,S^\{b\\to 2\}\}from \(W5\) is the identity onE2=\{1,2,3\}E\_\{2\}=\\\{1,2,3\\\}: coordinatekkof the catalystSb→2S^\{b\\to 2\}produces coordinatekkofP2→aP^\{2\\to a\}\. Both bundles have cardinality33because both are associated with variable22\.

Product bundleP2→bP^\{2\\to b\}\(ne​\(2\)∖b=\{a\}\\mathrm\{ne\}\(2\)\\setminus b=\\\{a\\\}; catalyst:Sa→2S^\{a\\to 2\}\)\. Three reactions:

P02→b\+S1a→2\\displaystyle P^\{2\\to b\}\_\{0\}\+S^\{a\\to 2\}\_\{1\}→κprodP12→b\+S1a→2,\\displaystyle\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{2\\to b\}\_\{1\}\+S^\{a\\to 2\}\_\{1\},P02→b\+S2a→2\\displaystyle P^\{2\\to b\}\_\{0\}\+S^\{a\\to 2\}\_\{2\}→κprodP22→b\+S2a→2,\\displaystyle\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{2\\to b\}\_\{2\}\+S^\{a\\to 2\}\_\{2\},P02→b\+S3a→2\\displaystyle P^\{2\\to b\}\_\{0\}\+S^\{a\\to 2\}\_\{3\}→κprodP32→b\+S3a→2\.\\displaystyle\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{2\\to b\}\_\{3\}\+S^\{a\\to 2\}\_\{3\}\.
Product bundleP3→bP^\{3\\to b\}\(ne​\(3\)=\{b\}\\mathrm\{ne\}\(3\)=\\\{b\\\}, so no catalysts\)\. Two reactions:

P03→b→κprodP13→b,P03→b→κprodP23→b\.P^\{3\\to b\}\_\{0\}\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{3\\to b\}\_\{1\},\\qquad P^\{3\\to b\}\_\{0\}\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{3\\to b\}\_\{2\}\.
Product\-bundle production count:2\+3\+3\+2=𝟏𝟎2\+3\+3\+2=\\mathbf\{10\}reactions\.

#### CRN summary\.

Species:28Recycling reactions:20Sum\-bundle production reactions:24Product\-bundle production reactions:10Total reactions:𝟓𝟒\\boxed\{\\begin\{aligned\} &\\text\{Species:\}&&28\\\\ &\\text\{Recycling reactions:\}&&20\\\\ &\\text\{Sum\-bundle production reactions:\}&&24\\\\ &\\text\{Product\-bundle production reactions:\}&&10\\\\ &\\text\{Total reactions:\}&&\\mathbf\{54\}\\end\{aligned\}\}

#### Verification of the recognition conditions\.

We verify that the compiled CRN𝒩​\(ℱ\)\\mathcal\{N\}\(\\mathcal\{F\}\)satisfies \(W1\)–\(W6\) and \(R1\)\.

- \(W1\)Every reaction has net stoichiometry supported in a single bundle\. Recycling reactions change onlyBk→B0B\_\{k\}\\to B\_\{0\}within their bundle\. Sum\- and product\-production reactions convertB0→BkB\_\{0\}\\to B\_\{k\}within a single bundle; catalytic species appear on both sides with zero net change\.
- \(W2\)Each bundle has a distinguished zero species \(S0a→1S^\{a\\to 1\}\_\{0\},P01→aP^\{1\\to a\}\_\{0\}, etc\.\)\.
- \(W3\)All internal net stoichiometries are of the form−XB,0\+XB,k\-X\_\{B,0\}\+X\_\{B,k\}\(production\) orXB,0−XB,kX\_\{B,0\}\-X\_\{B,k\}\(recycling\)\.
- \(W4\)ConsiderSa→1S^\{a\\to 1\}withCat​\(Sa→1\)=\{P2→a\}\\mathrm\{Cat\}\(S^\{a\\to 1\}\)=\\\{P^\{2\\to a\}\\\}\. Its positive production is dd​t​\[Ska→1\]\+=\[S0a→1\]​∑k2∈E2ψa​\(k,k2\)​\[Pk22→a\],\\frac\{d\}\{dt\}\[S^\{a\\to 1\}\_\{k\}\]^\{\+\}=\[S^\{a\\to 1\}\_\{0\}\]\\sum\_\{k\_\{2\}\\in E\_\{2\}\}\\psi\_\{a\}\(k,k\_\{2\}\)\\,\[P^\{2\\to a\}\_\{k\_\{2\}\}\],which is multilinear in the catalytic coordinates, with assignment tableCSa→1​\(k;k2\)=ψa​\(k,k2\)C\_\{S^\{a\\to 1\}\}\(k;\\,k\_\{2\}\)=\\psi\_\{a\}\(k,k\_\{2\}\), confirming the separability requirement\. The other sum bundles are analogous\.
- \(W5\)ConsiderP2→aP^\{2\\to a\}withCat​\(P2→a\)=\{Sb→2\}\\mathrm\{Cat\}\(P^\{2\\to a\}\)=\\\{S^\{b\\to 2\}\\\}\. For eachk∈E2k\\in E\_\{2\}, the unique supporting assignment isa\(k\)​\(Sb→2\)=ka^\{\(k\)\}\(S^\{b\\to 2\}\)=k\(the identity bijection\), and the rate isκprod\\kappa\_\{\\mathrm\{prod\}\}independent ofkk\. This is the diagonal/bijection structure\. The endpoint bundlesP1→aP^\{1\\to a\}andP3→bP^\{3\\to b\}haveCat=∅\\mathrm\{Cat\}=\\varnothing, so \(W5\) holds vacuously\.
- \(W6\)The partition of sum bundles into factor classes isJa=\{Sa→1,Sa→2\}J\_\{a\}=\\\{S^\{a\\to 1\},S^\{a\\to 2\}\\\},Jb=\{Sb→2,Sb→3\}J\_\{b\}=\\\{S^\{b\\to 2\},S^\{b\\to 3\}\\\}\. ForJaJ\_\{a\}:\|Ja\|=\|VJa\|=2\|J\_\{a\}\|=\|V\_\{J\_\{a\}\}\|=2;Cat​\(Sa→1\)=VJa∖\{P1→a\}=\{P2→a\}\\mathrm\{Cat\}\(S^\{a\\to 1\}\)=V\_\{J\_\{a\}\}\\setminus\\\{P^\{1\\to a\}\\\}=\\\{P^\{2\\to a\}\\\}\(missing\-one\-neighbor\)\. The state\-space identificationιJa→P1→a:E1→ESa→1\\iota\_\{J\_\{a\}\\to P^\{1\\to a\}\}:E\_\{1\}\\to E\_\{S^\{a\\to 1\}\}is the identity, andιJa→P2→a:E2→ESa→2\\iota\_\{J\_\{a\}\\to P^\{2\\to a\}\}:E\_\{2\}\\to E\_\{S^\{a\\to 2\}\}is the identity, confirming \(W6\.2\) with\|E1\|=2≠3=\|E2\|\|E\_\{1\}\|=2\\neq 3=\|E\_\{2\}\|—the mixed\-cardinality case\. The clamped\-table property \(W6\.3\) holds:CSa→1​\(k;k2\)=ψa​\(k,k2\)C\_\{S^\{a\\to 1\}\}\(k;\\,k\_\{2\}\)=\\psi\_\{a\}\(k,k\_\{2\}\)as computed above\.
- \(R1\)All recycling reactions use the global rateκr\\kappa\_\{r\}, and all product\-production reactions use the global rateκprod\\kappa\_\{\\mathrm\{prod\}\}, both independent of the statekk\.

#### Steady\-state equations \(Theorem 5\.4\)\.

At a positive steady state of𝒩​\(ℱ\)\\mathcal\{N\}\(\\mathcal\{F\}\), the sum\-bundle concentrations satisfy

κr\[S0a→1\]​\[Ska→1\]\\displaystyle\\frac\{\\kappa\_\{r\}\}\{\[S^\{a\\to 1\}\_\{0\}\]\}\\,\[S^\{a\\to 1\}\_\{k\}\]=∑k2=13ψa​\(k,k2\)​\[Pk22→a\],\\displaystyle=\\sum\_\{k\_\{2\}=1\}^\{3\}\\psi\_\{a\}\(k,k\_\{2\}\)\\,\[P^\{2\\to a\}\_\{k\_\{2\}\}\],k\\displaystyle k∈\{1,2\},\\displaystyle\\in\\\{1,2\\\},\(7\)κr\[S0a→2\]​\[Ska→2\]\\displaystyle\\frac\{\\kappa\_\{r\}\}\{\[S^\{a\\to 2\}\_\{0\}\]\}\\,\[S^\{a\\to 2\}\_\{k\}\]=∑k1=12ψa​\(k1,k\)​\[Pk11→a\],\\displaystyle=\\sum\_\{k\_\{1\}=1\}^\{2\}\\psi\_\{a\}\(k\_\{1\},k\)\\,\[P^\{1\\to a\}\_\{k\_\{1\}\}\],k\\displaystyle k∈\{1,2,3\},\\displaystyle\\in\\\{1,2,3\\\},\(8\)κr\[S0b→2\]​\[Skb→2\]\\displaystyle\\frac\{\\kappa\_\{r\}\}\{\[S^\{b\\to 2\}\_\{0\}\]\}\\,\[S^\{b\\to 2\}\_\{k\}\]=∑k3=12ψb​\(k,k3\)​\[Pk33→b\],\\displaystyle=\\sum\_\{k\_\{3\}=1\}^\{2\}\\psi\_\{b\}\(k,k\_\{3\}\)\\,\[P^\{3\\to b\}\_\{k\_\{3\}\}\],k\\displaystyle k∈\{1,2,3\},\\displaystyle\\in\\\{1,2,3\\\},\(9\)κr\[S0b→3\]​\[Skb→3\]\\displaystyle\\frac\{\\kappa\_\{r\}\}\{\[S^\{b\\to 3\}\_\{0\}\]\}\\,\[S^\{b\\to 3\}\_\{k\}\]=∑k2=13ψb​\(k2,k\)​\[Pk22→b\],\\displaystyle=\\sum\_\{k\_\{2\}=1\}^\{3\}\\psi\_\{b\}\(k\_\{2\},k\)\\,\[P^\{2\\to b\}\_\{k\_\{2\}\}\],k\\displaystyle k∈\{1,2\},\\displaystyle\\in\\\{1,2\\\},\(10\)and the product\-bundle concentrations satisfy

κrκprod​\[P01→a\]​\[Pk1→a\]\\displaystyle\\frac\{\\kappa\_\{r\}\}\{\\kappa\_\{\\mathrm\{prod\}\}\\,\[P^\{1\\to a\}\_\{0\}\]\}\\,\[P^\{1\\to a\}\_\{k\}\]=1,\\displaystyle=1,k\\displaystyle k∈\{1,2\},\\displaystyle\\in\\\{1,2\\\},\(11\)κrκprod​\[P02→a\]​\[Pk2→a\]\\displaystyle\\frac\{\\kappa\_\{r\}\}\{\\kappa\_\{\\mathrm\{prod\}\}\\,\[P^\{2\\to a\}\_\{0\}\]\}\\,\[P^\{2\\to a\}\_\{k\}\]=\[Skb→2\],\\displaystyle=\[S^\{b\\to 2\}\_\{k\}\],k\\displaystyle k∈\{1,2,3\},\\displaystyle\\in\\\{1,2,3\\\},\(12\)κrκprod​\[P02→b\]​\[Pk2→b\]\\displaystyle\\frac\{\\kappa\_\{r\}\}\{\\kappa\_\{\\mathrm\{prod\}\}\\,\[P^\{2\\to b\}\_\{0\}\]\}\\,\[P^\{2\\to b\}\_\{k\}\]=\[Ska→2\],\\displaystyle=\[S^\{a\\to 2\}\_\{k\}\],k\\displaystyle k∈\{1,2,3\},\\displaystyle\\in\\\{1,2,3\\\},\(13\)κrκprod​\[P03→b\]​\[Pk3→b\]\\displaystyle\\frac\{\\kappa\_\{r\}\}\{\\kappa\_\{\\mathrm\{prod\}\}\\,\[P^\{3\\to b\}\_\{0\}\]\}\\,\[P^\{3\\to b\}\_\{k\}\]=1,\\displaystyle=1,k\\displaystyle k∈\{1,2\}\.\\displaystyle\\in\\\{1,2\\\}\.\(14\)
Equations \([11](https://arxiv.org/html/2606.27737#A5.E11)\) and \([14](https://arxiv.org/html/2606.27737#A5.E14)\) say thatP1→aP^\{1\\to a\}andP3→bP^\{3\\to b\}are uniform at steady state \(no incoming sum messages, since variables11and33each touch only one factor\)\. Equations \([12](https://arxiv.org/html/2606.27737#A5.E12)\) and \([13](https://arxiv.org/html/2606.27737#A5.E13)\) express the product\-message update for variable22: the message from variable22to factoraais proportional to the sum message arriving from factorbb, and vice versa, matching the BP product\-message equation \([9](https://arxiv.org/html/2606.27737#A5.E9)\) withne​\(2\)∖a=\{b\}\\mathrm\{ne\}\(2\)\\setminus a=\\\{b\\\}\.

Together, equations \([7](https://arxiv.org/html/2606.27737#A5.E7)\)–\([14](https://arxiv.org/html/2606.27737#A5.E14)\) are exactly the sum\-product BP fixed\-point equations on the chain11–aa–22–bb–33with mixed cardinalities\(\|E1\|,\|E2\|,\|E3\|\)=\(2,3,2\)\(\|E\_\{1\}\|,\|E\_\{2\}\|,\|E\_\{3\}\|\)=\(2,3,2\), confirming Theorem 5\.4 on this instance\.

#### Remark \(effect of mixed cardinalities on CRN size\)\.

In the uniform\-cardinality case\|Ei\|=K\|E\_\{i\}\|=Kfor allii, a chain onnnvariables produces4​\(n−1\)4\(n\{\-\}1\)bundles each of sizeK\+1K\{\+\}1, giving4​\(n−1\)​\(K\+1\)4\(n\{\-\}1\)\(K\{\+\}1\)species and4​\(n−1\)​K\+\(n−1\)​K2\+\(n−1\)​K24\(n\{\-\}1\)K\+\(n\{\-\}1\)K^\{2\}\+\(n\{\-\}1\)K^\{2\}reactions \(recycling \+ sum\-production \+ product\-production\)\. With mixed cardinalities, these counts depend on the per\-variable\|Ei\|\|E\_\{i\}\|and the per\-factor product∏i∈ne​\(j\)\|Ei\|\\prod\_\{i\\in\\mathrm\{ne\}\(j\)\}\|E\_\{i\}\|\. For our instance, the asymmetry\|E2\|=3\|E\_\{2\}\|=3versus\|E1\|=\|E3\|=2\|E\_\{1\}\|=\|E\_\{3\}\|=2is visible in the reaction tables above: factoraagenerates\|E1\|⋅\|E2\|=6\|E\_\{1\}\|\\cdot\|E\_\{2\}\|=6sum\-production reactions per sum bundle, while a hypothetical uniform binary chain would generate2⋅2=42\\cdot 2=4\. The example below shows that the reduction deletes entire message bundles associated with retractable tendrils while leaving the core computation unchanged\. On this small instance, the reduced CRN therefore has fewer species and reactions but still computes the same BP information on the surviving variables\.

###### Example E\.2\(Reduction of the mixed\-cardinality chain\)\.

We continue Example[E\.1](https://arxiv.org/html/2606.27737#A5.Thmexample1)and apply SP–B retractions to the chain11–aa–22–bb–33, tracking the induced CRN morphism𝒩​\(r\)\\mathcal\{N\}\(r\)at each step\.

#### Poset structure\.

The associated posetAAhas five elements\{1,2,3,a,b\}\\\{1,2,3,a,b\\\}with cover relations1<a1<a,2<a2<a,2<b2<b,3<b3<b\. We check for linear and colinear points\.

*Variable11*has upper covers\{a\}\\\{a\\\}\. Candidate1↑=a1^\{\\uparrow\}=a: for everyx≥1x\\geq 1inAA, the only suchxxisaa, anda≥aa\\geq a\. So the condition\(∀x≥1\)​x≥a\(\\forall x\\geq 1\)\\;x\\geq aholds\. Thereforevariable11is linearwith1↑=a1^\{\\uparrow\}=a\.

*Variable33*has upper covers\{b\}\\\{b\\\}\. By the same argument,variable33is linearwith3↑=b3^\{\\uparrow\}=b\.

*Variable22*has upper covers\{a,b\}\\\{a,b\\\}\. For candidate2↑=a2^\{\\uparrow\}=a: we needb≥ab\\geq a, butb≱ab\\not\\geq a\. For candidate2↑=b2^\{\\uparrow\}=b: we needa≥ba\\geq b, buta≱ba\\not\\geq b\. So variable22isnot linear\.

No factor is colinear in the original poset \(bothaaandbbare binary factors, and neither scope is contained in the other\)\.

We choose to retract variable11\.

#### Step 1: Linear retraction of variable11\.

##### Factor\-graph level\.

Removing variable11from the poset givesA′=\{2,3,a,b\}A^\{\\prime\}=\\\{2,3,a,b\\\}with relations2<a2<a,2<b2<b,3<b3<b\. Factoraanow hasne​\(a\)=\{2\}\\mathrm\{ne\}\(a\)=\\\{2\\\}\(it has become unary\)\. By the SP–B linear\-retraction rule \(Appendix[D\.1](https://arxiv.org/html/2606.27737#A4.SS1)\), the Hamiltonians on survivors are unchanged, so the surviving factor tables are:

ψa′​\(k2\)\\displaystyle\\psi^\{\\prime\}\_\{a\}\(k\_\{2\}\)=∑k1∈E1ψa​\(k1,k2\)=ψa​\(1,k2\)\+ψa​\(2,k2\),k2∈\{1,2,3\},\\displaystyle=\\sum\_\{k\_\{1\}\\in E\_\{1\}\}\\psi\_\{a\}\(k\_\{1\},k\_\{2\}\)=\\psi\_\{a\}\(1,k\_\{2\}\)\+\\psi\_\{a\}\(2,k\_\{2\}\),\\qquad k\_\{2\}\\in\\\{1,2,3\\\},ψb′\\displaystyle\\psi^\{\\prime\}\_\{b\}=ψb\(unchanged\)\.\\displaystyle=\\psi\_\{b\}\\quad\\text\{\(unchanged\)\}\.The marginalization overk1k\_\{1\}arises because variable11had no other incident factors, soP1→aP^\{1\\to a\}was uniform at steady state \(cf\. equation \([11](https://arxiv.org/html/2606.27737#A5.E11)\)\)\.

##### CRN morphism𝒩​\(r1\)=\(ϕ0,ϕ1,ψ\)\\mathcal\{N\}\(r\_\{1\}\)=\(\\phi\_\{0\},\\phi\_\{1\},\\psi\)\.

- ϕ0\\phi\_\{0\}:Deleteall bundles involving variable11: Sa→1,P1→a\.S^\{a\\to 1\},\\quad P^\{1\\to a\}\.This removes3\+3=63\+3=6species:S0a→1,S1a→1,S2a→1S^\{a\\to 1\}\_\{0\},S^\{a\\to 1\}\_\{1\},S^\{a\\to 1\}\_\{2\}andP01→a,P11→a,P21→aP^\{1\\to a\}\_\{0\},P^\{1\\to a\}\_\{1\},P^\{1\\to a\}\_\{2\}\.
- ϕ1\\phi\_\{1\}:Deleteall reactions involving deleted species: - –Recycling ofSa→1S^\{a\\to 1\}: 2 reactions \(S1a→1→S0a→1S^\{a\\to 1\}\_\{1\}\\to S^\{a\\to 1\}\_\{0\},S2a→1→S0a→1S^\{a\\to 1\}\_\{2\}\\to S^\{a\\to 1\}\_\{0\}\)\. - –Recycling ofP1→aP^\{1\\to a\}: 2 reactions \(P11→a→P01→aP^\{1\\to a\}\_\{1\}\\to P^\{1\\to a\}\_\{0\},P21→a→P01→aP^\{1\\to a\}\_\{2\}\\to P^\{1\\to a\}\_\{0\}\)\. - –Sum\-production ofSa→1S^\{a\\to 1\}: 6 reactions \(the fullSa→1S^\{a\\to 1\}table from Example[E\.1](https://arxiv.org/html/2606.27737#A5.Thmexample1)\)\. - –Sum\-production ofSa→2S^\{a\\to 2\}: all 6 reactions are deleted, because they usePk11→aP^\{1\\to a\}\_\{k\_\{1\}\}as catalyst\. - –Product\-production ofP1→aP^\{1\\to a\}: 2 reactions \(P01→a→P11→aP^\{1\\to a\}\_\{0\}\\to P^\{1\\to a\}\_\{1\},P01→a→P21→aP^\{1\\to a\}\_\{0\}\\to P^\{1\\to a\}\_\{2\}\)\. Total deleted:2\+2\+6\+6\+2=𝟏𝟖2\+2\+6\+6\+2=\\mathbf\{18\}reactions\.
- ψ\\psi:Updatesum\-production rates forSa→2S^\{a\\to 2\}\. After removingP1→aP^\{1\\to a\}from the catalytic set, the factor classJaJ\_\{a\}now contains only the sum bundleSa→2S^\{a\\to 2\}\(sinceSa→1S^\{a\\to 1\}was deleted\), andVJa=\{P2→a\}V\_\{J\_\{a\}\}=\\\{P^\{2\\to a\}\\\}\. The sum bundleSa→2S^\{a\\to 2\}hasCat​\(Sa→2\)=∅\\mathrm\{Cat\}\(S^\{a\\to 2\}\)=\\varnothing\(no remaining catalytic neighbors\), and the factor table is now the unary tableψa′​\(k2\)=∑k1ψa​\(k1,k2\)\\psi^\{\\prime\}\_\{a\}\(k\_\{2\}\)=\\sum\_\{k\_\{1\}\}\\psi\_\{a\}\(k\_\{1\},k\_\{2\}\)\. New sum\-production reactions forSa→2S^\{a\\to 2\}\(no catalysts, one reaction per state\): S0a→2\\displaystyle S^\{a\\to 2\}\_\{0\}→ψa′​\(1\)S1a→2,\\displaystyle\\xrightarrow\{\\;\\psi^\{\\prime\}\_\{a\}\(1\)\\;\}S^\{a\\to 2\}\_\{1\},S0a→2\\displaystyle S^\{a\\to 2\}\_\{0\}→ψa′​\(2\)S2a→2,\\displaystyle\\xrightarrow\{\\;\\psi^\{\\prime\}\_\{a\}\(2\)\\;\}S^\{a\\to 2\}\_\{2\},S0a→2\\displaystyle S^\{a\\to 2\}\_\{0\}→ψa′​\(3\)S3a→2,\\displaystyle\\xrightarrow\{\\;\\psi^\{\\prime\}\_\{a\}\(3\)\\;\}S^\{a\\to 2\}\_\{3\},whereψa′​\(k2\)=ψa​\(1,k2\)\+ψa​\(2,k2\)\\psi^\{\\prime\}\_\{a\}\(k\_\{2\}\)=\\psi\_\{a\}\(1,k\_\{2\}\)\+\\psi\_\{a\}\(2,k\_\{2\}\)\. These replace the 6 deletedSa→2S^\{a\\to 2\}reactions with 3 new ones\. All other surviving reactions \(recycling ofSa→2S^\{a\\to 2\},Sb→2S^\{b\\to 2\},Sb→3S^\{b\\to 3\},P2→aP^\{2\\to a\},P2→bP^\{2\\to b\},P3→bP^\{3\\to b\}; sum\-production ofSb→2S^\{b\\to 2\},Sb→3S^\{b\\to 3\}; product\-production ofP2→aP^\{2\\to a\},P2→bP^\{2\\to b\},P3→bP^\{3\\to b\}\) retain their original rate constants\.

##### Intermediate CRNΓ′\\Gamma^\{\\prime\}after Step 1\.

After Step 1:22species,54−18\+3=39reactions111We deleted 18 reactions and added 3 new sum\-production reactions for the updated unary factorψ′a\. Equivalently:=\-20416recycling\+=\+\-2412315sum\-production\+=\-1028product\-production=39\.\.\\boxed\{\\text\{After Step~1:\}\\quad 22\\text\{ species\},\\quad 54\-18\+3=39\\text\{ reactions\}\.\}

#### Step 2: Colinear retraction of unary factoraa\.

##### Poset analysis ofA′=\{2,3,a,b\}A^\{\\prime\}=\\\{2,3,a,b\\\}\.

After Step 1, factoraais unary withne​\(a\)=\{2\}\\mathrm\{ne\}\(a\)=\\\{2\\\}\. Its scope\{2\}\\\{2\\\}is contained in the scope\{2,3\}\\\{2,3\\\}of factorbb\. Thereforefactoraais colinearwitha↓=2a^\{\\downarrow\}=2\(the unique lower cover ofaa\)\.

We check the conditions of Lemma B\.3\. LetB=A′∖\{a\}=\{2,3,b\}B=A^\{\\prime\}\\setminus\\\{a\\\}=\\\{2,3,b\\\}\. InBB, variable22has a single upper coverb↑=bb^\{\\uparrow\}=b\. So variable22is linear inBB, and Lemma B\.3 applies\.

##### Factor\-table update \(Lemma B\.3\)\.

Witha↓=2a^\{\\downarrow\}=2andb↑=bb^\{\\uparrow\}=b, the updated factor table is

ψb′​\(k2,k3\)∝ψb​\(k2,k3\)⋅∑z∈Ea:za↓=k2ψa​\(z\)ψa↓​\(k2\)\.\\psi^\{\\prime\}\_\{b\}\(k\_\{2\},k\_\{3\}\)\\;\\propto\\;\\psi\_\{b\}\(k\_\{2\},k\_\{3\}\)\\cdot\\frac\{\\displaystyle\\sum\_\{z\\in E\_\{a\}:\\,z\_\{a^\{\\downarrow\}\}=k\_\{2\}\}\\psi\_\{a\}\(z\)\}\{\\psi\_\{a^\{\\downarrow\}\}\(k\_\{2\}\)\}\.Since factoraais already unary on variable22\(i\.e\.Ea=E2E\_\{a\}=E\_\{2\}andza↓=z=k2z\_\{a^\{\\downarrow\}\}=z=k\_\{2\}\), the sum has a single term:

∑z∈Ea:z2=k2ψa′​\(z\)=ψa′​\(k2\)\.\\sum\_\{z\\in E\_\{a\}:\\,z\_\{2\}=k\_\{2\}\}\\psi^\{\\prime\}\_\{a\}\(z\)=\\psi^\{\\prime\}\_\{a\}\(k\_\{2\}\)\.Furthermore, the “variable region”ψa↓\\psi\_\{a^\{\\downarrow\}\}is absent here \(variable22carries no separate unary factor in the current posetA′A^\{\\prime\}; equivalently,ψ2≡1\\psi\_\{2\}\\equiv 1\)\. So the update simplifies to

ψb′​\(k2,k3\)=ψb​\(k2,k3\)⋅ψa′​\(k2\)=ψb​\(k2,k3\)⋅\[ψa​\(1,k2\)\+ψa​\(2,k2\)\]\.\\psi^\{\\prime\}\_\{b\}\(k\_\{2\},k\_\{3\}\)=\\psi\_\{b\}\(k\_\{2\},k\_\{3\}\)\\cdot\\psi^\{\\prime\}\_\{a\}\(k\_\{2\}\)=\\psi\_\{b\}\(k\_\{2\},k\_\{3\}\)\\cdot\\bigl\[\\psi\_\{a\}\(1,k\_\{2\}\)\+\\psi\_\{a\}\(2,k\_\{2\}\)\\bigr\]\.\(15\)This is still a3×23\\times 2table \(unchanged shape\), but the entries have been modulated row\-wise by the marginalized unary factor from variable11\.

##### Reduced poset\.

After removing factoraa, the poset isA′′=\{2,3,b\}A^\{\\prime\\prime\}=\\\{2,3,b\\\}with relations2<b2<b,3<b3<b\. This is the factor graph consisting of a single factorbbon variables\{2,3\}\\\{2,3\\\}with the updated tableψb′\\psi^\{\\prime\}\_\{b\}\.

##### CRN morphism𝒩​\(r2\)=\(ϕ0,ϕ1,ψ\)\\mathcal\{N\}\(r\_\{2\}\)=\(\\phi\_\{0\},\\phi\_\{1\},\\psi\)\.

- ϕ0\\phi\_\{0\}:Deleteall bundles involving factoraa\(as sender or receiver\): Sa→2,P2→a\.S^\{a\\to 2\},\\quad P^\{2\\to a\}\.This removes4\+4=84\+4=8species\.
- ϕ1\\phi\_\{1\}:Deleteall reactions involving deleted species: - –Recycling ofSa→2S^\{a\\to 2\}: 3 reactions\. - –Recycling ofP2→aP^\{2\\to a\}: 3 reactions\. - –Sum\-production ofSa→2S^\{a\\to 2\}: 3 reactions \(the uncatalyzed reactions from Step 1\)\. - –Product\-production ofP2→aP^\{2\\to a\}: 3 reactions \(P02→a\+Skb→2→⋯P^\{2\\to a\}\_\{0\}\+S^\{b\\to 2\}\_\{k\}\\to\\cdots\)\. - –Product\-production ofP2→bP^\{2\\to b\}: These usedSka→2S^\{a\\to 2\}\_\{k\}as catalyst, so all 3 are deleted\. Total deleted:3\+3\+3\+3\+3=𝟏𝟓3\+3\+3\+3\+3=\\mathbf\{15\}reactions\.
- ψ\\psi:Updaterates\. *Sum\-production ofSb→2S^\{b\\to 2\}*: Factor classJbJ\_\{b\}now hasVJb=\{P2→b,P3→b\}V\_\{J\_\{b\}\}=\\\{P^\{2\\to b\},P^\{3\\to b\}\\\}\(as before\), but the table changes fromψb\\psi\_\{b\}toψb′\\psi^\{\\prime\}\_\{b\}\. The 6 reactions forSb→2S^\{b\\to 2\}retain their structure but their rates change fromψb​\(k2,k3\)\\psi\_\{b\}\(k\_\{2\},k\_\{3\}\)toψb′​\(k2,k3\)\\psi^\{\\prime\}\_\{b\}\(k\_\{2\},k\_\{3\}\): *Sum\-production ofSb→3S^\{b\\to 3\}*: Same structure, updated rates: *Product\-production ofP2→bP^\{2\\to b\}*: After deletion ofSa→2S^\{a\\to 2\}, the catalytic set ofP2→bP^\{2\\to b\}becomesCat​\(P2→b\)=∅\\mathrm\{Cat\}\(P^\{2\\to b\}\)=\\varnothing\(variable22now touches only factorbb\)\. The product\-production reactions become uncatalyzed: P02→b→κprodP12→b,P02→b→κprodP22→b,P02→b→κprodP32→b\.P^\{2\\to b\}\_\{0\}\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{2\\to b\}\_\{1\},\\quad P^\{2\\to b\}\_\{0\}\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{2\\to b\}\_\{2\},\\quad P^\{2\\to b\}\_\{0\}\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{2\\to b\}\_\{3\}\.These replace the 3 deleted catalyzed reactions with 3 new uncatalyzed ones\. All other surviving reactions \(recycling ofSb→2S^\{b\\to 2\},Sb→3S^\{b\\to 3\},P2→bP^\{2\\to b\},P3→bP^\{3\\to b\}; product\-production ofP3→bP^\{3\\to b\}\) are unchanged\.

#### Final reduced CRNΓ′′=𝒩​\(A′′,H′′\)\\Gamma^\{\\prime\\prime\}=\\mathcal\{N\}\(A^\{\\prime\\prime\},H^\{\\prime\\prime\}\)\.

The reduced factor graph has variables\{2,3\}\\\{2,3\\\}with state spacesE2=\{1,2,3\}E\_\{2\}=\\\{1,2,3\\\},E3=\{1,2\}E\_\{3\}=\\\{1,2\\\}, a single factorbbwithne​\(b\)=\{2,3\}\\mathrm\{ne\}\(b\)=\\\{2,3\\\}, and updated tableψb′\\psi^\{\\prime\}\_\{b\}from equation \([15](https://arxiv.org/html/2606.27737#A5.E15)\)\. The compiled CRN has four bundles:

#### Complete reaction list ofΓ′′\\Gamma^\{\\prime\\prime\}\.

*Recycling \(8 reactions\):*

S1b→2→κrS0b→2,\\displaystyle S^\{b\\to 2\}\_\{1\}\\xrightarrow\{\\kappa\_\{r\}\}S^\{b\\to 2\}\_\{0\},\\quadS2b→2→κrS0b→2,\\displaystyle S^\{b\\to 2\}\_\{2\}\\xrightarrow\{\\kappa\_\{r\}\}S^\{b\\to 2\}\_\{0\},\\quadS3b→2→κrS0b→2,\\displaystyle S^\{b\\to 2\}\_\{3\}\\xrightarrow\{\\kappa\_\{r\}\}S^\{b\\to 2\}\_\{0\},S1b→3→κrS0b→3,\\displaystyle S^\{b\\to 3\}\_\{1\}\\xrightarrow\{\\kappa\_\{r\}\}S^\{b\\to 3\}\_\{0\},\\quadS2b→3→κrS0b→3,\\displaystyle S^\{b\\to 3\}\_\{2\}\\xrightarrow\{\\kappa\_\{r\}\}S^\{b\\to 3\}\_\{0\},P12→b→κrP02→b,\\displaystyle P^\{2\\to b\}\_\{1\}\\xrightarrow\{\\kappa\_\{r\}\}P^\{2\\to b\}\_\{0\},\\quadP22→b→κrP02→b,\\displaystyle P^\{2\\to b\}\_\{2\}\\xrightarrow\{\\kappa\_\{r\}\}P^\{2\\to b\}\_\{0\},\\quadP32→b→κrP02→b,\\displaystyle P^\{2\\to b\}\_\{3\}\\xrightarrow\{\\kappa\_\{r\}\}P^\{2\\to b\}\_\{0\},P13→b→κrP03→b,\\displaystyle P^\{3\\to b\}\_\{1\}\\xrightarrow\{\\kappa\_\{r\}\}P^\{3\\to b\}\_\{0\},\\quadP23→b→κrP03→b\.\\displaystyle P^\{3\\to b\}\_\{2\}\\xrightarrow\{\\kappa\_\{r\}\}P^\{3\\to b\}\_\{0\}\.
*Sum\-production ofSb→2S^\{b\\to 2\}\(6 reactions, catalystP3→bP^\{3\\to b\}\):*

*Sum\-production ofSb→3S^\{b\\to 3\}\(6 reactions, catalystP2→bP^\{2\\to b\}\):*

*Product\-production ofP2→bP^\{2\\to b\}\(3 reactions, no catalysts\):*

P02→b→κprodP12→b,P02→b→κprodP22→b,P02→b→κprodP32→b\.P^\{2\\to b\}\_\{0\}\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{2\\to b\}\_\{1\},\\qquad P^\{2\\to b\}\_\{0\}\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{2\\to b\}\_\{2\},\\qquad P^\{2\\to b\}\_\{0\}\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{2\\to b\}\_\{3\}\.
*Product\-production ofP3→bP^\{3\\to b\}\(2 reactions, no catalysts\):*

P03→b→κprodP13→b,P03→b→κprodP23→b\.P^\{3\\to b\}\_\{0\}\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{3\\to b\}\_\{1\},\\qquad P^\{3\\to b\}\_\{0\}\\xrightarrow\{\\;\\kappa\_\{\\mathrm\{prod\}\}\\;\}P^\{3\\to b\}\_\{2\}\.

#### Reduced CRN summary\.

Species:4\+3\+4\+3=𝟏𝟒\(vs\. 28 original,𝟓𝟎%reduction\)Recycling reactions:𝟏𝟎Sum\-bundle production:6\+6=𝟏𝟐Product\-bundle production:3\+2=𝟓Total reactions:𝟐𝟕\(vs\. 54 original,𝟓𝟎%reduction\)\\boxed\{\\begin\{aligned\} &\\text\{Species:\}&&4\+3\+4\+3=\\mathbf\{14\}&&\\quad\\text\{\(vs\.\\ 28 original, $\\mathbf\{50\\%\}$ reduction\)\}\\\\ &\\text\{Recycling reactions:\}&&\\mathbf\{10\}\\\\ &\\text\{Sum\-bundle production:\}&&6\+6=\\mathbf\{12\}\\\\ &\\text\{Product\-bundle production:\}&&3\+2=\\mathbf\{5\}\\\\ &\\text\{Total reactions:\}&&\\mathbf\{27\}&&\\quad\\text\{\(vs\.\\ 54 original, $\\mathbf\{50\\%\}$ reduction\)\}\\end\{aligned\}\}

#### Steady\-state equations of the reduced CRN\.

At a positive steady state ofΓ′′\\Gamma^\{\\prime\\prime\}:

κr\[S0b→2\]​\[Skb→2\]\\displaystyle\\frac\{\\kappa\_\{r\}\}\{\[S^\{b\\to 2\}\_\{0\}\]\}\\,\[S^\{b\\to 2\}\_\{k\}\]=∑k3=12ψb′​\(k,k3\)​\[Pk33→b\],\\displaystyle=\\sum\_\{k\_\{3\}=1\}^\{2\}\\psi^\{\\prime\}\_\{b\}\(k,k\_\{3\}\)\\,\[P^\{3\\to b\}\_\{k\_\{3\}\}\],k\\displaystyle k∈\{1,2,3\},\\displaystyle\\in\\\{1,2,3\\\},\(16\)κr\[S0b→3\]​\[Skb→3\]\\displaystyle\\frac\{\\kappa\_\{r\}\}\{\[S^\{b\\to 3\}\_\{0\}\]\}\\,\[S^\{b\\to 3\}\_\{k\}\]=∑k2=13ψb′​\(k2,k\)​\[Pk22→b\],\\displaystyle=\\sum\_\{k\_\{2\}=1\}^\{3\}\\psi^\{\\prime\}\_\{b\}\(k\_\{2\},k\)\\,\[P^\{2\\to b\}\_\{k\_\{2\}\}\],k\\displaystyle k∈\{1,2\},\\displaystyle\\in\\\{1,2\\\},\(17\)κrκprod​\[P02→b\]​\[Pk2→b\]\\displaystyle\\frac\{\\kappa\_\{r\}\}\{\\kappa\_\{\\mathrm\{prod\}\}\\,\[P^\{2\\to b\}\_\{0\}\]\}\\,\[P^\{2\\to b\}\_\{k\}\]=1,\\displaystyle=1,k\\displaystyle k∈\{1,2,3\},\\displaystyle\\in\\\{1,2,3\\\},\(18\)κrκprod​\[P03→b\]​\[Pk3→b\]\\displaystyle\\frac\{\\kappa\_\{r\}\}\{\\kappa\_\{\\mathrm\{prod\}\}\\,\[P^\{3\\to b\}\_\{0\}\]\}\\,\[P^\{3\\to b\}\_\{k\}\]=1,\\displaystyle=1,k\\displaystyle k∈\{1,2\}\.\\displaystyle\\in\\\{1,2\\\}\.\(19\)
Equations \([18](https://arxiv.org/html/2606.27737#A5.E18)\) and \([19](https://arxiv.org/html/2606.27737#A5.E19)\) say that both product bundles are uniform at steady state \(both variables now touch a single factor\)\. The sum\-bundle equations \([16](https://arxiv.org/html/2606.27737#A5.E16)\) and \([17](https://arxiv.org/html/2606.27737#A5.E17)\) implement BP on the single\-factor graph\{2,3,b\}\\\{2,3,b\\\}with the updated tableψb′\\psi^\{\\prime\}\_\{b\}, which encodes the influence of the retracted variable11through the factorψa′​\(k2\)=ψa​\(1,k2\)\+ψa​\(2,k2\)\\psi^\{\\prime\}\_\{a\}\(k\_\{2\}\)=\\psi\_\{a\}\(1,k\_\{2\}\)\+\\psi\_\{a\}\(2,k\_\{2\}\)absorbed intoψb′\\psi^\{\\prime\}\_\{b\}\.

#### BP fixed\-point preservation\.

By Lemma 6\.3, the marginals of variables22and33computed from the reduced CRNΓ′′\\Gamma^\{\\prime\\prime\}coincide with those from the original CRNΓ\\Gamma\. Concretely, at any positive steady state:

\[Skb→2\]∑k′\[Sk′b→2\]\|Γ′′=\[Skb→2\]∑k′\[Sk′b→2\]\|Γ,k∈\{1,2,3\},\\frac\{\[S^\{b\\to 2\}\_\{k\}\]\}\{\\sum\_\{k^\{\\prime\}\}\[S^\{b\\to 2\}\_\{k^\{\\prime\}\}\]\}\\bigg\|\_\{\\Gamma^\{\\prime\\prime\}\}\\;=\\;\\frac\{\[S^\{b\\to 2\}\_\{k\}\]\}\{\\sum\_\{k^\{\\prime\}\}\[S^\{b\\to 2\}\_\{k^\{\\prime\}\}\]\}\\bigg\|\_\{\\Gamma\},\\qquad k\\in\\\{1,2,3\\\},and similarly for the messages involving variable33\. The retracted variable11is no longer represented in the reduced CRN, but its effect on the surviving marginals is faithfully encoded in the updated rate constantsψb′\\psi^\{\\prime\}\_\{b\}\.

#### Remark \(reduction on the chain graph is maximal\)\.

The reduced posetA′′=\{2,3,b\}A^\{\\prime\\prime\}=\\\{2,3,b\\\}has no linear or colinear points: variable22has upper covers\{b\}\\\{b\\\}and variable33has upper covers\{b\}\\\{b\\\}, but neither is linear becausebbhas two lower covers \(b↓=\{2,3\}b^\{\\downarrow\}=\\\{2,3\\\}, not a single element\)\. Factorbbis binary, so it is not colinear\. ThereforeA′′A^\{\\prime\\prime\}is the*core*of the original posetAA\. By symmetry, retracting variable33first \(instead of variable11\) would yield the same core\{1,2,a\}\\\{1,2,a\\\}up to relabeling, with an analogous updated factor tableψa′​\(k1,k2\)=ψa​\(k1,k2\)⋅\[ψb​\(k2,1\)\+ψb​\(k2,2\)\]\\psi^\{\\prime\}\_\{a\}\(k\_\{1\},k\_\{2\}\)=\\psi\_\{a\}\(k\_\{1\},k\_\{2\}\)\\cdot\[\\psi\_\{b\}\(k\_\{2\},1\)\+\\psi\_\{b\}\(k\_\{2\},2\)\]\.

#### Remark \(Reconstructing beliefs on retracted variable nodes\)

The retraction of a linear or colinear point induces a mapping that enables the reconstruction of beliefs for the retracted node from the node onto which it was retracted\. By recursively applying this mapping, one can reconstruct the belief of any retracted node by following the path from a node in the reduced factor graph back to the node in question\. This implies that from the steady\-state concentrations of the reduced CRN, one can infer the concentrations of the original CRN\. This reconstruction can be implemented using any CRN designed for filtering, such as Napp–Adams Belief Propagation \(BP\) CRNs\(Napp and Adams,[2013](https://arxiv.org/html/2606.27737#bib.bib5)\), or specific CRNs proposed for the E\-step of the Expectation\-Maximization \(EM\) algorithm for learning Hidden Markov Models \(HMMs\)\(Wiufet al\.,[2023](https://arxiv.org/html/2606.27737#bib.bib17)\)\. Doing so is arguably less computationally expensive than executing full BP and it’s associated CRN ODEs, particularly when only a few retracted belief nodes need to be reconstructed for downstream decision\-making\. A hypothetical example of this efficiency could arise when designing a synthetic chemotactic or phototactic cell\. Such a cell might use a message\-passing CRN to perform inference on sensory inputs, which then drive decision\-making through competition between chemical concentrations, similar to the mechanisms described in\(Tanget al\.,[2026](https://arxiv.org/html/2606.27737#bib.bib34)\)\. In this setting, retracted factor nodes that are not directly associated with observations possess factors that remain invariant to new sensory data\. Consequently, the reconstruction mapping outlined above can be structurally enforced by design\. These specific concentrations, which may play a critical role in the cell’s decision\-making, can thus be computed highly efficiently: first by performing the ODE associated to BP exclusively on the reduced CRN, and then by executing a targeted reconstruction only for the necessary nodes\.