When Do Causal World Models Help Modular LLM Agents

# When Do Causal World Models Help Modular LLM Agents
Source: [https://arxiv.org/html/2610.00012](https://arxiv.org/html/2610.00012)
Xinyuan Song1Zekun Cai2,3 1Emory University, Atlanta, GA, USA2The University of Tokyo, Tokyo, Japan 3LocationMind, Tokyo, Japan xinyuan\.song@emory\.edu, caizekun@csis\.u\-tokyo\.ac\.jp

###### Abstract

LLM agents increasingly act through modular systems, such as order, payment, inventory, and shipment services, where actions in one module change which transitions are valid in another\. Standard world models usually fit observational traces, but this is not the quantity needed for intervention\-time planning: a trace may show that payment precedes shipment without identifying whether payment authorizes shipment, inventory mediates the effect, or a hidden trigger explains both\. We study this gap throughFedCausalCompose, a causal world\-model framework for modular LLM agents in which local actions provide intervention\-response evidence for cross\-module interfaces\. We first show that observational world models incur an irreducible interventional error under unblocked back\-door paths, that interface recovery improves with intervention\-response coverage, and that an oracle causal composition can beat the non\-causal lower bound when coverage and local mechanism errors are controlled\. We then test the resulting prediction in diagnostic agent settings\. Causal interfaces help most in structured tool environments, where API signatures expose preconditions and downstream effects\. In contrast, dialogue and narrative environments often ignore raw edge lists unless a short attention anchor makes the causal information decision\-relevant\. These results identify a concrete condition for causal world models in LLM agents: causal structure helps when cross\-module interfaces are both statistically identifiable and presented in a form the agent can use at action time\. Our code is available at[https://github\.com/Hik289/fed\-causal\-world\.git](https://github.com/Hik289/fed-causal-world.git)\.

When Do Causal World Models Help Modular LLM Agents

Xinyuan Song1Zekun Cai2,31Emory University, Atlanta, GA, USA2The University of Tokyo, Tokyo, Japan3LocationMind, Tokyo, Japanxinyuan\.song@emory\.edu, caizekun@csis\.u\-tokyo\.ac\.jp

## 1Introduction

LLM agents increasingly act across systems that are modular by design\. A retail agent authenticates a user, queries orders, modifies payment or inventory state, dispatches shipment, and only then resolves refunds\. A household agent in ALFWorld moves between navigation, containers, appliances, and task monitors\. These components are not independent environments; they are functional modules inside one coupled system\. Existing model\-based RL, LLM world\-model, and agent\-benchmark work usually treats dynamics as a centralized sequence model or as independent environment replicas\(Janneret al\.,[2019](https://arxiv.org/html/2610.00012#bib.bib1); Hafneret al\.,[2019](https://arxiv.org/html/2610.00012#bib.bib2),[2020](https://arxiv.org/html/2610.00012#bib.bib3),[2021](https://arxiv.org/html/2610.00012#bib.bib4),[2023](https://arxiv.org/html/2610.00012#bib.bib5); Wanget al\.,[2024](https://arxiv.org/html/2610.00012#bib.bib21); Meiet al\.,[2026](https://arxiv.org/html/2610.00012#bib.bib24); Yuet al\.,[2025](https://arxiv.org/html/2610.00012#bib.bib26); Huet al\.,[2025](https://arxiv.org/html/2610.00012#bib.bib27); Baiet al\.,[2026](https://arxiv.org/html/2610.00012#bib.bib28); Yaoet al\.,[2024](https://arxiv.org/html/2610.00012#bib.bib37); Barreset al\.,[2025](https://arxiv.org/html/2610.00012#bib.bib38); Shridharet al\.,[2021](https://arxiv.org/html/2610.00012#bib.bib39); Rawles and others,[2024](https://arxiv.org/html/2610.00012#bib.bib40); Liet al\.,[2023](https://arxiv.org/html/2610.00012#bib.bib42); Luet al\.,[2024](https://arxiv.org/html/2610.00012#bib.bib43); Yaoet al\.,[2022](https://arxiv.org/html/2610.00012#bib.bib44)\)\. We instead ask whether each client can be treated as afunctional modulewith local state, local actions, and causal interfaces to other modules\.

The obstacle is causal non\-identifiability\. A global sequence model may observe that payment precedes shipment, but it cannot determine from order alone whether payment authorizes shipment, inventory mediates the effect, or a latent trigger creates a spurious association\. Under interventions, the relevant object isP\(⋅∣do\(Ak=a\)\)P\(\\cdot\\mid\\mathrm\{do\}\(A\_\{k\}=a\)\), notP\(⋅∣Ak=a\)P\(\\cdot\\mid A\_\{k\}=a\); back\-door, invariant\-prediction, and causal\-representation theory formalize why observational prediction can fail under such shifts\(Pearl,[2009](https://arxiv.org/html/2610.00012#bib.bib20); Peterset al\.,[2016](https://arxiv.org/html/2610.00012#bib.bib48); Schölkopfet al\.,[2021](https://arxiv.org/html/2610.00012#bib.bib15); Brouillardet al\.,[2020](https://arxiv.org/html/2610.00012#bib.bib10); Perryet al\.,[2022](https://arxiv.org/html/2610.00012#bib.bib11); Arjovskyet al\.,[2019](https://arxiv.org/html/2610.00012#bib.bib36)\)\. Federated causal\-discovery work studies privacy\-preserving structure learning\(Mianet al\.,[2023](https://arxiv.org/html/2610.00012#bib.bib13); Baldo and Assaad,[2026](https://arxiv.org/html/2610.00012#bib.bib12); Hahnet al\.,[2026](https://arxiv.org/html/2610.00012#bib.bib14)\), but usually assumes shared variables or passive data\. Our setting has disjoint module states, and local actions provide natural intervention\-response evidence\.

We proposeFedCausalCompose, a six\-step framework for building a federated causal world model: identify local variables, actions, and interfaces; discover candidate outgoing–incoming event matches; match local interventions to downstream responses; validate cross\-module edges; compose local mechanisms; and optionally dispatch local causal\-control constraints\. The present experiments evaluate a centralized prototype because the available LLM harness exposes traces centrally\. The federated contribution is therefore the evidence model: each client would expose only interface\-level intervention summaries rather than raw traces or parameters\. Figure[1](https://arxiv.org/html/2610.00012#S1.F1)summarizes this pipeline\.

![Refer to caption](https://arxiv.org/html/2610.00012v1/x1.png)Figure 1:FedCausalCompose pipeline\.Each functional module contributes local actions, interface events, and intervention\-response summaries\. The aggregator validates cross\-module causal edges and composes them into a global world model without treating all client traces as one undifferentiated sequence\.Our contributions are deliberately matched to the evidence\. First, we formulate federated causal world modeling for modular LLM agents, where clients are functional modules rather than independent environment replicas\. Second, we give rigid oracle\-side theory: observational world models have an interventional lower bound under unblocked back\-door paths; interface recovery follows an exponential coverage law; and oracle causal composition strictly improves over the non\-causal lower bound once coverage is sufficient\. Third, we provide small but diagnostic experiments showing when this theory becomes useful in prompts: structured tool environments can use causal ordering directly, while dialogue and narrative environments often require attention anchoring before edge information affects action choice\. Finally, we identify an important boundary case: if observed action features already proxy the confounder, the empirical causal advantage can shrink even though the causal graph remains the correct object\.

## 2Related Work

#### Federated reinforcement learning and world models\.

Federated RL studies how clients can share policies, values, or surrogate models across heterogeneous environments\(Sunet al\.,[2024](https://arxiv.org/html/2610.00012#bib.bib6)\)\. Model\-based RL learns latent dynamics for planning and policy improvement\(Janneret al\.,[2019](https://arxiv.org/html/2610.00012#bib.bib1); Hafneret al\.,[2019](https://arxiv.org/html/2610.00012#bib.bib2),[2020](https://arxiv.org/html/2610.00012#bib.bib3),[2021](https://arxiv.org/html/2610.00012#bib.bib4),[2023](https://arxiv.org/html/2610.00012#bib.bib5)\)\. These settings typically interpret a client as an environment instance, such as a robot, simulator replica, or user partition\.FedCausalComposeuses a different modularization: a client is a functional component inside one coupled system\. The shared object is therefore not a parameter vector or a latent transition model, but an interface graph connecting local mechanisms\.

#### Causal discovery with interventions and environments\.

Observational DAG\-learning methods such as NOTEARS recover structure from centralized samples under acyclicity and faithfulness assumptions\(Zhenget al\.,[2018](https://arxiv.org/html/2610.00012#bib.bib9)\)\. Interventional and multi\-environment methods use distribution changes to improve identifiability\(Peterset al\.,[2016](https://arxiv.org/html/2610.00012#bib.bib48); Brouillardet al\.,[2020](https://arxiv.org/html/2610.00012#bib.bib10); Perryet al\.,[2022](https://arxiv.org/html/2610.00012#bib.bib11)\)\. Federated causal\-discovery methods add privacy and decentralization constraints\(Mianet al\.,[2023](https://arxiv.org/html/2610.00012#bib.bib13); Baldo and Assaad,[2026](https://arxiv.org/html/2610.00012#bib.bib12); Hahnet al\.,[2026](https://arxiv.org/html/2610.00012#bib.bib14)\)\. Our setting differs in two ways: module state spaces are disjoint, and agent actions supply naturally occurring intervention evidence\. The recovered graph is evaluated through world\-model prediction and downstream control, not only through structural accuracy\.

#### Causal and language\-agent world models\.

Causal representation learning argues that causal structure can support transfer and systematic generalization\(Schölkopfet al\.,[2021](https://arxiv.org/html/2610.00012#bib.bib15)\); causal world models in RL make related claims in control domains\(Yuet al\.,[2024](https://arxiv.org/html/2610.00012#bib.bib30); Muttiet al\.,[2023](https://arxiv.org/html/2610.00012#bib.bib31)\)\. Recent LLM\-agent work asks whether language models can simulate text environments, retrieve executable state transitions, or interleave reasoning and acting\(Wanget al\.,[2024](https://arxiv.org/html/2610.00012#bib.bib21); Meiet al\.,[2026](https://arxiv.org/html/2610.00012#bib.bib24); Yuet al\.,[2025](https://arxiv.org/html/2610.00012#bib.bib26); Huet al\.,[2025](https://arxiv.org/html/2610.00012#bib.bib27); Baiet al\.,[2026](https://arxiv.org/html/2610.00012#bib.bib28); Chenet al\.,[2023](https://arxiv.org/html/2610.00012#bib.bib29); Yaoet al\.,[2023](https://arxiv.org/html/2610.00012#bib.bib46)\)\. These methods motivate our global sequence and edge\-prompt baselines\. They do not, however, separate observational correlations from interventional effects across private functional modules\.

#### Agent benchmarks and modularization\.

τ\\tau\-bench,τ2\\tau^\{2\}\-bench, APIBank, ToolSandbox, WebShop, ALFWorld, ScienceWorld, and AndroidWorld expose stateful interactions\(Yaoet al\.,[2024](https://arxiv.org/html/2610.00012#bib.bib37); Barreset al\.,[2025](https://arxiv.org/html/2610.00012#bib.bib38); Liet al\.,[2023](https://arxiv.org/html/2610.00012#bib.bib42); Luet al\.,[2024](https://arxiv.org/html/2610.00012#bib.bib43); Yaoet al\.,[2022](https://arxiv.org/html/2610.00012#bib.bib44); Shridharet al\.,[2021](https://arxiv.org/html/2610.00012#bib.bib39); Wanget al\.,[2022](https://arxiv.org/html/2610.00012#bib.bib41); Rawles and others,[2024](https://arxiv.org/html/2610.00012#bib.bib40)\)\. They contain implicit modules: accounts, orders, payments, inventory, household containers, appliances, apps, permissions, and files\. Prior benchmark papers evaluate end\-to\-end task success\. We make the module interfaces explicit and ask which dependencies must be learned causally for prediction and control\.

#### Prompt attention as a deployment bottleneck\.

The experiments also connect to long\-context and tool\-use prompting\. Models often fail to use information that is present but poorly positioned or weakly signaled in long contexts\(Liuet al\.,[2024](https://arxiv.org/html/2610.00012#bib.bib47)\)\. Our attention\-anchor result should be read in this light: causal edges can be correct and still unused unless the prompt makes them operationally salient for the current goal\.

## 3Why Non\-Causal World Models Fail

The central failure is not that observational world models are weak predictors on their training distribution\. The failure is that a modular agent is often deployed under interventions\. A payment tool is not merely correlated with shipment; it changes which downstream transitions are valid\. A household action is not merely a token in a trajectory; it opens, closes, heats, moves, or blocks later state changes\. When an unobserved trigger affects both the action context and the downstream module, a global sequence model can learn the right observational conditional mean while learning the wrong interventional object\.

#### Modular SCM\.

For modulesℳ=\{ℳ1,…,ℳK\}\\mathcal\{M\}=\\\{\\mathcal\{M\}\_\{1\},\\ldots,\\mathcal\{M\}\_\{K\}\\\}, let

ℳk=\(Xk,Ak,Ikin,Ikout,Fk∗,Uk\),\\mathcal\{M\}\_\{k\}=\(X\_\{k\},A\_\{k\},I\_\{k\}^\{\\mathrm\{in\}\},I\_\{k\}^\{\\mathrm\{out\}\},F\_\{k\}^\{\*\},U\_\{k\}\),with local structural equation and cross\-module interface

Xk,t\+1\\displaystyle X\_\{k,t\+1\}=Fk∗​\(Xk,t,Ak,t,Ik,tin,Uk,t\),\\displaystyle=F\_\{k\}^\{\*\}\(X\_\{k,t\},A\_\{k,t\},I\_\{k,t\}^\{\\mathrm\{in\}\},U\_\{k,t\}\),\(1\)Ij,t\+ℓin\\displaystyle I\_\{j,t\+\\ell\}^\{\\mathrm\{in\}\}=Hi→j​\(Ii,tout\),\(i,j\)∈EM∗\.\\displaystyle=H\_\{i\\to j\}\(I\_\{i,t\}^\{\\mathrm\{out\}\}\),\\qquad\(i,j\)\\in E\_\{M\}^\{\*\}\.\(2\)LetGM∗=\(ℳ,EM∗\)G\_\{M\}^\{\*\}=\(\\mathcal\{M\},E\_\{M\}^\{\*\}\),Zt=\(X1:K,t,Ak,t\)Z\_\{t\}=\(X\_\{1:K,t\},A\_\{k,t\}\), and let a non\-causal world model be any predictor fitted only from samples underPobsP\_\{\\mathrm\{obs\}\}and measurable with respect toσ​\(Zt\)\\sigma\(Z\_\{t\}\)\.

###### Assumption 1\(Back\-door separation\)\.

For some\(i,j\)∈EM∗\(i,j\)\\in E\_\{M\}^\{\*\}and interventiondo​\(Ak,t=a\)\\mathrm\{do\}\(A\_\{k,t\}=a\), no observedZ⊆X1:K,tZ\\subseteq X\_\{1:K,t\}blocks the back\-door path

Ak,t←⋯←Ui​j→Xj,t\+1\.A\_\{k,t\}\\leftarrow\\cdots\\leftarrow U\_\{ij\}\\rightarrow X\_\{j,t\+1\}\.With

μobs​\(z\)\\displaystyle\\mu^\{\\mathrm\{obs\}\}\(z\):=𝔼Pobs​\[Xj,t\+1∣Zt=z\],\\displaystyle:=\\mathbb\{E\}\_\{P\_\{\\mathrm\{obs\}\}\}\[X\_\{j,t\+1\}\\mid Z\_\{t\}=z\],\(3\)μint​\(z\)\\displaystyle\\mu^\{\\mathrm\{int\}\}\(z\):=𝔼Pint​\[Xj,t\+1∣Zt=z\],\\displaystyle:=\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\[X\_\{j,t\+1\}\\mid Z\_\{t\}=z\],\(4\)the interventional separation is

δint2:=𝔼Pint​\[\(μint​\(Zt\)−μobs​\(Zt\)\)2\]\>0\.\\delta\_\{\\mathrm\{int\}\}^\{2\}:=\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\\\!\\left\[\(\\mu^\{\\mathrm\{int\}\}\(Z\_\{t\}\)\-\\mu^\{\\mathrm\{obs\}\}\(Z\_\{t\}\)\)^\{2\}\\right\]\>0\.\(5\)In the linear\-Gaussian synthetic SCM,

μint​\(z\)−μobs​\(z\)\\displaystyle\\mu^\{\\mathrm\{int\}\}\(z\)\-\\mu^\{\\mathrm\{obs\}\}\(z\)=γ​Vj​μshift,\\displaystyle=\\gamma V\_\{j\}\\mu\_\{\\mathrm\{shift\}\},\(6\)δint2\\displaystyle\\delta\_\{\\mathrm\{int\}\}^\{2\}=γ2​‖Vj​μshift‖2\.\\displaystyle=\\gamma^\{2\}\\left\\lVert V\_\{j\}\\mu\_\{\\mathrm\{shift\}\}\\right\\rVert^\{2\}\.\(7\)

###### Theorem 1\(Back\-door gap for observational world models\)\.

IfF^obs\\hat\{F\}^\{\\mathrm\{obs\}\}minimizes observational squared risk overσ​\(Zt\)\\sigma\(Z\_\{t\}\)\-measurable predictors, then

Rint​\(F^obs\)\\displaystyle R\_\{\\mathrm\{int\}\}\(\\hat\{F\}^\{\\mathrm\{obs\}\}\):=𝔼Pint​\[\(Xj,t\+1−F^obs​\(Zt\)\)2\]\\displaystyle:=\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\\\!\\left\[\(X\_\{j,t\+1\}\-\\hat\{F\}^\{\\mathrm\{obs\}\}\(Z\_\{t\}\)\)^\{2\}\\right\]=𝔼Pint​\[\(Xj,t\+1−μint​\(Zt\)\)2\]\+δint2\\displaystyle=\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\\\!\\left\[\(X\_\{j,t\+1\}\-\\mu^\{\\mathrm\{int\}\}\(Z\_\{t\}\)\)^\{2\}\\right\]\+\\delta\_\{\\mathrm\{int\}\}^\{2\}≥δint2\>0\.\\displaystyle\\geq\\delta\_\{\\mathrm\{int\}\}^\{2\}\>0\.\(8\)

The theorem gives the first story point: more observational data cannot remove a back\-door gap, because the predictor has already converged to the wrong conditional mean\. This is precisely where a causal interface graph helps\. It tells the agent which downstream transition should be evaluated underdo​\(⋅\)\\mathrm\{do\}\(\\cdot\)rather than under the passive trace distribution\.

###### Proposition 1\(Compounding across interface hops\)\.

LetSℓS\_\{\\ell\}be the event that the non\-causal predictor is correct at hopℓ\\ell, and assumeP​\(Sℓ∣S1:ℓ−1\)≤1−δintP\(S\_\{\\ell\}\\mid S\_\{1:\\ell\-1\}\)\\leq 1\-\\delta\_\{\\mathrm\{int\}\}\. For a depth\-ddrollout,

Pok​\(d\)\\displaystyle P\_\{\\mathrm\{ok\}\}\(d\):=P​\(⋂ℓ=1d−1Sℓ\)≤\(1−δint\)d−1,\\displaystyle:=P\\\!\\left\(\\bigcap\_\{\\ell=1\}^\{d\-1\}S\_\{\\ell\}\\right\)\\leq\(1\-\\delta\_\{\\mathrm\{int\}\}\)^\{d\-1\},\(9\)ℰdobs\\displaystyle\\mathcal\{E\}^\{\\mathrm\{obs\}\}\_\{d\}:=1−Pok​\(d\)≥1−\(1−δint\)d−1\.\\displaystyle:=1\-P\_\{\\mathrm\{ok\}\}\(d\)\\geq 1\-\(1\-\\delta\_\{\\mathrm\{int\}\}\)^\{d\-1\}\.\(10\)Thusℰd\+1obs−ℰdobs≥\(1−δint\)d−1​δint\>0\\mathcal\{E\}^\{\\mathrm\{obs\}\}\_\{d\+1\}\-\\mathcal\{E\}^\{\\mathrm\{obs\}\}\_\{d\}\\geq\(1\-\\delta\_\{\\mathrm\{int\}\}\)^\{d\-1\}\\delta\_\{\\mathrm\{int\}\}\>0\.

The proposition is the second story point: in modular systems the cost of a wrong interface is not local\. An early observational mistake changes the state presented to later modules, so rollout error compounds along the interface path\.

#### Why the causal model can win\.

FedCausalComposeattacks exactly the two terms above\. Local interventions supply evidence forP\(⋅∣do\(Ak=a\)\)P\(\\cdot\\mid\\mathrm\{do\}\(A\_\{k\}=a\)\), and the federated graph identifies which local mechanism should consume that evidence\. The coverage and oracle\-composition results in[section˜4](https://arxiv.org/html/2610.00012#S4)show that, once true interfaces are sufficiently covered and local mechanism errors are controlled, the causal composition can fall below the non\-causal lower bound\. Full proofs of[theorems˜1](https://arxiv.org/html/2610.00012#Thmtheorem1)and[1](https://arxiv.org/html/2610.00012#Thmproposition1)are in Appendix[A](https://arxiv.org/html/2610.00012#A1)\.

#### Synthetic theory checks\.

The synthetic diagnostics instantiate the same quantities:

Δγ\\displaystyle\\Delta\_\{\\gamma\}=MSESeqWM​\(γ\)−MSECausalOracle​\(γ\),\\displaystyle=\\mathrm\{MSE\}\_\{\\mathrm\{SeqWM\}\}\(\\gamma\)\-\\mathrm\{MSE\}\_\{\\mathrm\{CausalOracle\}\}\(\\gamma\),\(11\)Δh\\displaystyle\\Delta\_\{h\}=MSESeq​\(h\)−MSECausalOracle​\(h\)\.\\displaystyle=\\mathrm\{MSE\}\_\{\\mathrm\{Seq\}\}\(h\)\-\\mathrm\{MSE\}\_\{\\mathrm\{CausalOracle\}\}\(h\)\.\(12\)The observedΔγ\\Delta\_\{\\gamma\}grows with confounder shift andΔh\\Delta\_\{h\}grows with rollout depth\.

![Refer to caption](https://arxiv.org/html/2610.00012v1/figures/fig2_backdoor_gap.png)Figure 2:Back\-door gap\.Observational sequence models learnμobs\\mu^\{\\mathrm\{obs\}\}, while intervention\-time rollouts requireμint\\mu^\{\\mathrm\{int\}\}\. The resulting gap grows with confounding strength and compounds across interface hops\.

## 4FedCausalCompose

FedCausalComposeestimatesGM∗G\_\{M\}^\{\*\}from interface\-level intervention\-response summaries rather than raw trajectories\. For each candidate edgee=\(i,j\)e=\(i,j\), define Bernoulli matches

Me\(n\)\\displaystyle M\_\{e\}^\{\(n\)\}∼Bernoulli​\(qi​rj\),\\displaystyle\\sim\\mathrm\{Bernoulli\}\(q\_\{i\}r\_\{j\}\),\(13\)Ne\\displaystyle N\_\{e\}=∑n=1NMe\(n\),\\displaystyle=\\sum\_\{n=1\}^\{N\}M\_\{e\}^\{\(n\)\},\(14\)Nmin\\displaystyle N\_\{\\min\}=mine∈EM∗⁡Ne\.\\displaystyle=\\min\_\{e\\in E\_\{M\}^\{\*\}\}N\_\{e\}\.\(15\)
###### Assumption 2\(Bernoulli verification\)\.

For everye=\(i,j\)∈EM∗e=\(i,j\)\\in E\_\{M\}^\{\*\},\{Me\(n\)\}n=1N\\\{M\_\{e\}^\{\(n\)\}\\\}\_\{n=1\}^\{N\}are independent conditional on shared confounders, withqi≥qmin\>0q\_\{i\}\\geq q\_\{\\min\}\>0andrj≥rmin\>0r\_\{j\}\\geq r\_\{\\min\}\>0\.

###### Theorem 2\(False\-negative coverage\)\.

Under[Assumption˜2](https://arxiv.org/html/2610.00012#Thmassumption2),

P​\(EM∗⊈E^M\)≤\|EM∗\|​exp⁡\(−Nmin​qmin​rmin\)\.P\(E\_\{M\}^\{\*\}\\nsubseteq\\hat\{E\}\_\{M\}\)\\leq\|E\_\{M\}^\{\*\}\|\\exp\(\-N\_\{\\min\}q\_\{\\min\}r\_\{\\min\}\)\.\(16\)ThusP​\(EM∗⊆E^M\)≥1−δP\(E\_\{M\}^\{\*\}\\subseteq\\hat\{E\}\_\{M\}\)\\geq 1\-\\deltawhenever

Nmin≥log⁡\(\|EM∗\|/δ\)qmin​rmin\.N\_\{\\min\}\\geq\\frac\{\\log\(\|E\_\{M\}^\{\*\}\|/\\delta\)\}\{q\_\{\\min\}r\_\{\\min\}\}\.\(17\)

###### Assumption 3\(Oracle composition\)\.

For allkk, local mechanism error is at mostε0\\varepsilon\_\{0\}; direction, lag, and composition errors sum to at mostε1\\varepsilon\_\{1\}; and all interface maps areLL\-Lipschitz\.

###### Theorem 3\(Oracle FCC error upper bound\)\.

Under[Assumptions˜2](https://arxiv.org/html/2610.00012#Thmassumption2)and[3](https://arxiv.org/html/2610.00012#Thmassumption3), an oracleFedCausalComposepredictor using causal adjustment onG^M\\hat\{G\}\_\{M\}satisfies

εGFCC≤K\(Kε0\+L2\|EM∗\|e−Nmin​qmin​rmin\+ε1\)\.\\varepsilon\_\{G\}^\{\\mathrm\{FCC\}\}\\leq K\\\!\\left\(K\\varepsilon\_\{0\}\+L^\{2\}\|E\_\{M\}^\{\*\}\|e^\{\-N\_\{\\min\}q\_\{\\min\}r\_\{\\min\}\}\\right\.\\\\ \\left\.\+\\varepsilon\_\{1\}\\right\)\.\(18\)

###### Proposition 2\(Strict oracle dominance\)\.

Ifγ\>0\\gamma\>0,δint2\>K​\(K​ε0\+ε1\)\\delta\_\{\\mathrm\{int\}\}^\{2\}\>K\(K\\varepsilon\_\{0\}\+\\varepsilon\_\{1\}\), and

Nmin\>1qmin​rmin​log⁡K​L2​\|EM∗\|δint2−K​\(K​ε0\+ε1\),N\_\{\\min\}\>\\frac\{1\}\{q\_\{\\min\}r\_\{\\min\}\}\\log\\frac\{KL^\{2\}\|E\_\{M\}^\{\*\}\|\}\{\\delta\_\{\\mathrm\{int\}\}^\{2\}\-K\(K\\varepsilon\_\{0\}\+\\varepsilon\_\{1\}\)\},\(19\)then the oracleFedCausalComposeerror is strictly below the non\-causal lower bound:εGFCC<δint2\\varepsilon\_\{G\}^\{\\mathrm\{FCC\}\}<\\delta\_\{\\mathrm\{int\}\}^\{2\}\.

Proofs are given in Appendix[A](https://arxiv.org/html/2610.00012#A1)\. The implementation diagnostics used in the main experiments are

MSENaive​\(γ=0,1\)\\displaystyle\\mathrm\{MSE\}\_\{\\mathrm\{Naive\}\}\(\\gamma=0,1\)=\(0\.040,24\.633\),\\displaystyle=\(040,4633\),Δγ\\displaystyle\\Delta\_\{\\gamma\}=\(0\.003,1\.318,4\.593,\\displaystyle=\(003,318,593,9\.497,16\.269\),\\displaystyle\\quad 497,6269\),MSESeq​\(h\)\\displaystyle\\mathrm\{MSE\}\_\{\\mathrm\{Seq\}\}\(h\)=\(3\.293,6\.466,9\.128,\\displaystyle=\(293,466,128,12\.602,14\.132\),\\displaystyle\\quad 2602,4132\),EdgeF1​\(p\)\\displaystyle\\mathrm\{EdgeF1\}\(p\)=\(0\.000,0\.000,0\.190,\\displaystyle=\(000,000,190,0\.707,0\.775,0\.785\)\.\\displaystyle\\quad 707,775,785\)\.HereNaive\-CausalomitsUconfU\_\{\\mathrm\{conf\}\},Causal\-Oracleincludes the structural confounder, and the horizon sequence is the three\-seed aggregate\. The edge\-F1 sequence corresponds top∈\{0,0\.05,0\.10,0\.25,0\.50,1\.0\}p\\in\\\{0,0\.05,0\.10,0\.25,0\.50,1\.0\\\}\.

![Refer to caption](https://arxiv.org/html/2610.00012v1/figures/fig3_federated_causal_loop.png)Figure 3:Federated causal loop\.Local modules send intervention\-response summaries to an aggregator, which updates the interface graph and dispatches causal rollout constraints without centralizing raw trajectories\.![Refer to caption](https://arxiv.org/html/2610.00012v1/figures/fig4_local_causal_control.png)Figure 4:Local causal control\.A client checks upstream interfaces, predicts downstream effects, gates unsafe actions, and verifies residuals after execution\.
## 5Experiments

#### Setup\.

The experiments are diagnostic cells rather than a large benchmark sweep\. We combine the seed\-0 synthetic SCM study with targeted cells spanningτ\\tau\-bench retail, ALFWorld, ScienceWorld, APIBank Stateful, and stateful tool\-use settings\(Yaoet al\.,[2024](https://arxiv.org/html/2610.00012#bib.bib37); Shridharet al\.,[2021](https://arxiv.org/html/2610.00012#bib.bib39); Wanget al\.,[2022](https://arxiv.org/html/2610.00012#bib.bib41); Liet al\.,[2023](https://arxiv.org/html/2610.00012#bib.bib42); Luet al\.,[2024](https://arxiv.org/html/2610.00012#bib.bib43)\)\. The prompt baselines are also tied to standard language\-agent and tool\-use practice:Seq\-WMfollows observational trace and ReAct\-style world\-model prompting\(Yaoet al\.,[2023](https://arxiv.org/html/2610.00012#bib.bib46); Schicket al\.,[2023](https://arxiv.org/html/2610.00012#bib.bib45); Wanget al\.,[2024](https://arxiv.org/html/2610.00012#bib.bib21)\),Discovered\-Causaladds recovered interface edges, andOracle\-Causalexposes ground\-truth causal interfaces\. Values are point estimates; several cells have wide Wilson intervals, so the goal is to identify where causal interface information is used, ignored, or harmful\.

Table 1:Key diagnostic findings\. We organize results by the empirical question each cell tests\. Success values are percentages unless otherwise noted; ROUGE\-L is reported for APIBank because it is a standard API\-response quality metric\.
#### Added diagnostic detail\.

The added cells sharpen the table in three ways\. First, split\-level Wilson half\-widths range from roughly±7%\\pm 7\\%to±21%\\pm 21\\%, so we read these cells mechanistically rather than as powered benchmark claims\. Second, APIBank Stateful aggregates 582 evaluations and reports both task success and ROUGE\-L; its monotone success pattern is the strongest structured\-tool signal\. Third, the ALFWorld edge\-correctness sweep varies the correct\-edge fraction from 0 to 1 and remains nearly uncorrelated with task success; correct raw edges alone are therefore not a reliable control signal in narrative tasks\.

#### The useful case: structured tools expose causal order\.

Table[1](https://arxiv.org/html/2610.00012#S5.T1)shows that APIBank Stateful is the cleanest non\-synthetic cell\. The benchmark’s API calls naturally follow authentication, query, modification, and verification dependencies\(Liet al\.,[2023](https://arxiv.org/html/2610.00012#bib.bib42); Luet al\.,[2024](https://arxiv.org/html/2610.00012#bib.bib43)\), so causal ordering aligns with the action format\. In this setting, performance improves monotonically from the non\-causal sequence baseline to discovered\-edge prompting and then to the oracle\-edge condition\. This is the main positive empirical story: when the environment already presents state transitions as explicit tool dependencies, causal world\-model information is directly actionable\.

#### The failure mode: raw edges are not automatically attended\.

Theτ\\tau\-bench retail cell reproduces the deployment concern\(Yaoet al\.,[2024](https://arxiv.org/html/2610.00012#bib.bib37)\)\. Without an attention anchor, causal\-edge prompts underperform the simpler sequence baseline\. Adding a short instruction to stay focused on the user’s primary goal reverses the pattern, improving both discovered\-edge and oracle\-edge variants while the non\-causal baseline no longer benefits\. This is consistent with long\-context findings that models may fail to use information that is present but weakly signaled\(Liuet al\.,[2024](https://arxiv.org/html/2610.00012#bib.bib47)\)\. The lesson is not that causal edges are useless; it is that causal edges must be made attendable\.

#### Narrative environments are the hardest setting\.

ALFWorld and ScienceWorld encode causal structure in natural language rather than typed API signatures\(Shridharet al\.,[2021](https://arxiv.org/html/2610.00012#bib.bib39); Wanget al\.,[2022](https://arxiv.org/html/2610.00012#bib.bib41)\)\. ALFWorld’s edge\-correctness sweep directly tests whether more correct edges lead to better execution\. The pattern is nearly noise, suggesting that narrative agents often ignore raw edge lists embedded in prompt text\. Split checks on retail, ALFWorld, and ScienceWorld show the same fragility\. Table[2](https://arxiv.org/html/2610.00012#S5.T2)adds an efficiency view: causal prompts can reduce or preserve call counts in some cells, but they do not automatically improve cost per win unless the agent actually uses the edge information\.

Table 2:Representative efficiency metrics\. Lower cost per successful run is better\. The table shows that causal prompting is not free: it helps only when the added structure improves decisions enough to offset longer prompts or extra calls\.
#### Theory\-facing checks remain stable\.

The synthetic cells evaluate the terms in \([8](https://arxiv.org/html/2610.00012#S3.E8)\), \([10](https://arxiv.org/html/2610.00012#S3.E10)\), and \([16](https://arxiv.org/html/2610.00012#S4.E16)\)\. The confounding sweep increasesΔγ\\Delta\_\{\\gamma\}, the horizon sweep increasesΔh\\Delta\_\{h\}, and edge recovery saturates oncepint≥0\.25p\_\{\\mathrm\{int\}\}\\geq 0\.25\. The conflict sweep givesρ=−1\.000\\rho=\-1\.000, i\.e\.,

∂∂c​\(MSESeqWM​\(c\)−MSECausalOracle​\(c\)\)<0,\\frac\{\\partial\}\{\\partial c\}\\left\(\\mathrm\{MSE\}\_\{\\mathrm\{SeqWM\}\}\(c\)\-\\mathrm\{MSE\}\_\{\\mathrm\{CausalOracle\}\}\(c\)\\right\)<0,whereccis the action\-confounder proxy strength\. This is the single\-proxy boundary: asAAcarries more information aboutUi​jU\_\{ij\}, the observational predictor partially adjusts throughAA\.

## 6When Does a Causal World Model Help?

The answer is conditional\. Causal interface information helps when the environment makes the interface graph operational\. Structured tool tasks are the clearest case: API calls have typed arguments, visible preconditions, and explicit verification steps, so causal edges align with the format in which the agent chooses actions\. In this regime,FedCausalComposesupplies the right inductive bias: it tells the agent which upstream state must be satisfied before a downstream operation is meaningful\.

Narrative and dialogue environments behave differently\. ALFWorld,τ\\tau\-bench dialogue, and SciWorld encode causal structure in natural language, so raw edge lists compete with goals, history, and tool text\. The attention\-anchor result explains why the same edge information can fail or help depending on prompt form\. A short metaprompt makes causal edges salient to the decision, while an unanchored edge list becomes extra context that the model may ignore\. This connects the causal\-world\-model question to the practical prompt\-attention problem: identifiability is necessary, but not sufficient, for deployment\.

The conflict sweep adds an identifiability boundary\. As action increasingly carries confounder signal, theSeq\-WM–Causal\-Oraclegap decreases monotonically: a single action proxy partially adjusts for the latent confounder\. This is consistent with Pearl’s back\-door analysis\(Pearl,[2009](https://arxiv.org/html/2610.00012#bib.bib20)\)and single\-proxy non\-identifiability\. Causal gains are therefore largest when confounding is real, relevant, and not already encoded in observed action features\.

## 7Limitations

The empirical scope is limited\. Most LLM\-agent cells are single\-seed point estimates, and several split\-level intervals are wide\. We therefore treat the results as mechanistic evidence about when causal information is used, not as powered benchmark claims\. The negative split results are part of the finding: raw causal edges should not be deployed without attention anchoring, structured\-tool validation, or a policy that explicitly consumes the graph\.

The system is also not yet a full privacy\-preserving federated deployment\. The experiments use a centralized trace harness, while the theory describes the federated evidence object that would be exchanged: intervention\-response summaries rather than raw trajectories\. A deployed system would need privacy accounting, secure aggregation, communication constraints, and asynchronous client updates\. AndroidWorld is deferred because the required Android emulator support was unavailable in our compute environment\.

Finally, modularization and oracle edges are author\-designed in the current prototype\. This is acceptable for testing the causal object and its prompt usability, but future work must learn module boundaries, edge lags, and response matches with less supervision before claiming broad autonomy\.

## 8Conclusion

We presentedFedCausalComposeand studied when causal interface information helps modular LLM agents\. The theory gives a clean oracle target: observational world models suffer irreducible interventional error under unblocked back\-door paths, while a covered and correctly composed causal interface graph can beat that lower bound\. The experiments show why this target is not enough by itself\. Structured tool environments can use causal edges directly; narrative and dialogue agents often need attention anchoring before the same edges influence behavior\. Federated causal world models therefore require two ingredients: identifiable cross\-module interfaces and deployment mechanisms that make those interfaces attendable at decision time\.

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## Appendix ARigid Theory Proofs

### A\.1Notation and Standing Assumptions

Letℳ=\{ℳ1,…,ℳK\}\\mathcal\{M\}=\\\{\\mathcal\{M\}\_\{1\},\\ldots,\\mathcal\{M\}\_\{K\}\\\}\. Modulekkhas stateXk∈ℝnkX\_\{k\}\\in\\mathbb\{R\}^\{n\_\{k\}\}, actionAk∈ℝmkA\_\{k\}\\in\\mathbb\{R\}^\{m\_\{k\}\}, incoming interfaceIkinI\_\{k\}^\{\\mathrm\{in\}\}, outgoing interfaceIkoutI\_\{k\}^\{\\mathrm\{out\}\}, local mechanismFk∗F\_\{k\}^\{\*\}, and exogenous noiseUkU\_\{k\}\.

###### Assumption 4\(Modular SCM\)\.

ℳk\\displaystyle\\mathcal\{M\}\_\{k\}=\(Xk,Ak,Fk∗,Uk,Ikin,Ikout\),\\displaystyle=\(X\_\{k\},A\_\{k\},F\_\{k\}^\{\*\},U\_\{k\},I\_\{k\}^\{\\mathrm\{in\}\},I\_\{k\}^\{\\mathrm\{out\}\}\),\(20\)Xk,t\+1\\displaystyle X\_\{k,t\+1\}=Fk∗​\(Xk,t,Ak,t,Ik,tin,Uk,t\),\\displaystyle=F\_\{k\}^\{\*\}\(X\_\{k,t\},A\_\{k,t\},I\_\{k,t\}^\{\\mathrm\{in\}\},U\_\{k,t\}\),\(21\)Ij,t\+ℓin\\displaystyle I\_\{j,t\+\\ell\}^\{\\mathrm\{in\}\}=Hi→j​\(Ii,tout\),\(i,j\)∈EM∗,\\displaystyle=H\_\{i\\to j\}\(I\_\{i,t\}^\{\\mathrm\{out\}\}\),\\qquad\(i,j\)\\in E\_\{M\}^\{\*\},\(22\)GM∗\\displaystyle G\_\{M\}^\{\*\}=\(ℳ,EM∗\)​is a DAG,\|EM∗\|≥1\.\\displaystyle=\(\\mathcal\{M\},E\_\{M\}^\{\*\}\)\\ \\text\{is a DAG\},\\qquad\|E\_\{M\}^\{\*\}\|\\geq 1\.\(23\)Conditional on shared cross\-module confounders,Uk⟂⟂Uk′U\_\{k\}\\perp\\\!\\\!\\\!\\perp U\_\{k^\{\\prime\}\}fork≠k′k\\neq k^\{\\prime\}\.

###### Assumption 5\(Unblocked back\-door path\)\.

For some\(i,j\)∈EM∗\(i,j\)\\in E\_\{M\}^\{\*\}and interventiondo​\(Ak,t=a\)\\mathrm\{do\}\(A\_\{k,t\}=a\),

Ak,t←⋯←Ui​j→Xj,t\+1,Z⊆\{X1:K,t\}⇒Z⟂⟂Ui​j\.A\_\{k,t\}\\leftarrow\\cdots\\leftarrow U\_\{ij\}\\rightarrow X\_\{j,t\+1\},\\qquad Z\\subseteq\\\{X\_\{1:K,t\}\\\}\\Rightarrow Z\\not\\\!\\perp\\\!\\\!\\\!\\perp U\_\{ij\}\.Equivalently, no observed admissible adjustment set blocks every such path\.

###### Assumption 6\(Non\-degenerate linear\-Gaussian confounding\)\.

In the synthetic SCM,

Uconf,t\\displaystyle U\_\{\\mathrm\{conf\},t\}∼𝒩​\(μconf,Idc\),\\displaystyle\\sim\\mathcal\{N\}\(\\mu\_\{\\mathrm\{conf\}\},I\_\{d\_\{c\}\}\),\(24\)𝔼​‖Vi​Uconf,t‖2\\displaystyle\\mathbb\{E\}\\left\\lVert V\_\{i\}U\_\{\\mathrm\{conf\},t\}\\right\\rVert^\{2\}\>0,γ\>0,Vj≠0,\\displaystyle\>0,\\qquad\\gamma\>0,\\qquad V\_\{j\}\\neq 0,\(25\)μconfint\\displaystyle\\mu\_\{\\mathrm\{conf\}\}^\{\\mathrm\{int\}\}≠0,Pint​\(Uconf\)≠Pobs​\(Uconf\)\.\\displaystyle\\neq 0,\\qquad P\_\{\\mathrm\{int\}\}\(U\_\{\\mathrm\{conf\}\}\)\\neq P\_\{\\mathrm\{obs\}\}\(U\_\{\\mathrm\{conf\}\}\)\.\(26\)

###### Assumption 7\(Bernoulli intervention matching\)\.

For each ordered pair\(i,j\)\(i,j\)and roundnn,

Mi​j\(n\)\\displaystyle M\_\{ij\}^\{\(n\)\}=𝟏​\{matched intervention\-response pair for​i→j\},\\displaystyle=\\mathbf\{1\}\\\{\\text\{matched intervention\-response pair for \}i\\to j\\\},\(27\)Mi​j\(n\)\\displaystyle M\_\{ij\}^\{\(n\)\}∼Bernoulli​\(pi​j\),pi​j=qi​rj,\\displaystyle\\sim\\mathrm\{Bernoulli\}\(p\_\{ij\}\),\\qquad p\_\{ij\}=q\_\{i\}r\_\{j\},\(28\)qi\\displaystyle q\_\{i\}=P​\[ℳi​executes​do​\(Ai=a∗\)\],\\displaystyle=P\[\\mathcal\{M\}\_\{i\}\\text\{ executes \}\\mathrm\{do\}\(A\_\{i\}=a^\{\*\}\)\],\(29\)rj\\displaystyle r\_\{j\}=P​\[ℳj​’s response is observed\],\\displaystyle=P\[\\mathcal\{M\}\_\{j\}\\text\{'s response is observed\}\],\(30\)qi\\displaystyle q\_\{i\}≥qmin\>0,rj≥rmin\>0\.\\displaystyle\\geq q\_\{\\min\}\>0,\\qquad r\_\{j\}\\geq r\_\{\\min\}\>0\.\(31\)Trials are independent across rounds and pairs conditional on shared confounders\.

### A\.2Back\-Door Lower Bound

Let

Y\\displaystyle Y:=Xj,t\+1,Z:=Zt=\(X1:K,t,Ak,t\),\\displaystyle:=X\_\{j,t\+1\},\\qquad Z:=Z\_\{t\}=\(X\_\{1:K,t\},A\_\{k,t\}\),\(32\)μobs​\(z\)\\displaystyle\\mu^\{\\mathrm\{obs\}\}\(z\):=𝔼Pobs​\[Y∣Z=z\],\\displaystyle:=\\mathbb\{E\}\_\{P\_\{\\mathrm\{obs\}\}\}\[Y\\mid Z=z\],\(33\)μint​\(z\)\\displaystyle\\mu^\{\\mathrm\{int\}\}\(z\):=𝔼Pint​\[Y∣Z=z\],\\displaystyle:=\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\[Y\\mid Z=z\],\(34\)b​\(z\)\\displaystyle b\(z\):=μint​\(z\)−μobs​\(z\),\\displaystyle:=\\mu^\{\\mathrm\{int\}\}\(z\)\-\\mu^\{\\mathrm\{obs\}\}\(z\),\(35\)δint2\\displaystyle\\delta\_\{\\mathrm\{int\}\}^\{2\}:=𝔼Pint​\[b​\(Z\)2\]\.\\displaystyle:=\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\[b\(Z\)^\{2\}\]\.\(36\)For squared loss, the observational Bayes predictor overσ​\(Z\)\\sigma\(Z\)isF^obs​\(z\)=μobs​\(z\)\\hat\{F\}^\{\\mathrm\{obs\}\}\(z\)=\\mu^\{\\mathrm\{obs\}\}\(z\)\. Therefore

Rint​\(F^obs\)\\displaystyle R\_\{\\mathrm\{int\}\}\(\\hat\{F\}^\{\\mathrm\{obs\}\}\)=𝔼Pint​\[\(Y−F^obs​\(Z\)\)2\]\\displaystyle=\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\\\!\\left\[\(Y\-\\hat\{F\}^\{\\mathrm\{obs\}\}\(Z\)\)^\{2\}\\right\]\(37\)=𝔼Pint​\[\(Y−μobs​\(Z\)\)2\]\\displaystyle=\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\\\!\\left\[\(Y\-\\mu^\{\\mathrm\{obs\}\}\(Z\)\)^\{2\}\\right\]\(38\)=𝔼Pint​\[\(Y−μint​\(Z\)\+μint​\(Z\)−μobs​\(Z\)\)2\]\\displaystyle=\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\\\!\\left\[\(Y\-\\mu^\{\\mathrm\{int\}\}\(Z\)\+\\mu^\{\\mathrm\{int\}\}\(Z\)\-\\mu^\{\\mathrm\{obs\}\}\(Z\)\)^\{2\}\\right\]\(39\)=𝔼Pint​\[\(Y−μint​\(Z\)\+b​\(Z\)\)2\]\\displaystyle=\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\\\!\\left\[\(Y\-\\mu^\{\\mathrm\{int\}\}\(Z\)\+b\(Z\)\)^\{2\}\\right\]\(40\)=𝔼Pint​\[\(Y−μint​\(Z\)\)2\]\+2​𝔼Pint​\[\(Y−μint​\(Z\)\)​b​\(Z\)\]\+𝔼Pint​\[b​\(Z\)2\]\\displaystyle=\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\\\!\\left\[\(Y\-\\mu^\{\\mathrm\{int\}\}\(Z\)\)^\{2\}\\right\]\+2\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\\\!\\left\[\(Y\-\\mu^\{\\mathrm\{int\}\}\(Z\)\)b\(Z\)\\right\]\+\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\\\!\\left\[b\(Z\)^\{2\}\\right\]\(41\)=𝔼Pint​\[\(Y−μint​\(Z\)\)2\]\+2​𝔼Pint​\[b​\(Z\)​𝔼Pint​\[Y−μint​\(Z\)∣Z\]\]\+δint2\\displaystyle=\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\\\!\\left\[\(Y\-\\mu^\{\\mathrm\{int\}\}\(Z\)\)^\{2\}\\right\]\+2\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\\\!\\left\[b\(Z\)\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\[Y\-\\mu^\{\\mathrm\{int\}\}\(Z\)\\mid Z\]\\right\]\+\\delta\_\{\\mathrm\{int\}\}^\{2\}\(42\)=𝔼Pint​\[\(Y−μint​\(Z\)\)2\]\+δint2\\displaystyle=\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\\\!\\left\[\(Y\-\\mu^\{\\mathrm\{int\}\}\(Z\)\)^\{2\}\\right\]\+\\delta\_\{\\mathrm\{int\}\}^\{2\}\(43\)≥δint2\.\\displaystyle\\geq\\delta\_\{\\mathrm\{int\}\}^\{2\}\.\(44\)Under[Assumption˜6](https://arxiv.org/html/2610.00012#Thmassumption6), for componentqq,

Yq\\displaystyle Y\_\{q\}=\[Wjself​Xj,t\]q\+\[Bj​Aj,t\]q\+α​\[Wi​j​tanh⁡\(Xi,t−ℓ\)\]q\+γ​\[Vj​Uconf,t\]q\+Uj,t,q\\displaystyle=\\left\[W\_\{j\}^\{\\mathrm\{self\}\}X\_\{j,t\}\\right\]\_\{q\}\+\\left\[B\_\{j\}A\_\{j,t\}\\right\]\_\{q\}\+\\alpha\\left\[W\_\{ij\}\\tanh\(X\_\{i,t\-\\ell\}\)\\right\]\_\{q\}\+\\gamma\\left\[V\_\{j\}U\_\{\\mathrm\{conf\},t\}\\right\]\_\{q\}\+U\_\{j,t,q\}\(45\)=:gq\(Z\)\+γ\[VjUconf,t\]q\+Uj,t,q,\\displaystyle=:g\_\{q\}\(Z\)\+\\gamma\[V\_\{j\}U\_\{\\mathrm\{conf\},t\}\]\_\{q\}\+U\_\{j,t,q\},\(46\)𝔼Pobs​\[Uconf,t\]\\displaystyle\\mathbb\{E\}\_\{P\_\{\\mathrm\{obs\}\}\}\[U\_\{\\mathrm\{conf\},t\}\]=0,𝔼Pint​\[Uconf,t\]=μshift≠0,\\displaystyle=0,\\qquad\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\[U\_\{\\mathrm\{conf\},t\}\]=\\mu\_\{\\mathrm\{shift\}\}\\neq 0,\(47\)μqobs​\(z\)\\displaystyle\\mu^\{\\mathrm\{obs\}\}\_\{q\}\(z\)=gq​\(z\)\+γ​\[Vj​𝔼Pobs​\(Uconf∣Z=z\)\]q\\displaystyle=g\_\{q\}\(z\)\+\\gamma\[V\_\{j\}\\mathbb\{E\}\_\{P\_\{\\mathrm\{obs\}\}\}\(U\_\{\\mathrm\{conf\}\}\\mid Z=z\)\]\_\{q\}\(48\)=gq​\(z\),\\displaystyle=g\_\{q\}\(z\),\(49\)μqint​\(z\)\\displaystyle\\mu^\{\\mathrm\{int\}\}\_\{q\}\(z\)=gq​\(z\)\+γ​\[Vj​𝔼Pint​\(Uconf∣Z=z\)\]q\\displaystyle=g\_\{q\}\(z\)\+\\gamma\[V\_\{j\}\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\(U\_\{\\mathrm\{conf\}\}\\mid Z=z\)\]\_\{q\}\(50\)=gq​\(z\)\+γ​\[Vj​μshift\]q,\\displaystyle=g\_\{q\}\(z\)\+\\gamma\[V\_\{j\}\\mu\_\{\\mathrm\{shift\}\}\]\_\{q\},\(51\)bq​\(z\)\\displaystyle b\_\{q\}\(z\)=μqint​\(z\)−μqobs​\(z\)\\displaystyle=\\mu^\{\\mathrm\{int\}\}\_\{q\}\(z\)\-\\mu^\{\\mathrm\{obs\}\}\_\{q\}\(z\)\(52\)=γ​\[Vj​μshift\]q,\\displaystyle=\\gamma\[V\_\{j\}\\mu\_\{\\mathrm\{shift\}\}\]\_\{q\},\(53\)δint2\\displaystyle\\delta\_\{\\mathrm\{int\}\}^\{2\}=𝔼Pint​‖b​\(Z\)‖2\\displaystyle=\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\\left\\lVert b\(Z\)\\right\\rVert^\{2\}\(54\)=γ2​‖Vj​μshift‖2\\displaystyle=\\gamma^\{2\}\\left\\lVert V\_\{j\}\\mu\_\{\\mathrm\{shift\}\}\\right\\rVert^\{2\}\(55\)\>0\.\\displaystyle\>0\.\(56\)Hence

infF^obs𝔼Pint​\[\(Y−F^obs​\(Z\)\)2\]≥δint2\>0\.\\inf\_\{\\hat\{F\}^\{\\mathrm\{obs\}\}\}\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\\\!\\left\[\(Y\-\\hat\{F\}^\{\\mathrm\{obs\}\}\(Z\)\)^\{2\}\\right\]\\geq\\delta\_\{\\mathrm\{int\}\}^\{2\}\>0\.Component\-normalized form:

δint2=γ2nj​‖Vj​μshift‖2\.\\delta\_\{\\mathrm\{int\}\}^\{2\}=\\frac\{\\gamma^\{2\}\}\{n\_\{j\}\}\\left\\lVert V\_\{j\}\\mu\_\{\\mathrm\{shift\}\}\\right\\rVert^\{2\}\.

### A\.3Horizon Compounding

LetSℓS\_\{\\ell\}be correctness at hopℓ\\ell\. Under the per\-hop abstraction,

P​\(Sℓ∣S1:ℓ−1\)≤1−δint,δint∈\(0,1\)\.P\(S\_\{\\ell\}\\mid S\_\{1:\\ell\-1\}\)\\leq 1\-\\delta\_\{\\mathrm\{int\}\},\\qquad\\delta\_\{\\mathrm\{int\}\}\\in\(0,1\)\.Then

Pok​\(d\)\\displaystyle P\_\{\\mathrm\{ok\}\}\(d\)=P​\(⋂ℓ=1d−1Sℓ\)\\displaystyle=P\\\!\\left\(\\bigcap\_\{\\ell=1\}^\{d\-1\}S\_\{\\ell\}\\right\)\(57\)=∏ℓ=1d−1P​\(Sℓ∣S1,…,Sℓ−1\)\\displaystyle=\\prod\_\{\\ell=1\}^\{d\-1\}P\(S\_\{\\ell\}\\mid S\_\{1\},\\ldots,S\_\{\\ell\-1\}\)\(58\)≤∏ℓ=1d−1\(1−δint\)\\displaystyle\\leq\\prod\_\{\\ell=1\}^\{d\-1\}\(1\-\\delta\_\{\\mathrm\{int\}\}\)\(59\)=\(1−δint\)d−1,\\displaystyle=\(1\-\\delta\_\{\\mathrm\{int\}\}\)^\{d\-1\},\(60\)ℰdobs\\displaystyle\\mathcal\{E\}^\{\\mathrm\{obs\}\}\_\{d\}=1−Pok​\(d\)\\displaystyle=1\-P\_\{\\mathrm\{ok\}\}\(d\)\(61\)≥1−\(1−δint\)d−1\.\\displaystyle\\geq 1\-\(1\-\\delta\_\{\\mathrm\{int\}\}\)^\{d\-1\}\.\(62\)The binomial lower envelope is

1−\(1−δint\)d−1\\displaystyle 1\-\(1\-\\delta\_\{\\mathrm\{int\}\}\)^\{d\-1\}=1−∑m=0d−1\(d−1m\)​\(−δint\)m\\displaystyle=1\-\\sum\_\{m=0\}^\{d\-1\}\\binom\{d\-1\}\{m\}\(\-\\delta\_\{\\mathrm\{int\}\}\)^\{m\}\(63\)=\(d−1\)​δint−\(d−12\)​δint2\+∑m=3d−1\(−1\)m\+1​\(d−1m\)​δintm\\displaystyle=\(d\-1\)\\delta\_\{\\mathrm\{int\}\}\-\\binom\{d\-1\}\{2\}\\delta\_\{\\mathrm\{int\}\}^\{2\}\+\\sum\_\{m=3\}^\{d\-1\}\(\-1\)^\{m\+1\}\\binom\{d\-1\}\{m\}\\delta\_\{\\mathrm\{int\}\}^\{m\}\(64\)≥\(d−1\)​δint−\(d−12\)​δint2\.\\displaystyle\\geq\(d\-1\)\\delta\_\{\\mathrm\{int\}\}\-\\binom\{d\-1\}\{2\}\\delta\_\{\\mathrm\{int\}\}^\{2\}\.\(65\)Strict monotonicity:

ℰd\+1obs−ℰdobs\\displaystyle\\mathcal\{E\}^\{\\mathrm\{obs\}\}\_\{d\+1\}\-\\mathcal\{E\}^\{\\mathrm\{obs\}\}\_\{d\}≥\[1−\(1−δint\)d\]−\[1−\(1−δint\)d−1\]\\displaystyle\\geq\[1\-\(1\-\\delta\_\{\\mathrm\{int\}\}\)^\{d\}\]\-\[1\-\(1\-\\delta\_\{\\mathrm\{int\}\}\)^\{d\-1\}\]\(66\)=\(1−δint\)d−1−\(1−δint\)d\\displaystyle=\(1\-\\delta\_\{\\mathrm\{int\}\}\)^\{d\-1\}\-\(1\-\\delta\_\{\\mathrm\{int\}\}\)^\{d\}\(67\)=\(1−δint\)d−1​δint\\displaystyle=\(1\-\\delta\_\{\\mathrm\{int\}\}\)^\{d\-1\}\\delta\_\{\\mathrm\{int\}\}\(68\)\>0\.\\displaystyle\>0\.\(69\)Ifℰobs​\[c​a​u​s​a​l\]d≤ε0\\mathcal\{E\}^\{\\mathrm\{obs\}\}\[causal\]\_\{d\}\\leq\\varepsilon\_\{0\}, then

Δ​\(d\)\\displaystyle\\Delta\(d\):=ℰdobs−ℰobs​\[c​a​u​s​a​l\]d\\displaystyle:=\\mathcal\{E\}^\{\\mathrm\{obs\}\}\_\{d\}\-\\mathcal\{E\}^\{\\mathrm\{obs\}\}\[causal\]\_\{d\}\(70\)≥1−\(1−δint\)d−1−ε0,\\displaystyle\\geq 1\-\(1\-\\delta\_\{\\mathrm\{int\}\}\)^\{d\-1\}\-\\varepsilon\_\{0\},\(71\)Δ​\(d\+1\)−Δ​\(d\)\\displaystyle\\Delta\(d\+1\)\-\\Delta\(d\)≥\(1−δint\)d−1​δint\>0\.\\displaystyle\\geq\(1\-\\delta\_\{\\mathrm\{int\}\}\)^\{d\-1\}\\delta\_\{\\mathrm\{int\}\}\>0\.\(72\)Ford=10d=10andδint=0\.06\\delta\_\{\\mathrm\{int\}\}=0\.06,

ℰ10obs≥1−0\.949≃0\.427,Δ​\(10\)≥0\.427−ε0\.\\mathcal\{E\}^\{\\mathrm\{obs\}\}\_\{10\}\\geq 1\-0\.94^\{9\}\\simeq 0\.427,\\qquad\\Delta\(10\)\\geq 0\.427\-\\varepsilon\_\{0\}\.

### A\.4Edge Verification and Coverage

Fore=\(i,j\)e=\(i,j\), writepe=pi​j=qi​rjp\_\{e\}=p\_\{ij\}=q\_\{i\}r\_\{j\}andNe=∑n=1NMe\(n\)N\_\{e\}=\\sum\_\{n=1\}^\{N\}M\_\{e\}^\{\(n\)\}\. The verification probability is

Pverify​\(e,N\)\\displaystyle P\_\{\\mathrm\{verify\}\}\(e,N\)=P​\(Ne≥1\)\\displaystyle=P\(N\_\{e\}\\geq 1\)\(73\)=1−P​\(Ne=0\)\\displaystyle=1\-P\(N\_\{e\}=0\)\(74\)=1−P​\(Me\(1\)=0,…,Me\(N\)=0\)\\displaystyle=1\-P\(M\_\{e\}^\{\(1\)\}=0,\\ldots,M\_\{e\}^\{\(N\)\}=0\)\(75\)=1−∏n=1NP​\(Me\(n\)=0\)\\displaystyle=1\-\\prod\_\{n=1\}^\{N\}P\(M\_\{e\}^\{\(n\)\}=0\)\(76\)=1−\(1−pe\)N\\displaystyle=1\-\(1\-p\_\{e\}\)^\{N\}\(77\)=1−\(1−qi​rj\)N\.\\displaystyle=1\-\(1\-q\_\{i\}r\_\{j\}\)^\{N\}\.\(78\)Therefore

P​\(e∉E^M\)\\displaystyle P\(e\\notin\\hat\{E\}\_\{M\}\)=1−Pverify​\(e,N\)\\displaystyle=1\-P\_\{\\mathrm\{verify\}\}\(e,N\)\(79\)=\(1−qi​rj\)N\\displaystyle=\(1\-q\_\{i\}r\_\{j\}\)^\{N\}\(80\)≤e−N​qi​rj\\displaystyle\\leq e^\{\-Nq\_\{i\}r\_\{j\}\}\(81\)≤e−N​qmin​rmin\.\\displaystyle\\leq e^\{\-Nq\_\{\\min\}r\_\{\\min\}\}\.\(82\)WithNmin=mine∈EM∗⁡NeN\_\{\\min\}=\\min\_\{e\\in E\_\{M\}^\{\*\}\}N\_\{e\},

P​\(EM∗⊈E^M\)\\displaystyle P\(E\_\{M\}^\{\*\}\\nsubseteq\\hat\{E\}\_\{M\}\)=P​\(⋃e∈EM∗\{e∉E^M\}\)\\displaystyle=P\\\!\\left\(\\bigcup\_\{e\\in E\_\{M\}^\{\*\}\}\\\{e\\notin\\hat\{E\}\_\{M\}\\\}\\right\)\(83\)≤∑e∈EM∗P​\(e∉E^M\)\\displaystyle\\leq\\sum\_\{e\\in E\_\{M\}^\{\*\}\}P\(e\\notin\\hat\{E\}\_\{M\}\)\(84\)≤∑e∈EM∗e−Nmin​qmin​rmin\\displaystyle\\leq\\sum\_\{e\\in E\_\{M\}^\{\*\}\}e^\{\-N\_\{\\min\}q\_\{\\min\}r\_\{\\min\}\}\(85\)=\|EM∗\|​e−Nmin​qmin​rmin\.\\displaystyle=\|E\_\{M\}^\{\*\}\|e^\{\-N\_\{\\min\}q\_\{\\min\}r\_\{\\min\}\}\.\(86\)Thus

\|EM∗\|​e−Nmin​qmin​rmin\\displaystyle\|E\_\{M\}^\{\*\}\|e^\{\-N\_\{\\min\}q\_\{\\min\}r\_\{\\min\}\}≤δ\\displaystyle\\leq\\delta\(87\)⟺−Nmin​qmin​rmin\+log⁡\|EM∗\|\\displaystyle\\Longleftrightarrow\\quad\-N\_\{\\min\}q\_\{\\min\}r\_\{\\min\}\+\\log\|E\_\{M\}^\{\*\}\|≤log⁡δ\\displaystyle\\leq\\log\\delta\(88\)⟺Nmin\\displaystyle\\Longleftrightarrow\\quad N\_\{\\min\}≥log⁡\(\|EM∗\|/δ\)qmin​rmin\\displaystyle\\geq\\frac\{\\log\(\|E\_\{M\}^\{\*\}\|/\\delta\)\}\{q\_\{\\min\}r\_\{\\min\}\}\(89\)implies

P​\(EM∗⊆E^M\)≥1−δ\.P\(E\_\{M\}^\{\*\}\\subseteq\\hat\{E\}\_\{M\}\)\\geq 1\-\\delta\.If complete detection is paired with the identifiability conditions of invariant/interventional discovery\(Peterset al\.,[2016](https://arxiv.org/html/2610.00012#bib.bib48); Eberhardtet al\.,[2006](https://arxiv.org/html/2610.00012#bib.bib49)\), then

EM∗⊆E^M,false\-positive control,lag disambiguation⟹G^M=GM∗\.E\_\{M\}^\{\*\}\\subseteq\\hat\{E\}\_\{M\},\\quad\\text\{false\-positive control\},\\quad\\text\{lag disambiguation\}\\quad\\Longrightarrow\\quad\\hat\{G\}\_\{M\}=G\_\{M\}^\{\*\}\.Numerical scales:

\|EM∗\|=14,qmin​rmin=0\.48,δ=0\.05\\displaystyle\|E\_\{M\}^\{\*\}\|=14,\\quad q\_\{\\min\}r\_\{\\min\}=0\.48,\\quad\\delta=0\.05⟹Nmin≥⌈log⁡\(280\)0\.48⌉=12,\\displaystyle\\Longrightarrow N\_\{\\min\}\\geq\\left\\lceil\\frac\{\\log\(280\)\}\{0\.48\}\\right\\rceil=12,\(90\)qmin​rmin=0\.135\\displaystyle q\_\{\\min\}r\_\{\\min\}=0\.135⟹Nmin≥⌈log⁡\(280\)0\.135⌉=42\.\\displaystyle\\Longrightarrow N\_\{\\min\}\\geq\\left\\lceil\\frac\{\\log\(280\)\}\{0\.135\}\\right\\rceil=42\.\(91\)

### A\.5Global Error Decomposition

Let

L\\displaystyle L:=maxk,t⁡‖Wkself‖2​\(1\+α​maxi,k⁡‖Wi​k‖2\),\\displaystyle:=\\max\_\{k,t\}\\left\\lVert W\_\{k\}^\{\\mathrm\{self\}\}\\right\\rVert\_\{2\}\\bigl\(1\+\\alpha\\max\_\{i,k\}\\left\\lVert W\_\{ik\}\\right\\rVert\_\{2\}\\bigr\),\(92\)εklocal\\displaystyle\\varepsilon\_\{k\}^\{\\mathrm\{local\}\}:=𝔼​‖F^kcaus​\(Xk,Ak,Ikin\)−Fk∗​\(Xk,Ak,Ikin,Uk\)‖2,\\displaystyle:=\\mathbb\{E\}\\left\\lVert\\hat\{F\}\_\{k\}^\{\\mathrm\{caus\}\}\(X\_\{k\},A\_\{k\},I\_\{k\}^\{\\mathrm\{in\}\}\)\-F\_\{k\}^\{\*\}\(X\_\{k\},A\_\{k\},I\_\{k\}^\{\\mathrm\{in\}\},U\_\{k\}\)\\right\\rVert^\{2\},\(93\)εinterface\\displaystyle\\varepsilon^\{\\mathrm\{interface\}\}:=L2​P​\(EM∗⊈E^M\),\\displaystyle:=L^\{2\}P\(E\_\{M\}^\{\*\}\\nsubseteq\\hat\{E\}\_\{M\}\),\(94\)εdirection\\displaystyle\\varepsilon^\{\\mathrm\{direction\}\}:=L2​𝔼​‖G^V−GV∗‖F2‖GV∗‖F2,\\displaystyle:=L^\{2\}\\frac\{\\mathbb\{E\}\\left\\lVert\\hat\{G\}\_\{V\}\-G\_\{V\}^\{\*\}\\right\\rVert\_\{F\}^\{2\}\}\{\\left\\lVert G\_\{V\}^\{\*\}\\right\\rVert\_\{F\}^\{2\}\},\(95\)εlag\\displaystyle\\varepsilon^\{\\mathrm\{lag\}\}:=L2​𝔼​\[\(ℓ^−ℓ∗\)2\],\\displaystyle:=L^\{2\}\\mathbb\{E\}\[\(\\hat\{\\ell\}\-\\ell^\{\*\}\)^\{2\}\],\(96\)εcomposition\\displaystyle\\varepsilon^\{\\mathrm\{composition\}\}:=L2​𝔼​‖H^−H∗‖2,\\displaystyle:=L^\{2\}\\mathbb\{E\}\\left\\lVert\\hat\{H\}\-H^\{\*\}\\right\\rVert^\{2\},\(97\)ε1\\displaystyle\\varepsilon\_\{1\}:=εdirection\+εlag\+εcomposition,\\displaystyle:=\\varepsilon^\{\\mathrm\{direction\}\}\+\\varepsilon^\{\\mathrm\{lag\}\}\+\\varepsilon^\{\\mathrm\{composition\}\},\(98\)ε0\\displaystyle\\varepsilon\_\{0\}:=maxk⁡εklocal\.\\displaystyle:=\\max\_\{k\}\\varepsilon\_\{k\}^\{\\mathrm\{local\}\}\.\(99\)For each module,

Xk,t\+1−X^k,t\+1FCC\\displaystyle X\_\{k,t\+1\}\-\\hat\{X\}\_\{k,t\+1\}^\{\\mathrm\{FCC\}\}=Fk∗​\(Xk,Ak,Ikin,Uk\)−F^kcaus​\(Xk,Ak,I^kin\)\\displaystyle=F\_\{k\}^\{\*\}\(X\_\{k\},A\_\{k\},I\_\{k\}^\{\\mathrm\{in\}\},U\_\{k\}\)\-\\hat\{F\}\_\{k\}^\{\\mathrm\{caus\}\}\(X\_\{k\},A\_\{k\},\\hat\{I\}\_\{k\}^\{\\mathrm\{in\}\}\)\(100\)=Fk∗​\(Xk,Ak,Ikin,Uk\)−F^kcaus​\(Xk,Ak,Ikin\)⏟Δklocal\\displaystyle=\\underbrace\{F\_\{k\}^\{\*\}\(X\_\{k\},A\_\{k\},I\_\{k\}^\{\\mathrm\{in\}\},U\_\{k\}\)\-\\hat\{F\}\_\{k\}^\{\\mathrm\{caus\}\}\(X\_\{k\},A\_\{k\},I\_\{k\}^\{\\mathrm\{in\}\}\)\}\_\{\\Delta\_\{k\}^\{\\mathrm\{local\}\}\}\(101\)\+F^kcaus​\(Xk,Ak,Ikin\)−F^kcaus​\(Xk,Ak,I^kin\)⏟Δkgraph\.\\displaystyle\\quad\+\\underbrace\{\\hat\{F\}\_\{k\}^\{\\mathrm\{caus\}\}\(X\_\{k\},A\_\{k\},I\_\{k\}^\{\\mathrm\{in\}\}\)\-\\hat\{F\}\_\{k\}^\{\\mathrm\{caus\}\}\(X\_\{k\},A\_\{k\},\\hat\{I\}\_\{k\}^\{\\mathrm\{in\}\}\)\}\_\{\\Delta\_\{k\}^\{\\mathrm\{graph\}\}\}\.\(102\)Using the squared triangle/Minkowski bound and the definitions above,

𝔼​‖Xk,t\+1−X^k,t\+1FCC‖2\\displaystyle\\mathbb\{E\}\\left\\lVert X\_\{k,t\+1\}\-\\hat\{X\}\_\{k,t\+1\}^\{\\mathrm\{FCC\}\}\\right\\rVert^\{2\}≤K​\(𝔼​‖Δklocal‖2\+𝔼​‖Δkgraph‖2\)\\displaystyle\\leq K\\left\(\\mathbb\{E\}\\left\\lVert\\Delta\_\{k\}^\{\\mathrm\{local\}\}\\right\\rVert^\{2\}\+\\mathbb\{E\}\\left\\lVert\\Delta\_\{k\}^\{\\mathrm\{graph\}\}\\right\\rVert^\{2\}\\right\)\(103\)≤K​\(εklocal\+εinterface\+εdirection\+εlag\+εcomposition\)\.\\displaystyle\\leq K\\left\(\\varepsilon\_\{k\}^\{\\mathrm\{local\}\}\+\\varepsilon^\{\\mathrm\{interface\}\}\+\\varepsilon^\{\\mathrm\{direction\}\}\+\\varepsilon^\{\\mathrm\{lag\}\}\+\\varepsilon^\{\\mathrm\{composition\}\}\\right\)\.\(104\)Hence

ℰGFCC\\displaystyle\\mathcal\{E\}^\{\\mathrm\{FCC\}\}\_\{G\}:=𝔼Pint​‖X1:K,t\+1−X^1:K,t\+1FCC‖2\\displaystyle:=\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\\left\\lVert X\_\{1:K,t\+1\}\-\\hat\{X\}\_\{1:K,t\+1\}^\{\\mathrm\{FCC\}\}\\right\\rVert^\{2\}\(105\)=𝔼Pint​∑k=1K‖Xk,t\+1−X^k,t\+1FCC‖2\\displaystyle=\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\\sum\_\{k=1\}^\{K\}\\left\\lVert X\_\{k,t\+1\}\-\\hat\{X\}\_\{k,t\+1\}^\{\\mathrm\{FCC\}\}\\right\\rVert^\{2\}\(106\)=∑k=1K𝔼Pint​‖Xk,t\+1−X^k,t\+1FCC‖2\\displaystyle=\\sum\_\{k=1\}^\{K\}\\mathbb\{E\}\_\{P\_\{\\mathrm\{int\}\}\}\\left\\lVert X\_\{k,t\+1\}\-\\hat\{X\}\_\{k,t\+1\}^\{\\mathrm\{FCC\}\}\\right\\rVert^\{2\}\(107\)≤K​\(∑k=1Kεklocal\+εinterface\+εdirection\+εlag\+εcomposition\)\\displaystyle\\leq K\\left\(\\sum\_\{k=1\}^\{K\}\\varepsilon\_\{k\}^\{\\mathrm\{local\}\}\+\\varepsilon^\{\\mathrm\{interface\}\}\+\\varepsilon^\{\\mathrm\{direction\}\}\+\\varepsilon^\{\\mathrm\{lag\}\}\+\\varepsilon^\{\\mathrm\{composition\}\}\\right\)\(108\)=K​\(∑k=1Kεklocal\+εinterface\+ε1\)\.\\displaystyle=K\\left\(\\sum\_\{k=1\}^\{K\}\\varepsilon\_\{k\}^\{\\mathrm\{local\}\}\+\\varepsilon^\{\\mathrm\{interface\}\}\+\\varepsilon\_\{1\}\\right\)\.\(109\)The interface term satisfies

εinterface\\displaystyle\\varepsilon^\{\\mathrm\{interface\}\}=L2​P​\(EM∗⊈E^M\)\\displaystyle=L^\{2\}P\(E\_\{M\}^\{\*\}\\nsubseteq\\hat\{E\}\_\{M\}\)\(110\)≤L2​\|EM∗\|​e−Nmin​qmin​rmin\.\\displaystyle\\leq L^\{2\}\|E\_\{M\}^\{\*\}\|e^\{\-N\_\{\\min\}q\_\{\\min\}r\_\{\\min\}\}\.\(111\)Therefore

ℰGFCC\\displaystyle\\mathcal\{E\}^\{\\mathrm\{FCC\}\}\_\{G\}≤K​\(∑k=1Kεklocal\+L2​\|EM∗\|​e−Nmin​qmin​rmin\+ε1\)\\displaystyle\\leq K\\left\(\\sum\_\{k=1\}^\{K\}\\varepsilon\_\{k\}^\{\\mathrm\{local\}\}\+L^\{2\}\|E\_\{M\}^\{\*\}\|e^\{\-N\_\{\\min\}q\_\{\\min\}r\_\{\\min\}\}\+\\varepsilon\_\{1\}\\right\)\(112\)≤K​\(K​ε0\+L2​\|EM∗\|​e−Nmin​qmin​rmin\+ε1\)\.\\displaystyle\\leq K\\left\(K\\varepsilon\_\{0\}\+L^\{2\}\|E\_\{M\}^\{\*\}\|e^\{\-N\_\{\\min\}q\_\{\\min\}r\_\{\\min\}\}\+\\varepsilon\_\{1\}\\right\)\.\(113\)Taking limits,

limNmin→∞ℰGFCC\\displaystyle\\lim\_\{N\_\{\\min\}\\to\\infty\}\\mathcal\{E\}^\{\\mathrm\{FCC\}\}\_\{G\}≤K​\(K​ε0\+ε1\),\\displaystyle\\leq K\(K\\varepsilon\_\{0\}\+\\varepsilon\_\{1\}\),\(114\)limNmin→∞ε0,ε1→0ℰGFCC\\displaystyle\\lim\_\{\\begin\{subarray\}\{c\}N\_\{\\min\}\\to\\infty\\\\ \\varepsilon\_\{0\},\\varepsilon\_\{1\}\\to 0\\end\{subarray\}\}\\mathcal\{E\}^\{\\mathrm\{FCC\}\}\_\{G\}=0\.\\displaystyle=0\.\(115\)ForK=6K=6,L=1L=1,\|EM∗\|=14\|E\_\{M\}^\{\*\}\|=14,Nmin=50N\_\{\\min\}=50, andqmin​rmin=0\.48q\_\{\\min\}r\_\{\\min\}=0\.48,

K​L2​\|EM∗\|​e−Nmin​qmin​rmin=6⋅14⋅e−24≃3\.2×10−9\.KL^\{2\}\|E\_\{M\}^\{\*\}\|e^\{\-N\_\{\\min\}q\_\{\\min\}r\_\{\\min\}\}=6\\cdot 14\\cdot e^\{\-24\}\\simeq 3\.2\\times 10^\{\-9\}\.

### A\.6Strict Oracle Dominance

Assume

γ\>0,δint2\>K​\(K​ε0\+ε1\)\.\\gamma\>0,\\qquad\\delta\_\{\\mathrm\{int\}\}^\{2\}\>K\(K\\varepsilon\_\{0\}\+\\varepsilon\_\{1\}\)\.The lower bound above gives

infF^obsRint​\(F^obs\)≥δint2\.\\inf\_\{\\hat\{F\}^\{\\mathrm\{obs\}\}\}R\_\{\\mathrm\{int\}\}\(\\hat\{F\}^\{\\mathrm\{obs\}\}\)\\geq\\delta\_\{\\mathrm\{int\}\}^\{2\}\.If

Nmin\>1qmin​rmin​log⁡K​L2​\|EM∗\|δint2−K​\(K​ε0\+ε1\),N\_\{\\min\}\>\\frac\{1\}\{q\_\{\\min\}r\_\{\\min\}\}\\log\\frac\{KL^\{2\}\|E\_\{M\}^\{\*\}\|\}\{\\delta\_\{\\mathrm\{int\}\}^\{2\}\-K\(K\\varepsilon\_\{0\}\+\\varepsilon\_\{1\}\)\},then

Nmin​qmin​rmin\\displaystyle N\_\{\\min\}q\_\{\\min\}r\_\{\\min\}\>log⁡K​L2​\|EM∗\|δint2−K​\(K​ε0\+ε1\)\\displaystyle\>\\log\\frac\{KL^\{2\}\|E\_\{M\}^\{\*\}\|\}\{\\delta\_\{\\mathrm\{int\}\}^\{2\}\-K\(K\\varepsilon\_\{0\}\+\\varepsilon\_\{1\}\)\}\(116\)e−Nmin​qmin​rmin\\displaystyle e^\{\-N\_\{\\min\}q\_\{\\min\}r\_\{\\min\}\}<δint2−K​\(K​ε0\+ε1\)K​L2​\|EM∗\|\\displaystyle<\\frac\{\\delta\_\{\\mathrm\{int\}\}^\{2\}\-K\(K\\varepsilon\_\{0\}\+\\varepsilon\_\{1\}\)\}\{KL^\{2\}\|E\_\{M\}^\{\*\}\|\}\(117\)K​L2​\|EM∗\|​e−Nmin​qmin​rmin\\displaystyle KL^\{2\}\|E\_\{M\}^\{\*\}\|e^\{\-N\_\{\\min\}q\_\{\\min\}r\_\{\\min\}\}<δint2−K​\(K​ε0\+ε1\)\.\\displaystyle<\\delta\_\{\\mathrm\{int\}\}^\{2\}\-K\(K\\varepsilon\_\{0\}\+\\varepsilon\_\{1\}\)\.\(118\)Substitution into the oracle upper bound gives

ℰGFCC\\displaystyle\\mathcal\{E\}^\{\\mathrm\{FCC\}\}\_\{G\}≤K​\(K​ε0\+L2​\|EM∗\|​e−Nmin​qmin​rmin\+ε1\)\\displaystyle\\leq K\\left\(K\\varepsilon\_\{0\}\+L^\{2\}\|E\_\{M\}^\{\*\}\|e^\{\-N\_\{\\min\}q\_\{\\min\}r\_\{\\min\}\}\+\\varepsilon\_\{1\}\\right\)\(119\)=K​\(K​ε0\+ε1\)\+K​L2​\|EM∗\|​e−Nmin​qmin​rmin\\displaystyle=K\(K\\varepsilon\_\{0\}\+\\varepsilon\_\{1\}\)\+KL^\{2\}\|E\_\{M\}^\{\*\}\|e^\{\-N\_\{\\min\}q\_\{\\min\}r\_\{\\min\}\}\(120\)<K​\(K​ε0\+ε1\)\+δint2−K​\(K​ε0\+ε1\)\\displaystyle<K\(K\\varepsilon\_\{0\}\+\\varepsilon\_\{1\}\)\+\\delta\_\{\\mathrm\{int\}\}^\{2\}\-K\(K\\varepsilon\_\{0\}\+\\varepsilon\_\{1\}\)\(121\)=δint2\\displaystyle=\\delta\_\{\\mathrm\{int\}\}^\{2\}\(122\)≤infF^obsRint​\(F^obs\)\.\\displaystyle\\leq\\inf\_\{\\hat\{F\}^\{\\mathrm\{obs\}\}\}R\_\{\\mathrm\{int\}\}\(\\hat\{F\}^\{\\mathrm\{obs\}\}\)\.\(123\)Thus the oracle causal composition is strictly below the non\-causal interventional lower bound\.

Equivalent slack parametrization: choose

δ0=δint24​K2,δ1=δint24​K\.\\delta\_\{0\}=\\frac\{\\delta\_\{\\mathrm\{int\}\}^\{2\}\}\{4K^\{2\}\},\\qquad\\delta\_\{1\}=\\frac\{\\delta\_\{\\mathrm\{int\}\}^\{2\}\}\{4K\}\.Then

K​δ0\+δ1\\displaystyle K\\delta\_\{0\}\+\\delta\_\{1\}=δint24​K\+δint24​K=δint22​K,\\displaystyle=\\frac\{\\delta\_\{\\mathrm\{int\}\}^\{2\}\}\{4K\}\+\\frac\{\\delta\_\{\\mathrm\{int\}\}^\{2\}\}\{4K\}=\\frac\{\\delta\_\{\\mathrm\{int\}\}^\{2\}\}\{2K\},\(124\)δint2K−K​δ0−δ1\\displaystyle\\frac\{\\delta\_\{\\mathrm\{int\}\}^\{2\}\}\{K\}\-K\\delta\_\{0\}\-\\delta\_\{1\}=δint22​K\>0,\\displaystyle=\\frac\{\\delta\_\{\\mathrm\{int\}\}^\{2\}\}\{2K\}\>0,\(125\)N∗\\displaystyle N^\{\*\}=1qmin​rmin​log⁡L2​\|EM∗\|δint2/K−K​δ0−δ1\\displaystyle=\\frac\{1\}\{q\_\{\\min\}r\_\{\\min\}\}\\log\\frac\{L^\{2\}\|E\_\{M\}^\{\*\}\|\}\{\\delta\_\{\\mathrm\{int\}\}^\{2\}/K\-K\\delta\_\{0\}\-\\delta\_\{1\}\}\(126\)=1qmin​rmin​log⁡2​K​L2​\|EM∗\|δint2\.\\displaystyle=\\frac\{1\}\{q\_\{\\min\}r\_\{\\min\}\}\\log\\frac\{2KL^\{2\}\|E\_\{M\}^\{\*\}\|\}\{\\delta\_\{\\mathrm\{int\}\}^\{2\}\}\.\(127\)Forδint=0\.12\\delta\_\{\\mathrm\{int\}\}=0\.12,K=6K=6,L=1L=1,\|EM∗\|=14\|E\_\{M\}^\{\*\}\|=14, andqmin​rmin=0\.48q\_\{\\min\}r\_\{\\min\}=0\.48,

N∗=log⁡\(84/\(0\.122/12\)\)0\.48=log⁡\(70000\)0\.48≃24\.N^\{\*\}=\\frac\{\\log\(84/\(0\.12^\{2\}/12\)\)\}\{0\.48\}=\\frac\{\\log\(70000\)\}\{0\.48\}\\simeq 24\.Ifγ=0\\gamma=0, thenδint=0\\delta\_\{\\mathrm\{int\}\}=0and the dominance denominator is non\-positive; no strict dominance threshold exists\.