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Causal Retention in Interactive Agents: Interface Factorization and Selective Adaptation
Shengjun Zhang, Tingyi Liu, Dong Xie, Yunlong Dong, Xiang Wang, Cheng Zeng
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 96%
Last extracted: 9/28/2026, 3:20:51 AM
Summary
This paper introduces 'causal retention,' a property where a frozen learned state in an interactive agent can accurately answer mechanism-probe maps (action, context, target, value, delay) fixed independently of training. The authors define causal retention via structural causal models (SCMs) and Bayes decision risk, proposing 'Causal Core' as an implementation using evidence-gated writing and readout filtering. Experiments on Qwen2.5-7B-Instruct and TD-MPC2 demonstrate that this approach achieves high accuracy in identifying causal mechanisms and supports selective adaptation without degrading stable responses, distinguishing causal retention from mere task sufficiency.
Entities (8)
Relation Signals (6)
Shengjun Zhang → affiliatedwith → Hubei University
confidence 99% · Shengjun Zhang sj.zhang@hubu.edu.cn Affiliation: School of Artificial Intelligence, Hubei University
Dong Xie → affiliatedwith → Baidu Inc.
confidence 99% · Dong Xie xiedong04@baidu.com Affiliation: Baidu Inc.
Causal Core → implements → Causal Retention
confidence 95% · Causal Core implements these conditions through evidence-gated writing, readout filtering, temporal credit, hidden-context setup, and local diagnostic updates.
Qwen2.5-7B-Instruct → usedinexperiment → Causal Retention
confidence 92% · A frozen Qwen last-layer probe reaches 0.958 balanced accuracy on source mechanisms but 0.583 on changed delays; the gated mechanism state reaches 1.000
TD-MPC2 → usedinexperiment → Causal Retention
confidence 92% · In TD-MPC2, five target states per actuator recover effect-sign accuracy from 0.057 to 0.948 without degrading stable responses.
Causal Retention → distinctfrom → Task-Sufficiency
confidence 90% · Causal retention is therefore distinct from task sufficiency and source-domain decodability.
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Abstract
Abstract:Task performance need not determine which intervention mechanism an agent retains. We study causal retention: whether a frozen learned state answers a mechanism-probe map fixed independently of training, including action, context, direct target, value, and delay. For finite structural causal model classes, the optimal probe error is a Bayes decision risk. It vanishes exactly when every learning-interface fiber lies within one probe-answer fiber; any state obtained by post-processing that interface inherits the same lower bound. A posterior-coverage theorem characterizes budgeted retesting, while an exact edit decomposition shows that the shifted set is the unique support of an error-free target update. Causal Core implements these conditions through evidence-gated writing, readout filtering, temporal credit, hidden-context setup, and local diagnostic updates. Experiments cover finite causal systems, continuous simulators, an official TD-MPC2 world model, and Qwen2.5-7B-Instruct. A frozen Qwen last-layer probe reaches 0.958 balanced accuracy on source mechanisms but 0.583 on changed delays; the gated mechanism state reaches 1.000 and accepts only 0.056 of synchronized-readout candidates. In TD-MPC2, five target states per actuator recover effect-sign accuracy from 0.057 to 0.948 without degrading stable responses. Causal retention is therefore distinct from task sufficiency and source-domain decodability.
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- Source: https://arxiv.org/abs/2609.30650v1
- Canonical: https://arxiv.org/abs/2609.30650v1
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Shengjun Zhang, Tingyi Liu, Dong Xie, Yunlong Dong, Xiang Wang, and Cheng Zeng Causal Retention in Interactive Agents: Interface Factorization and Selective Adaptation Shengjun Zhang sj.zhang@hubu.edu.cn Affiliation: School of Artificial Intelligence, Hubei University Affiliation: Key Laboratory of Intelligent Sensing System and Security (Hubei University), Ministry of Education Tingyi Liu tingyiL@whu.edu.cn Affiliation: School of Economics and Management, Wuhan University Dong Xie xiedong04@baidu.com Affiliation: Baidu Inc. Yunlong Dong yunlongdong@outlook.com Affiliation: Independent Researcher*Correspondence: xiedong04@baidu.com; wangxiang.whu@whu.edu.cn Xiang Wang wangxiang.whu@whu.edu.cn Affiliation: School of Artificial Intelligence, Hubei University Affiliation: Key Laboratory of Intelligent Sensing System and Security (Hubei University), Ministry of Education Cheng Zeng zc@hubu.edu.cn Affiliation: School of Artificial Intelligence, Hubei University Affiliation: Key Laboratory of Intelligent Sensing System and Security (Hubei University), Ministry of Education Abstract Task performance need not determine which intervention mechanism an agent retains. We study causal retention: whether a frozen learned state answers a mechanism-probe map fixed independently of training, including action, context, direct target, value, and delay. For finite structural causal model classes, the optimal probe error is a Bayes decision risk. It vanishes exactly when every learning-interface fiber lies within one probe-answer fiber; any state obtained by post-processing that interface inherits the same lower bound. A posterior-coverage theorem characterizes budgeted retesting, while an exact edit decomposition shows that the shifted set is the unique support of an error-free target update. Causal Core implements these conditions through evidence-gated writing, readout filtering, temporal credit, hidden-context setup, and local diagnostic updates. Experiments cover finite causal systems, continuous simulators, an official TD-MPC2 world model, and Qwen2.5-7B-Instruct. A frozen Qwen last-layer probe reaches 0.958 balanced accuracy on source mechanisms but 0.583 on changed delays; the gated mechanism state reaches 1.000 and accepts only 0.056 of synchronized-readout candidates. In TD-MPC2, five target states per actuator recover effect-sign accuracy from 0.057 to 0.948 without degrading stable responses. Causal retention is therefore distinct from task sufficiency and source-domain decodability. †heading: Preprint 2026 1– September 2026 †shortheadings: Causal Retention in Interactive Agents / Zhang, Liu, Xie, Dong, Wang, and Zeng†firstpage: 1 keywords causal retention, interactive agents, mechanism memory, causal adaptation, structural causal models 1 Introduction and Related Work Training objectives identify equivalence classes of environments, not complete intervention mechanisms. Two interactive systems may induce the same reward process, preferred action, or one-step prediction target while disagreeing about the direct target of an action, its active context, or its delay. A state that is sufficient for the training task can therefore be insufficient for an intervention query that the objective never had to answer. The ability of a frozen learned state to answer such queries is called causal retention. The evaluator fixes the probe semantics independently of the learned state, freezes that state after source learning, and then asks which action changes which target, under which context, to what value, and after what delay. Under a local mechanism shift, the same state must identify the changed entry without overwriting invariant ones. This is a property of what learning retained, not only of the behavior it produced. Several causal-learning traditions address neighboring but different estimands. Structural causal models define interventions and counterfactuals (Pearl, 2009; Peters et al., 2017). Treatment-effect methods estimate average, conditional, or individual effects on a designated outcome (Rubin, 1974; Imbens and Rubin, 2015; Chernozhukov et al., 2018; Wager and Athey, 2018; Künzel et al., 2019; Nie and Wager, 2021; Shalit et al., 2017). Causal discovery instead estimates a graph or an interventional equivalence class (Spirtes et al., 2000; Chickering, 2002; Hauser and Bühlmann, 2012; Shimizu et al., 2006; Zheng et al., 2018). A correct outcome effect or graph can remain silent about the action-indexed value, delay, and context that must be edited after transfer. Invariant prediction seeks predictors stable across environments (Peters et al., 2016; Arjovsky et al., 2019). Causal representation learning instead studies recovery of latent causal variables or graphs from indirect observations (Locatello et al., 2019; Schölkopf et al., 2021; Gamella et al., 2025; Varici et al., 2025). Intervention extrapolation uses an identifiable representation to predict the effect of unseen actions on an outcome (Saengkyongam et al., 2024). Causal retention fixes the downstream answer map and asks whether a completed learning process preserved it. It neither requires recovery of the latent SCM nor reduces the answer to one designated outcome. Reinforcement learning, world models, and language agents optimize return, transition prediction, or action quality (Sutton and Barto, 2018; Ha and Schmidhuber, 2018; Hafner et al., 2025; Hansen et al., 2024; Yao et al., 2023). Causal reinforcement learning and data-fusion methods improve decisions under interventions (Bareinboim and Pearl, 2016; Zhang et al., 2021), while ranking-style dynamics losses such as Hybrid2 supervise which intervention should be preferred (Zou et al., 2024). These objectives can be useful and exactly optimized without requiring the frozen state to expose every coordinate of the mechanism needed by a later probe. The distinction studied here is therefore between an objective’s behavioral sufficiency and a learned state’s causal sufficiency. Concurrent work on Explicit Symbolic Behavioral Models uses executable “mechanism memory,” adaptive questions, and active world-model branches to train editable symbolic policies (Shindo et al., 2026). There, mechanism memory predicts symbolic events and rewards and is optimized jointly with policy-specific questions. Here, the probe map and its direct-target semantics are fixed independently of the learner, scoring occurs after the state is frozen, and the main object is the smallest probe error permitted by a given learning interface. The two formulations therefore use similar language for different statistical questions. One realization of causal retention is a frozen, queryable mechanism memory with entries of the form θ=(a,c,y,v,δ),θ=(a,c,y,v,δ), where a is an action, c a context predicate, y the direct target, v the effect value, and δ the delay. After learning, the memory is frozen and queried on held-out mechanism probes. Equal reward laws, rankings, graphs, or action choices may still induce different probe answers. For finite structural causal model (SCM) classes, the optimal population probe error is an explicit Bayes risk. It is zero exactly when the interface quotient refines the probe quotient, is monotone under refinement, and lower-bounds every decoder of a post-processed state. The same construction with a metric loss covers continuous intervention responses. A posterior-coverage identity characterizes restricted target-domain retesting, and an exact edit decomposition characterizes selective adaptation. Standard concentration and testing inequalities are used only for finite-sample rates. Causal Core is the corresponding evidence-gated memory layer, with target admission, readout filtering, temporal credit, hidden-context setup, diagnostic localization, and entrywise updating. Experiments compare policy, ranking, conditional-discovery, tuple, latent-world-model, neural-probe, transfer, language-model proposal, readout-unsafe, and ungated baselines. Continuous responses are scored over held-out state, action-direction, and horizon distributions. An official pretrained TD-MPC2 checkpoint supplies a public implicit-world-model comparison. Frozen Qwen hidden states are evaluated with linear and nonlinear decoders under externally fixed probe semantics. Figure 1: Interface collapse and causal retention. A task interface can merge probe-distinct SCMs. A refined interface retains the answer map and supports selective adaptation. 2 Problem Formulation Let an interactive structural causal model (SCM) be Xt+1=F(Xt,Ht,At,εt+1),Ot=ψ(Xt,Ht,ηt),X_t+1=F(X_t,H_t,A_t, _t+1), O_t=ψ(X_t,H_t, _t), with physical state Xt∈X_t , optional latent context Ht∈ℋH_t , action At∈A_t , and observation OtO_t. Write Ξt=(Xt,Ht)∈×ℋ _t=(X_t,H_t) \!×\!H for the full evaluator state; the learner need not observe HtH_t. A protocol Π maps histories to actions and induces a trajectory law PMΠP_M for each SCM M. For an action intervention at time t, write PMdo(At=a),ΠP_M^do(A_t=a), for the law obtained by replacing the protocol action with a at t and leaving the remaining rollout policy fixed. A context predicate is a map c:×ℋ→0,1c:X×H→\0,1\. From the equations of M, the evaluator fixes a structural index σM(ξ,a)∈(×0,…,D)∪⊥. _M(ξ,a)∈ (C×Y×\0,…,D\ )∪\ \. When non-null, σM(ξ,a)=(c,y,δ) _M(ξ,a)=(c,y,δ) records the active context class, the variable directly acted on by the mechanism, and its first structural response delay. A deterministic descendant or sensor readout is not a direct target. In finite worlds σM _M is read from the structural assignment; in continuous simulators it is defined by the paired-pulse functional in Appendix B.4. For z=(ξ,a)z=(ξ,a) and r=(c,y,v,δ)r=(c,y,v,δ), define pM(r∣z):=σM(ξ,a)=(c,y,δ)ℙMdo(At=a),ΠXt+δy=v∣Ξt=ξ.p_M(r z):=1\ _M(ξ,a)=(c,y,δ)\P_M^do(A_t=a), \X_t+δ^y=v _t=ξ\. Let ℛR be the finite set of candidate quadruples (c,y,v,δ)(c,y,v,δ). If σM(ξ,a)=⊥ _M(ξ,a)= , set GM(z)=∅G_M(z)= ; otherwise define GM(z)=argmaxr∈ℛpM(r∣z),G_M(z)= *arg\,max_r p_M(r z), restricted to a probe class on which the maximizer is unique. In deterministic SCMs a tuple θ=(a,c,y,v,δ)θ=(a,c,y,v,δ) is correct exactly when pM((c,y,v,δ)∣z)=1p_M((c,y,v,δ) z)=1. In stochastic SCMs uniqueness is enforced by a margin κ>0κ>0: pM(r⋆∣z)−maxr≠r⋆pM(r∣z)≥κ.p_M(r z)- _r≠ r p_M(r z)≥κ. 2.1 Mechanism Space Let Θ=×0,…,D. =A×C×Y×V×\0,…,D\. A mechanism memory is a finite set μ⊂Θμ⊂ . For a query z=(ξ,a)z=(ξ,a), define Tμ(ξ,a)=(c,y,v,δ):(a,c,y,v,δ)∈μ,c(ξ)=1.T_μ(ξ,a)=\(c,y,v,δ):(a,c,y,v,δ)∈μ,\ c(ξ)=1\. Two answers are equivalent, written Tμ(ξ,a)≡Tμ′(ξ,a)T_μ(ξ,a)≡ T_μ (ξ,a), when they agree on the active context, direct target, value, and delay classes fixed by the evaluator. Exact tuple evaluation uses ordinary set equality. The constructive and empirical results use a representable mechanism schema: for every evaluated SCM M, there is a finite ground-truth memory μM⋆⊂Θ _M ⊂ such that TμM⋆(z)=GM(z)T_ _M (z)=G_M(z) for Q-almost every probe z. The lower bounds are stated directly for GMG_M and do not require such a representation. 2.2 Mechanism Risk For probe distribution Q and ground-truth memory μ⋆=μM⋆μ = _M , the exact mechanism risk is RQ(μ^,μ⋆)=ℙ(ξ,a)∼Q[Tμ^(ξ,a)≢Tμ⋆(ξ,a)].R_Q( μ,μ )=P_(ξ,a) Q\! [T_ μ(ξ,a) ≡ T_μ (ξ,a) ]. Graph error, reward regret, top-action accuracy, and prediction error are not substitutes for RQR_Q: each marginalizes at least one coordinate of (a,c,y,v,δ)(a,c,y,v,δ). Coordinate risks localize the source of an error. Let =ctx,tar,val,delK=\ctx,tar,val,del\ and let πkTμ(ξ,a) _kT_μ(ξ,a) be the projection of the answer set onto coordinate k. Define RQk(μ^,μ⋆)=ℙQ[πkTμ^(ξ,a)≠πkTμ⋆(ξ,a)].R_Q^k( μ,μ )=P_Q\! [ _kT_ μ(ξ,a)≠ _kT_μ (ξ,a) ]. When equivalence means equality of all four projections, maxk∈RQk≤RQ≤∑k∈RQk. _k R_Q^k≤ R_Q≤ _k R_Q^k. Thus the exact risk is the main target, while coordinate risks diagnose which part of the mechanism failed. Two additional operational risks are Rfp R_fp =ℙQ[Tμ^(ξ,a)≠∅,Tμ⋆(ξ,a)=∅], =P_Q[T_ μ(ξ,a)≠ ,\ T_μ (ξ,a)= ], Rfn R_fn =ℙQ[Tμ^(ξ,a)=∅,Tμ⋆(ξ,a)≠∅]. =P_Q[T_ μ(ξ,a)= ,\ T_μ (ξ,a)≠ ]. They distinguish spurious mechanisms from missed mechanisms. The fixed-memory protocol is essential. Exploration may use rewards, observations, auxiliary descriptions, or rankings; evaluation fixes μ μ and queries only the answer map Tμ^T_ μ. Thus a behaviorally competent learner receives no credit unless its stored mechanism answers the probe: write T^i=Tμ^(ξi,ai) T_i=T_ μ( _i,a_i) and Ti⋆=Tμ⋆(ξi,ai)T_i =T_μ ( _i,a_i). Then μ^↦Tμ^(ξi,ai)i=1m, μ \T_ μ( _i,a_i)\_i=1^m, score=1−1m∑i=1mℓi,ℓi=T^i≢Ti⋆.score=1- 1m _i=1^m _i, _i=1\ T_i ≡ T_i \. 2.3 Selective Adaptation Let μs=θjsj=1nμ^s=\ _j^s\_j=1^n be a source memory. A schema map ϕ=(ϕA,ϕC,ϕY)φ=( _A, _C, _Y) induces mapped entries ϕ(θjs)=(ϕA(ajs),ϕC(cjs),ϕY(yjs),vjs,δjs).φ( _j^s)=( _A(a_j^s), _C(c_j^s), _Y(y_j^s),v_j^s, _j^s). The target memory is μt=θjtj=1nμ^t=\ _j^t\_j=1^n. Define the stable and shifted index sets S=j:θjt=ϕ(θjs),B=[n]∖S.S=\j: _j^t=φ( _j^s)\, B=[n] S. For an adapted memory μ^t μ^t, Ladapt L_adapt =1n∑j=1nθ^jt≠θjt, = 1n _j=1^n1\ θ_j^t≠ _j^t\, Lstab L_stab =1|S|∑j∈Sθ^jt≠θjt, = 1|S| _j∈ S1\ θ_j^t≠ _j^t\, and LshiftL_shift is defined analogously over B; an empty-set loss is defined as zero. Selective adaptation requires Lstab=Lshift=0L_stab=L_shift=0 under a small target budget. For exploration and sampling randomness conditional on a fixed source-target pair, define ℛadapt=[Ladapt∣μs,μt],ℛstab=[Lstab∣μs,μt],R_adapt=E[L_adapt μ^s,μ^t], _stab=E[L_stab μ^s,μ^t], ℛshift=[Lshift∣μs,μt].R_shift=E[L_shift μ^s,μ^t]. The expectation decomposes as ℛadapt=|S|nℛstab+|B|nℛshift.R_adapt= |S|nR_stab+ |B|nR_shift. The operational failure probability is ℛop=ℙ(Lstab>0∨Lshift>0),R_op=P(L_stab>0\ \ L_shift>0), because a single missed shifted mechanism can be invisible in average loss when |B|≪n|B| n. Two limiting cases illustrate the requirement. Pure transfer keeps μ^t=ϕ(μs) μ^t=φ(μ^s) and therefore fails on B. Broad re-learning may fix B but can corrupt S. The desired operation is an entrywise map θ^jt=ϕ(θjs),j∈S,θjt,j∈B, θ_j^t= casesφ( _j^s),&j∈ S,\\ _j^t,&j∈ B, cases with evidence deciding membership in B. 2.4 Standing Conditions Each lower bound uses only the assumptions in its statement. The constructive guarantees for Causal Core use the following grouped conditions. (C1) The source and target memories contain n aligned keys. The evaluator’s probe distribution Q is fixed independently of target-domain samples. The schema map ϕφ is supplied or estimated from a disjoint alignment sample; its error is represented by the event FϕF_φ rather than assumed away. Target-coordinate values and visible context predicates in the memory schema are measurable from the learner’s observation history. Latent predicates are never supplied to the learner and are handled only through assigned setup and control regimes. (C2) A gate is estimated from independent, bounded outcomes collected under paired action and reference interventions at the same pre-intervention context. Equivalently, an observational implementation must satisfy consistency, conditional exchangeability, and positivity and use the corresponding propensity correction. An adaptive randomized policy may use the logged inverse-propensity contrast in Lemma 11. The finite agents execute interventions and are scored descriptively; the continuous and simulator protocols use explicit paired pulses. (C3) Every finite decision used by a gate is separated from its threshold. Write γg>0 _g>0 for target, context, readout, and temporal margins, Δ=p−q>0 =p-q>0 for diagnostic separation, and ηj>0 _j>0 for the excess loss of a wrong local-update candidate. The correct local candidate belongs to the finite class jN_j. (C4) Metric-valued probe answers lie in a compact space and use a bounded, continuous distortion. The probe distribution is fixed before fitting. A discrete implementation may additionally fix a measurable partition; its paired contrast routine fails with probability at most βM _M. Conditions C2–C3 are sufficient, not necessary. When they fail, the quotient and decision lower bounds still apply, but the finite-sample upper bounds need not. 2.5 Population Interfaces An interface ℐI induces a population descriptor τℐ,Π(M)=PMℐ,Π _I, (M)=P_M^I, : the data law generated by the protocol. Examples include reward trajectories, observations, rankings, prediction errors, diagnostic signals, or mechanism probes. For probe class Z with answer map GM(z)G_M(z), write M0∼ℐ,ΠM1 M_0 _I, M_1 ⟺PM0ℐ,Π=PM1ℐ,Π, P_M_0^I, =P_M_1^I, , M0∼M1 M_0 _ZM_1 ⟺GM0(z)=GM1(z)∀z∈. G_M_0(z)=G_M_1(z) ∀ z . The relevant condition is whether ∼ℐ,Π _I, refines ∼ _Z for mechanism probes. If not, the interface is too coarse. For the finite analysis, fix full-support distributions ν on ℳM and Q on Z. Let W∼ν,Z∼Q,ℐ=τℐ,Π(W),U=GW(Z),W ν, Z Q, T_I= _I, (W), U=G_W(Z), where Z is independent of W. The causal-retention risk of the interface is the probe Bayes risk ν,Q(ℐ,Π)=[1−maxu∈ℙ(U=u∣ℐ,Z)]. B_ν,Q(I, )=E\! [1- _u P(U=u T_I,Z) ]. (1) It will equal the smallest error of any decoder that sees the complete population interface. For U=(U1,…,Ud)U=(U_1,…,U_d), define j B_j by replacing U with UjU_j in Eq. (1). These risks depend on the interface and the fixed probe map rather than on a chosen decoder. The population minimax interface risk is ℜ(ℐ,)=infG^supM∈ℳG^(τℐ,Π(M))≠GM. R(I,Z)= _ G _M 1\ G( _I, (M))≠ G_M\. Theorem 2 gives the exact operational condition: ℜ(ℐ,)=0 R(I,Z)=0 only when the interface separates all probe-distinct SCMs. For a compact metric answer space (,ρ)(U,ρ), define the expected and worst-case metric risks ν,Qρ(ℐ,Π) B^ρ_ν,Q(I, ) =infgρ(U,g(ℐ,Z)), = _gEρ (U,g( T_I,Z) ), (2) ℜρ(ℐ,) R^ρ(I,Z) =infgsupM∈ℳ,z∈ρ(GM(z),g(τℐ,Π(M),z)). = _g _M ,\,z ρ (G_M(z),g( _I, (M),z) ). For Hamming loss, ρ= B^ρ= B. The metric version therefore changes the loss geometry, not the semantics of retention. 2.6 Exact Causal Retention A possibly random learned state S has exact causal retention for Z on ℳM if there exists a single decoder d such that ℙMd(S,z)=GM(z) for every z∈=1∀M∈ℳ.P_M\! \d(S,z)=G_M(z) for every z \=1 ∀ M . The probe class and its answer semantics are fixed independently of the learned state. This makes causal retention independent of the internal implementation: a memory table, recurrent state, or external module qualifies only to the extent that it supports the same decoder after learning has stopped. For ranking interfaces, the observed object is a distribution over ordered pairs rank=(i,j):UM(i)>UM(j),D_rank=\(i,j):U_M(i)>U_M(j)\, where UM(i)U_M(i) is the intervention utility. For reward interfaces it is a trajectory marginal of ∑trt _tr_t. For graph interfaces it is an edge set. For diagnostic adaptation it is instead an indexed surprise channel Xj,rj,r\X_j,r\_j,r tied to transferred memory entries. If two SCMs have the same interface law but different probe answers, no estimator using that interface can guarantee zero mechanism risk. 3 Causal Core Causal Core maintains mechanism memory independently of the policy optimizer. Its state consists of a candidate set CtC_t, readout set ℛtR_t, temporal buffer BtB_t, diagnostic surprise vector StS_t, and memory μt _t. It writes (a,c,y,v,δ)(a,c,y,v,δ) only when intervention evidence supports every coordinate. For a candidate θ=(a,c,y,v,δ)θ=(a,c,y,v,δ), define its memory key κ(θ)=(a,c,y,δ)κ(θ)=(a,c,y,δ). Let a0a^0 be the reference intervention for a. For a learner-visible context predicate c, let t1(a,c,δ) _t^1(a,c,δ) =s:s+δ≤t,As=a,c(Ξs)=1, =\s:s+δ≤ t,\ A_s=a,\ c( _s)=1\, t0(a,c,δ) _t^0(a,c,δ) =s:s+δ≤t,As=a0,c(Ξs)=1. =\s:s+δ≤ t,\ A_s=a^0,\ c( _s)=1\. Write ntu=|tu|n_t^u=|D_t^u| and Ysθ=Xs+δy=vY_s^θ=1\X_s+δ^y=v\. A candidate does not mature until nt0,nt1>0n_t^0,n_t^1>0. Its paired empirical frequencies are p^tu(θ)=1ntu∑s∈tu(a,c,δ)Ysθ,u∈0,1. p_t^u(θ)= 1n_t^u _s _t^u(a,c,δ)Y_s^θ, u∈\0,1\. (3) Under Condition C2, their difference estimates the interventional contrast between do(a)do(a) and do(a0)do(a^0) rather than an unadjusted action association. The target gate is Δ^t(θ) _t(θ) =p^t1(θ)−p^t0(θ), = p_t^1(θ)- p_t^0(θ), (4) Γttar(θ) _t^tar(θ) =nt0nt1>0,Δ^t(θ)≥λt. =1\n_t^0n_t^1>0,\ _t(θ)≥ _t\. The full write gate is Gt(θ) G_t(θ) =∏g∈tar,time,readout,ctxΓtg(θ), = _g∈\tar,time,readout,ctx\ _t^g(θ), (5) Ctadm C_t^adm =θ∈Ct:Gt(θ)=1. =\θ∈ C_t:G_t(θ)=1\. The memory transition is the key-preserving operator (μ,θ)=θ′∈μ:κ(θ′)≠κ(θ)∪θ. W(μ,θ)=\θ ∈μ:κ(θ )≠κ(θ)\∪\θ\. (6) If several candidates mature at the same time, they are applied in decreasing Δ^t _t with a fixed lexicographic tie-breaker. Thus the memory contains at most one value for each action-context-target-delay key. Admission is determined by the four gates; planner design does not enter the definition of mechanism memory. For a latent context, the sets in Eq. (3) are instead indexed by the learner-known assignment to the setup or control regime; membership never uses c(Ξs)c( _s) itself. The resulting context gate is defined in Section 3.4. Remark 1. Ground-truth mechanism labels are not supplied during training. Supervision is created by controlled interventions whose contrasts identify a target, context, value, or delay coordinate. One-step predictive self-supervision uses ot+1o_t+1 as the target given (ot,at)(o_t,a_t); here the retained target is the intervention-specific tuple (a,c,y,v,δ)(a,c,y,v,δ) queried after learning. 3.1 Readout Filtering If Y and R are synchronized, the two SCMs a→Y,R:=Yanda→R,Y:=Ra→ Y,\ R:=Y a→ R,\ Y:=R can produce the same observations. Causal Core therefore excludes readout candidates from direct-target writing unless an intervention separates them. Formally, for a candidate target u, define the direct-target set tdir=u∈:u∉ℛtorSept(u)=1,Y_t^dir=\u :u _t\ or\ Sep_t(u)=1\, Sept(u)=∃b:P(Xt+1u∣do(b))≠P(Xt+1r(u)∣do(b)),Sep_t(u)=1\∃ b:P(X_t+1^u do(b))≠ P(X_t+1^r(u) do(b))\, where r(u)r(u) denotes the synchronized readout paired with u when known. For formula readouts, the finite implementation uses, for each candidate coordinate u, a class ℱuF_u of formulas whose inputs exclude u. It computes A^t(f,u)=1mt∑r=1mtf(Xr)=Xru,u∈ℛt⟺maxf∈ℱuA^t(f,u)≥cro. A_t(f,u)= 1m_t _r=1^m_t1\f(X_r)=X_r^u\, u _t _f _u A_t(f,u)≥ c_ro. Proposition 12 controls both directions of this decision under a finite-class margin. A variable classified as a readout can re-enter the direct-target set only through a separating intervention. The readout gate is Γtreadout(a,c,y,v,δ)=y∈tdir. _t^readout(a,c,y,v,δ)=1\y _t^dir\. 3.2 Temporal Credit For delayed effects, Causal Core tests stability of the lag: δ^(a,c,y,v)∈argmax0≤δ≤DΔ^t(a,c,y,v,δ). δ(a,c,y,v)∈ _0≤δ≤ D _t(a,c,y,v,δ). An entry is written only if the selected lag beats competing recent actions by a margin. The analyzed margin form is Δ^t(a,c,y,v,δ)−max(a′,v′,δ′)≠(a,v,δ)Δ^t(a′,c,y,v′,δ′)≥γtime, _t(a,c,y,v,δ)- _(a ,v ,δ )≠(a,v,δ) _t(a ,c,y,v ,δ )≥ _time, where every contrast uses only matured paired trials from Eq. (3). This prevents a waiting action or a post-effect observation from being written as the cause. 3.3 Diagnostic Adaptation After transfer, mechanism j produces binary surprise samples Xj,r∈0,1X_j,r∈\0,1\. Causal Core computes X¯j(m)=1m∑r=1mXj,r X_j(m)= 1m _r=1^mX_j,r and retests entries above a threshold. The edit is local: θ^jt θ_j^t may change only for selected j. For τ=(p+q)/2τ=(p+q)/2, B^m=j:X¯j(m)≥τ. B_m=\j: X_j(m)≥τ\. (7) θ^jt=ϕ(θjs),j∉B^m,θ~j,j∈B^m. θ_j^t= casesφ( _j^s),&j∉ B_m,\\ θ_j,&j∈ B_m. cases (8) For a selected entry, the identifying retest gives an independent local sample Djre=(Zj,r,Uj,r)r=1rjD_j^re=\(Z_j,r,U_j,r)\_r=1^r_j, where Uj,rU_j,r is the observed response statistic specified by the intervention protocol, and a finite candidate class jN_j. Let gθg_θ be the probe-answer map predicted by candidate θ. Define L^j(θ) L_j(θ) =1rj∑r=1rjgθ(Zj,r)≠Uj,r, = 1r_j _r=1^r_j1\g_θ(Z_j,r)≠ U_j,r\, (9) θ~j θ_j ∈argminθ∈jL^j(θ). ∈ _θ _j L_j(θ). The corresponding local edit is j(μ,θ~j)=θℓ∈μ:ℓ≠j∪θ~j. U_j(μ, θ_j)=\ _ ∈μ: ≠ j\∪\ θ_j\. (10) All entries outside B^m B_m are held fixed. Under Eq. (10), stable corruption can occur only through the set-selection event B^m≠B B_m≠ B. 3.4 Hidden-Context Setup A hidden gate is not written from passive frequency alone. The module actively searches setup actions that increase eligibility, then tests a visible-separator class G. A hidden label survives only when a controlled setup regime and its matched control have separated success rates and no visible separator explains the pattern. For candidate i, let EiE_i be latent eligibility and ZiZ_i success. The learner observes the assigned regime, not EiE_i. It seeks a setup policy πiset _i^set and a matched control πi0 _i^0 such that ℙπiset(Ei=1)≥1−ϵset,ℙπi0(Ei=1)≤ϵ0,P_ _i^set(E_i=1)≥ 1- _set, _ _i^0(E_i=1)≤ _0, πisetZi−πi0Zi≥2γ.E_ _i^setZ_i-E_ _i^0Z_i≥ 2γ. With miset,mi0>0m_i^set,m_i^0>0 outcomes assigned to the two regimes, the context gate uses d^i=1miset∑r∈isetZi,r−1mi0∑r∈i0Zi,r,Γictx=d^i≥γ,g^i=∅, d_i= 1m_i^set _r _i^setZ_i,r- 1m_i^0 _r _i^0Z_i,r, _i^ctx=1\ d_i≥γ,\ g_i= \, where g^i g_i is a visible separator found in G. The label is retained only when the contrast passes and no visible predicate explains it. Lemma 10 controls the contrast error, while Proposition 13 converts repeated setup reachability into action cost when setup completion is observable. If passive access gives ℙ(Ei=1)=qiP(E_i=1)=q_i, then even an oracle that marks eligible trials needs m/qim/q_i attempts on average to collect m eligible outcomes. Algorithm 1 Causal Core memory update and adaptation 1: Initialize μ, readout set ℛR, buffer B, scores dj=0d_j=0. 2: for each interaction step t do 3: Observe oto_t, execute ata_t, and observe the next response. 4: Update ℛR using readout formulas and separating interventions. 5: Add candidate effects to B with action, context, target, value, and delay fields. 6: for each buffered candidate (a,c,y,v,δ)(a,c,y,v,δ) whose delay has matured do 7: if (a,c,y,v,δ)∈Ctadm(a,c,y,v,δ)∈ C_t^adm from Eq. (5) then 8: Update μ by Eq. (6). 9: end if 10: end for 11: For transferred entries, accumulate diagnostic surprise dj←dj+Xj,td_j← d_j+X_j,t. 12: end for 13: Select shifted candidates by Eq. (7) or top-s diagnostic scores. 14: for each selected mechanism index j do 15: Retest entry j and compute θ~j θ_j by Eq. (9). 16: Update μ by Eq. (10). 17: end for 18: return fixed memory μ. 4 Theoretical Results The results connect interface fibers, frozen-state probe error, and the number of target-domain retests. Theorem 2 gives an exact factorization criterion and the optimal probe risk. Corollary 3 gives the finite-sample two-model bound. Theorem 4 gives the exact posterior coverage of budgeted retesting, while Propositions 5 and 6 give constructive rates under the stated margins. Theorem 7 identifies the unique support of an exact target update and composes the finite-sample events. Theorem 8 gives the metric-valued extension. The core results follow by conditional decision minimization, fiber factorization, and an exact edit identity. Standard testing inequalities enter only the finite-sample corollaries. Auxiliary separations and consequences appear in Appendix A. Let M0∼ℐ,ΠM1M_0 _I, M_1 denote equality of the interface law under protocol Π , and let M0∼M1M_0 _ZM_1 denote equality of all fixed probe answers on Z. All logarithms are natural. Theorem 2. Let ℳM and Z be finite, let ν and Q have full support, and use the variables in Eq. (1). Then the following statements hold. (i) The probe Bayes risk has the exact representation ν,Q(ℐ,Π)=infgℙg(ℐ,Z)≠U. B_ν,Q(I, )= _gP\g( T_I,Z)≠ U\. The infimum is attained by a deterministic decoder. (i) ν,Q(ℐ,Π)=0 B_ν,Q(I, )=0 if and only if M0∼ℐ,ΠM1⟹M0∼M1∀M0,M1∈ℳ.M_0 _I, M_1 M_0 _ZM_1 ∀ M_0,M_1 . Hence a zero-risk population decoder exists exactly when the interface quotient refines the probe quotient. (i) If two interface-equivalent models disagree on a set of probes of Q-mass ρ, every population decoder has worst-case mechanism risk at least ρ/2ρ/2. (iv) Let DND_N be a training transcript whose conditional law given W depends on W only through ℐ T_I, and let S be any randomized function of DND_N. Every frozen-state decoder satisfies ℙd(S,Z)≠U≥ν,Q(ℐ,Π).P\d(S,Z)≠ U\≥ B_ν,Q(I, ). (v) If 1=h(2) T_1=h( T_2), then (2)≤(1) B( T_2)≤ B( T_1). For a vector answer, maxj∈[d]j≤ν,Q(ℐ,Π)≤∑j=1dj. _j∈[d] B_j≤ B_ν,Q(I, )≤ _j=1^d B_j. (11) Proof. Write =ℐ T= T_I. Conditional on (,Z)=(t,z)( T,Z)=(t,z), a decoder returning u has error 1−ℙ(U=u∣t,z)1-P(U=u t,z). Pointwise minimization over u, followed by expectation over (,Z)( T,Z), proves (i). The risk in (i) is zero exactly when every conditional distribution P(U∣=t,Z=z)P(U T=t,Z=z) is a point mass. Full support of ν and Q turns this into a pointwise statement. Thus, whenever τℐ,Π(M0)=τℐ,Π(M1) _I, (M_0)= _I, (M_1), GM0(z)=g(τℐ,Π(M0),z)=g(τℐ,Π(M1),z)=GM1(z)G_M_0(z)=g( _I, (M_0),z)=g( _I, (M_1),z)=G_M_1(z) for every z∈z . This proves the forward implication in (i). Conversely, suppose the quotient implication holds. Every interface class C=M:τℐ,Π(M)=tC=\M: _I, (M)=t\ has a unique probe map gCg_C, so define g(t,z)=gC(z)g(t,z)=g_C(z). Then U=g(,Z)U=g( T,Z) and the Bayes risk is zero. For (i), let A=z:GM0(z)≠GM1(z)A=\z:G_M_0(z)≠ G_M_1(z)\ with Q(A)=ρQ(A)=ρ. Under the common interface descriptor, any randomized decoder has the same output law in the two models. Pointwise on A, G^(z)≠GM0(z)+G^(z)≠GM1(z)≥1.1\ G(z)≠ G_M_0(z)\+1\ G(z)≠ G_M_1(z)\≥ 1. Expectation over decoder randomness and z∼Qz Q gives RQ(G^,M0)+RQ(G^,M1)≥ρ,R_Q( G,M_0)+R_Q( G,M_1)≥ρ, and hence maxiRQ(G^,Mi)≥ρ/2 _iR_Q( G,M_i)≥ρ/2. For (iv), the transcript condition makes S a randomized garbling of T. Composing d(S,Z)d(S,Z) with the garbling kernel produces a randomized decoder from (,Z)( T,Z). Randomization cannot improve the pointwise minimum in (i), proving the bound. For (v), every decoder based on 1=h(2) T_1=h( T_2) is also a decoder based on 2 T_2, so the latter infimum is no larger. Any joint decoder induces a decoder for coordinate j; therefore its vector error is at least j B_j, proving the lower bound in Eq. (11). Conversely, concatenate the coordinatewise Bayes decoders. Its vector error is contained in the union of their coordinate errors, and the union bound proves the upper inequality. ∎ For a uniform two-model fiber and a point-mass probe on which the answers differ, the probe Bayes risk is 1/21/2. A probe-revealing interface has risk zero. With finite samples, total variation controls the residual indistinguishability of two probe-distinct models. Corollary 3. Let P0NP_0^N and P1NP_1^N be the N-sample transcript laws of two SCMs whose probe answers differ on a set of Q-mass ρ. Then every estimator obeys maxi∈0,1RQ(G^,Mi)≥ρ2(1−∥P0N−P1N∥TV). _i∈\0,1\R_Q( G,M_i)≥ ρ2 (1- P_0^N-P_1^N _TV ). If the samples are conditionally independent with one-sample laws P0,P1P_0,P_1 and d=minKL(P0∥P1),KL(P1∥P0)d= \KL(P_0\|P_1),KL(P_1\|P_0)\, then maxiRQ(G^,Mi)≥ρ2[1−Nd2]+. _iR_Q( G,M_i)≥ ρ2 [1- Nd2 ]_+. Proof. Fix a probe z on which the answers g0(z)g_0(z) and g1(z)g_1(z) differ. Let Ai(z)=G^(z)=gi(z)A_i(z)=\ G(z)=g_i(z)\. The events A0(z)A_0(z) and A1(z)A_1(z) are disjoint, hence P0N(A0(z)c)+P1N(A1(z)c) P_0^N(A_0(z)^c)+P_1^N(A_1(z)^c) ≥P0N(A1(z))+P1N(A1(z)c) ≥ P_0^N(A_1(z))+P_1^N(A_1(z)^c) =1−P1N(A1(z))−P0N(A1(z)) =1-\P_1^N(A_1(z))-P_0^N(A_1(z))\ ≥1−∥P0N−P1N∥TV. ≥ 1- P_0^N-P_1^N _TV. Integrating over the disagreement set and dividing the sum of the two risks by two proves the first inequality. For product laws, KL(P0N∥P1N)=NKL(P0∥P1)KL(P_0^N\|P_1^N)=NKL(P_0\|P_1). Pinsker’s inequality (Tsybakov, 2009), applied in the better of the two directions, gives ∥P0N−P1N∥TV≤Nd/2 P_0^N-P_1^N _TV≤ Nd/2; truncation at zero gives the second display. ∎ The proof is decision-theoretic: it does not pass through entropy, mutual information, or a decoder-class restriction. Risk decompositions, standard-target lower bounds, and explicit two-world constructions for ranking, readout, and hidden-context interfaces appear in Appendix A. Theorem 4. Let ℬs=b⊆[n]:|b|=sB_s=\b [n]:|b|=s\, let B∈ℬsB _s, and let Z be any pre-retest diagnostic signal. For s≤k<ns≤ k<n, define Γk(z)=maxL⊆[n]:|L|≤k∑b∈ℬs:b⊆Lℙ(B=b∣Z=z). _k(z)= _L [n]:\,|L|≤ k _b _s:\,b LP(B=b Z=z). (12) Then the largest probability that a Z-measurable retest list contains every shifted entry is supℒ:|ℒ(Z)|≤kℙB⊆ℒ(Z)=Γk(Z). _L:\,|L(Z)|≤ kP\B (Z)\=E _k(Z). (13) Internal randomization cannot increase this value. Zero omission is possible if and only if, for almost every z, |⋃b:ℙ(B=b∣Z=z)>0|≤k. | \b:P(B=b Z=z)>0\ |≤ k. (14) If B is uniform on ℬsB_s and independent of Z, the optimal coverage is exactly (ks)/(ns) ks/ ns. Finally, if Z1=h(Z2)Z_1=h(Z_2), then Γk(Z1)≤Γk(Z2)E _k(Z_1) _k(Z_2). Proof. For a deterministic list L(z)L(z), conditional coverage is ℙB⊆L(z)∣Z=z=∑b⊆L(z)ℙ(B=b∣Z=z).P\B L(z) Z=z\= _b L(z)P(B=b Z=z). Maximizing this finite sum separately for every z gives Eq. (12); averaging gives Eq. (13). A randomized rule is a mixture of deterministic lists, so its conditional coverage is a convex combination of values no larger than Γk(z) _k(z). The equality Γk(z)=1 _k(z)=1 holds precisely when one list of size at most k contains every b with positive posterior mass. Such a list exists precisely when the union of those posterior-support sets has size at most k, proving Eq. (14). Under the uniform independent law, a list of size ℓ contains exactly (ℓs) s of the (ns) ns possible shifted sets. This is maximized at ℓ=k =k. Finally, every list rule based on Z1=h(Z2)Z_1=h(Z_2) is also available to a rule based on Z2Z_2; taking suprema in Eq. (13) proves monotonicity. ∎ Proposition 5. For each transferred entry j∈[n]j∈[n], suppose the diagnostic samples Xj,1,…,Xj,m∈[0,1]X_j,1,…,X_j,m∈[0,1] are independent. Suppose also that Xj,r≥pEX_j,r≥ p for shifted entries j∈Bj∈ B and Xj,r≤qEX_j,r≤ q for stable entries j∉Bj∉ B, with Δ=p−q>0 =p-q>0. Let τ=(p+q)/2τ=(p+q)/2 and X¯j=1m∑r=1mXj,r,B^=j:X¯j≥τ. X_j= 1m _r=1^mX_j,r, B=\j: X_j≥τ\. If m≥2Δ−2log(n/δ)m≥ 2 ^-2 (n/δ), then ℙ(B^=B)≥1−δP( B=B)≥ 1-δ. Moreover, in the symmetric one-shift Bernoulli family with p=1/2+Δ/2p=1/2+ /2, q=1/2−Δ/2q=1/2- /2, and 0<Δ≤1/20< ≤ 1/2, every estimator with localization error at most ε requires m≥(1−ε)logn−log28Δ2.m≥ (1- ) n- 28 ^2. Thus the Δ−2logn ^-2 n scaling is necessary and sufficient up to constants. Proof. For j∈Bj∈ B, ℙX¯j<τ≤exp−2m(p−τ)2=exp−mΔ2/2. \ X_j<τ\≤ \-2m(p-τ)^2\= \-m ^2/2\. For j∉Bj∉ B, ℙX¯j≥τ≤exp−2m(τ−q)2=exp−mΔ2/2. \ X_j≥τ\≤ \-2m(τ-q)^2\= \-m ^2/2\. These two applications of Hoeffding’s inequality (Hoeffding, 1963), followed by a union bound over the n entries, give ℙ(B^≠B)≤nexp−mΔ2/2≤δP( B≠ B)≤ n \-m ^2/2\≤δ. This is the selection guarantee used by Eq. (7); the top-s variant is proved in Appendix A.6. The matching lower-bound calculation is given in Appendix A.5. ∎ Theorem 4 characterizes every retest list of size k without reducing the diagnostic signal to a scalar information budget. In contrast, Proposition 5 recovers the shifted set with m=O(Δ−2log(n/δ))m=O( ^-2 (n/δ)) diagnostic samples per entry. Proposition 6. Fix a selected entry j and a finite candidate class jN_j. Let θj⋆ _j be the target-domain mechanism and define the population retest loss Lj(θ)= 1gθ(Zj)≠Uj,θ∈j.L_j(θ)=E\,1\g_θ(Z_j)≠ U_j\, θ _j. Suppose θj⋆∈j _j _j and Lj(θ)−Lj(θj⋆)≥ηj>0L_j(θ)-L_j( _j )≥ _j>0 for every θ≠θj⋆θ≠ _j , and suppose the rjr_j retest pairs are independent and identically distributed. If the retest sample size is rj≥2ηj2log|j|δj,r_j≥ 2 _j^2 |N_j| _j, then the empirical minimizer in Eq. (9) satisfies ℙ(θ~j=θj⋆)≥1−δjP( θ_j= _j )≥ 1- _j. Proof. For a wrong candidate θ, set Dθ=gθ(Zj)≠Uj−gθj⋆(Zj)≠Uj.D_θ=1\g_θ(Z_j)≠ U_j\-1\g_ _j (Z_j)≠ U_j\. Then Dθ∈[−1,1]D_θ∈[-1,1] and Dθ=Lj(θ)−Lj(θj⋆)≥ηjED_θ=L_j(θ)-L_j( _j )≥ _j. If L^j(θ)≤L^j(θj⋆) L_j(θ)≤ L_j( _j ), then the empirical mean D¯θ D_θ is nonpositive. For Aθ=L^j(θ)≤L^j(θj⋆)A_θ=\ L_j(θ)≤ L_j( _j )\, Hoeffding’s inequality (Hoeffding, 1963) gives ℙ(Aθ)≤ℙD¯θ−Dθ≤−ηj≤e−rjηj2/2.P(A_θ) \ D_θ-ED_θ≤- _j\≤ e^-r_j _j^2/2. A union bound over j∖θj⋆N_j \ _j \ yields ℙ(θ~j≠θj⋆)≤|j|e−rjηj2/2≤δjP( θ_j≠ _j )≤|N_j|e^-r_j _j^2/2≤ _j. ∎ Theorem 7. Fix an aligned source-target pair satisfying Condition C1. Write mj=ϕ(θjs)m_j=φ( _j^s) and B=j:mj≠θjtB=\j:m_j≠ _j^t\, with S=[n]∖BS=[n] B. For any selected set D⊆[n]D [n] and local estimates θ~j:j∈D\ θ_j:j∈ D\, let θ^jt(D)=mj,j∉D,θ~j,j∈D. θ_j^t(D)= casesm_j,&j∉ D,\\ θ_j,&j∈ D. cases Then nLadapt nL_adapt =|B∖D|+∑j∈Dθ~j≠θjt, =|B D|+ _j∈ D1\ θ_j≠ _j^t\, (15) |S|Lstab |S|L_stab =∑j∈D∩Sθ~j≠mj, = _j∈ D∩ S1\ θ_j≠ m_j\, (16) |B|Lshift |B|L_shift =|B∖D|+∑j∈D∩Bθ~j≠θjt. =|B D|+ _j∈ D∩ B1\ θ_j≠ _j^t\. (17) Consequently the adapted memory is exact if and only if B⊆DB D and θ~j=θjt θ_j= _j^t for every j∈Dj∈ D. Moreover, for any exact target memory θ¯=θt θ=θ^t, j:θ¯j≠mj=B;\j: θ_j≠ m_j\=B; (18) thus B is the unique realized edit support of an exact update relative to the mapped source memory. Let FsrcF_src be the event that the frozen source memory has a missing entry or a wrong target, context, value, or delay, including a target error caused by readout contamination. Let FϕF_φ be schema-map failure, Fdiag=B^≠BF_diag=\ B≠ B\, and FupdF_upd be failure of any local retest on B. If their probabilities are bounded by δsrc,δϕ,δdiag _src, _φ, _diag, and δupd _upd, then the update in Eq. (8) satisfies ℙ(Ladapt>0)≤δsrc+δϕ+δdiag+δupd.P(L_adapt>0)≤ _src+ _φ+ _diag+ _upd. On the complement of these events, Lstab=Lshift=Ladapt=0L_stab=L_shift=L_adapt=0. For every realization, Ladapt=|S|nLstab+|B|nLshift.EL_adapt= |S|nEL_stab+ |B|nEL_shift. Proof. For j∉Dj∉ D, the update retains mjm_j, and therefore θ^jt(D)≠θjt=j∈B.1\ θ_j^t(D)≠ _j^t\=1\j∈ B\. For j∈Dj∈ D, the same error indicator is θ~j≠θjt1\ θ_j≠ _j^t\. Summing the first identity over j∉Dj∉ D and the second over j∈Dj∈ D proves Eq. (15). On S, mj=θjtm_j= _j^t; on B, mj≠θjtm_j≠ _j^t. Splitting the two sums accordingly proves Eqs. (16) and (17). Every term on the right of Eq. (15) is nonnegative. It vanishes precisely when B∖D=∅B D= and every selected estimate equals its target value, giving the stated necessary and sufficient condition. If θ¯=θt θ=θ^t, then θ¯j≠mj θ_j≠ m_j holds precisely when θjt≠mj _j^t≠ m_j, which is the definition of j∈Bj∈ B. This proves Eq. (18). Now let E=(Fsrc∪Fϕ∪Fdiag∪Fupd)cE=(F_src∪ F_φ∪ F_diag∪ F_upd)^c. On E, the mapped source entries are correct, D=B^=BD= B=B, and θ~j=θjt θ_j= _j^t for every selected j. The exact condition just proved gives Ladapt=0L_adapt=0. Therefore Ladapt>0⊆Fsrc∪Fϕ∪Fdiag∪Fupd,\L_adapt>0\ F_src∪ F_φ∪ F_diag∪ F_upd, and the probability bound follows by the union bound. The decomposition of the mean loss follows from partitioning the exact coordinate errors: Ladapt L_adapt =1n∑j∈Sθ^jt≠θjt+1n∑j∈Bθ^jt≠θjt = 1n _j∈ S1\ θ_j^t≠ _j^t\+ 1n _j∈ B1\ θ_j^t≠ _j^t\ =|S|nLstab+|B|nLshift. = |S|nL_stab+ |B|nL_shift. Taking expectation gives the displayed identity. Appendix A.7 gives the coordinate-level loss expansion and decomposes FsrcF_src into its target, context, readout, and temporal gates. ∎ Substitution of the target-contrast, readout, hidden-setup, temporal, diagnostic, schema, and local-update bounds into Theorem 7 is given in Appendix A.7. The corresponding top-s diagnostic result appears in Appendix A.6; its selected empirical means recover the shifted set under the same Δ−2log(n/δ) ^-2 (n/δ) scaling. Theorem 8. Let ℳM and Z be finite, let ν and Q have full support, and let (,ρ)(U,ρ) be a compact metric space with continuous ρ. Use Eq. (2) and write V=(ℐ,Z)V=( T_I,Z). Then: (i) The expected metric risk has the pointwise representation ν,Qρ(ℐ,Π)=[mina∈ρ(U,a)∣V], B^ρ_ν,Q(I, )=E\! [ _a E\ρ(U,a) V\ ], and a deterministic decoder attains the minimum. (i) ν,Qρ=0 B^ρ_ν,Q=0 and ℜρ=0 R^ρ=0 if and only if every interface fiber has one probe answer map; equivalently, M0∼ℐ,ΠM1M_0 _I, M_1 implies M0∼M1M_0 _ZM_1. (i) If DND_N and S satisfy the transcript condition in Theorem 2, every frozen-state decoder obeys ρU,d(S,Z)≥ν,Qρ(ℐ,Π).Eρ\U,d(S,Z)\≥ B^ρ_ν,Q(I, ). If 1=h(2) T_1=h( T_2), then ρ(2)≤ρ(1) B^ρ( T_2)≤ B^ρ( T_1) and ℜρ(2)≤ℜρ(1) R^ρ( T_2)≤ R^ρ( T_1). (iv) If two interface-equivalent models M0,M1M_0,M_1 have answer maps G0,G1G_0,G_1, every possibly randomized decoder of their common interface satisfies maxi∈0,1Z∼QρG^(Z),Gi(Z)≥12Z∼QρG0(Z),G1(Z). _i∈\0,1\E_Z Qρ\ G(Z),G_i(Z)\≥ 12E_Z Qρ\G_0(Z),G_1(Z)\. In particular, separation by at least 2ε2 on a set of Q-mass q forces worst-case metric risk at least qεq . Proof. Condition on V=vV=v. A deterministic answer a has conditional loss ρ(U,a)∣V=vE\ρ(U,a) V=v\. Compactness and continuity give a minimizer; pointwise minimization and averaging prove (i). Randomization only averages these conditional losses and cannot improve their minimum. The risk in (i) is zero exactly when U is constant conditional on every (ℐ,Z)( T_I,Z) value with positive probability. Full support and the metric property turn this into equality of the complete probe maps inside each interface fiber. The same condition is plainly necessary and sufficient for zero worst-case risk, proving (i). For (i), composing a decoder of S with the random kernel from ℐ T_I to S gives a randomized decoder of the population interface. Part (i) lower-bounds its loss. If 1=h(2) T_1=h( T_2), every decoder based on 1 T_1 is available from 2 T_2; taking the two infima proves both monotonicity statements. For (iv), interface equivalence makes the output law of G^(z) G(z) the same under M0M_0 and M1M_1. The triangle inequality gives, pointwise in z and in decoder randomness, ρG0(z),G1(z)≤ρG0(z),G^(z)+ρG^(z),G1(z).ρ\G_0(z),G_1(z)\≤ρ\G_0(z), G(z)\+ρ\ G(z),G_1(z)\. Expectation over Z and decoder randomness shows that the sum of the two risks is at least QρG0(Z),G1(Z)E_Qρ\G_0(Z),G_1(Z)\. Their maximum is at least half this quantity, which proves Eq. ((iv)). The final statement follows by integrating the assumed separation over its probe set. ∎ 5 Experiments The evaluation covers finite SCM families, readout and hidden-context variants, few-shot renamed adaptation, metric-valued control responses, language-model hidden states, and language-model proposal policies. Every learned state is frozen before the mechanism probes are scored. Exact protocols, seeds, and auxiliary tables are in Appendices B.1–B.5. Proposition 9 supplies exact indistinguishable model pairs. The empirical runs measure the corresponding fixed-probe failures and the finite-sample behavior of the constructive rules. Renamed transfer uses shifted-set localization and entrywise retesting. Continuous responses are integrated over held-out state, action-direction, and horizon distributions; language-model results use a separately fixed probe map. Held-out state-action probes query target, value, delay, context, and stable preservation. Reward and top-action accuracy are not terms in the score. For probe set P, MechAcc=1||∑(ξ,a)∈Tμ^(ξ,a)≡Tμ⋆(ξ,a).MechAcc= 1|P| _(ξ,a) 1\T_ μ(ξ,a)≡ T_μ (ξ,a)\. (19) Thus R^mech=1−MechAcc R_mech=1-MechAcc estimates RQR_Q. The coordinate estimates are R^k=1||∑(ξ,a)∈πkTμ^(ξ,a)≠πkTμ⋆(ξ,a), R_k= 1|P| _(ξ,a) 1\ _kT_ μ(ξ,a)≠ _kT_μ (ξ,a)\, (20) with false-write risks obtained by replacing the indicator by the corresponding empty-set event. Few-shot renamed adaptation reports AdaptScore=Lstab=0,Lshift=0,AdaptScore=1\L_stab=0,\ L_shift=0\, (21) so a method must both preserve stable entries and correct shifted ones. The coordinate risks are empirical counterparts of j B_j; they identify which answer coordinate remains unresolved in the frozen state. The finite discovery and language-model tables additionally report the direct target projection E(μ)=(a,y):∃(c,v,δ),(a,c,y,v,δ)∈μ,E(μ)=\(a,y):∃(c,v,δ),(a,c,y,v,δ)∈μ\, (22) using set precision, recall, and F1 against E(μ⋆)E(μ ). This projected F1 is accompanied by context recall and false-write counts; it is not presented as the exact tuple accuracy in Eq. (19). The reported false-edge count is |E(μ^)∖E(μ⋆)||E( μ) E(μ )|; readout false edges are the subset whose target coordinate is a designated deterministic proxy. Figure 2: Fixed-probe target and adaptation outcomes. 5.1 Protocols and Baselines The suite contains finite worlds, readout and hidden-context variants, continuous paired interventions, and language-model proposal policies. Baselines match the available interface: passive and reward learners, ranking loss, conditional discovery, latent world models, tuple and neural probes, the official TD-MPC2 world model, ungated proposals, and transfer ablations. All are evaluated after freezing, using Eqs. (19)–(22) or the metric loss in Eq. (2). 5.2 Coarse Interfaces The complex noisy-hidden family realizes positive probe Bayes risk under Theorem 2. Passive correlation gives target-edge F1 0.0000.000, while a reward-optimized learner has target-edge F1 0.3260.326. A ranking-loss predictor receives supervised intervention-order labels; nevertheless, in the noisy-hidden ranking task it has target-edge F1 0.0000.000 even though Causal Core reaches 0.7640.764. Stronger trajectory-matched baselines control for output format. On the expanded matched run, a thresholded conditional-discovery baseline has target-edge F1 0.640.64, and its readout-oracle variant reaches 0.760.76 but still writes hidden labels with false-positive count 1.501.50. The readout-oracle tuple writer has target-edge F1 0.790.79 but 6.196.19 hidden-label false positives. A latent world model reaches next-bit accuracy 0.9050.905 yet has target-edge F1 0.240.24; with a readout oracle it reaches 0.320.32 and writes 39.8839.88 hidden-label false positives. Causal Core has target-edge F1 0.790.79 with zero readout and hidden-label false positives. High task or predictive performance therefore coexists with poor fixed-probe recovery. Under the uniform prior, each exact two-world separation in Proposition 9 has probe Bayes risk 1/21/2 and is decoder-independent. The neural rows support a narrower empirical statement: high transition accuracy does not make the mechanism answer recoverable by the specified frozen-state decoders, including readout-oracle variants. The hidden-state experiment below adds linear and nonlinear decoders of a public language model rather than inferring omission from its generated answer alone. 5.3 Write Validity and Local Adaptation Indiscriminate writing raises recall at the cost of validity. The ungated Qwen explorer writes 92.392.3 false entries and 37.037.0 readout false positives; the context-search and control-planner gates reduce these counts to 6.0/0.06.0/0.0 and 2.0/0.02.0/0.0, respectively. In renamed and semantic-readout adaptation, non-diagnostic and readout-unsafe baselines have AdaptScore 00, while Causal Core has AdaptScore 1.001.00. The evaluation uses the diagnostic selection and local update in Eqs. (7)–(10), whose finite-sample guarantees are Propositions 5 and 6. 5.4 Hidden Contexts and Continuous Responses Hidden gates test the setup part of the interface. Active setup improves hidden recall from 0.1250.125 after 80 interactions to 0.6250.625 after 160 interactions at zero hidden false positives in the four-family horizon sweep. Rare eligibility events leave some hidden contexts outside the observed quotient. In a continuous metric SCM, global linear regression reaches F1 0.6670.667 because it writes a deterministic readout and extends a context-specific effect to the wrong cell; metric Causal Core has F1 1.0001.000 with no readout false positives. We next evaluate the official 5.25.2M-parameter TD-MPC2 checkpoint (Hansen et al., 2024) on cheetah-run, walker-walk, and reacher-easy. For saved state x, unit action direction u, and horizon h∈1,2,4h∈\1,2,4\, the evaluator records the continuous response Δh(x,u)=Oh(x,+ϵu)−Oh(x,−ϵu)2ϵ,ϵ=0.25. _h(x,u)= O_h(x,+ε u)-O_h(x,-ε u)2ε, ε=0.25. (23) Linear and two-layer nonlinear decoders receive the frozen source latent state, latent intervention difference, u, and h. Thus a failure is not tied to a linear readout. The nonlinear decoder has source NRMSE 0.6290.629 and effect-sign accuracy 0.9190.919 over nine task-seed runs. At transfer, one actuator changes sign and its index varies across runs. The frozen model’s shifted-actuator sign accuracy falls to 0.0570.057. A global probe-head update using the target pairs raises it to 0.6940.694 but increases stable-response NRMSE from 0.6730.673 to 1.4601.460. The selective update compares the two actuator hypotheses on five target states per actuator, localizes the shift in all nine runs, and reaches sign accuracy 0.9480.948. Its random-direction NRMSE is 0.6300.630, equal to the oracle actuator map within reported precision, while its stable-response NRMSE remains 0.6730.673. Appendix B.4 gives per-task results, decoder training, and diagnostic-margin sensitivity. Figure 3: Distributional TD-MPC2 responses under one actuator remapping. 5.5 Language-Model States and Proposal Policies A frozen-state experiment tests whether mechanism answers are retained inside Qwen2.5-7B-Instruct (Yang et al., 2024), rather than judging generated explanations alone. Logistic probes are fit at five layers on 12 procedural families and then frozen for six disjoint families. A two-layer decoder and an RBF decoder test nonlinear accessibility at the final layer. Action and state names are independently replaced by neutral tokens in every family and condition. The final-layer linear decoder reaches balanced accuracy 0.9580.958 on source mechanisms and 0.8470.847 after renaming, but 0.5830.583 on the changed-delay subset. The nonlinear RBF decoder obtains 0.9440.944, 0.8750.875, and 0.2500.250, respectively. Thus the source result cannot be dismissed as complete absence of mechanism information, while that information does not support the held-out changed delay. A paired-evidence rule recovers every changed delay but accepts 0.9440.944 of synchronized readouts. Applying the admissible-write gate preserves changed-delay accuracy 1.0001.000 and lowers readout FPR to 0.0560.056. Table 5 gives all controls. Figure 4: Qwen hidden-state retention on held-out mechanism families. In the proposal-policy experiment, the same Qwen model chooses legal interventions and the evidence gate processes the resulting transitions. Causal prompting alone has target-edge F1 0.1630.163 and 32.032.0 readout false positives. Gating raises target-edge F1 to 0.6780.678–0.7270.727 and eliminates readout false positives. Table 6 gives the full breakdown. Figure 2 localizes four distinct errors in the frozen answer map: ranking loses target identity, ungated writing loses validity, source dynamics loses the shifted sign, and prompting loses readout control. These errors persist under relabeling, hidden gates, or readout corruption even when the corresponding task objective remains useful. 5.6 Reproducibility Code for generating the SCM families, running the baselines, reproducing the MuJoCo and language-model experiments, and rebuilding every figure is available at https://github.com/ShengjunZhang/mechanism-memory. The source-data tables and raw result files are included with the submission. Appendix B gives the random seeds, budgets, model variants, probe definitions, and implementation details used in the reported results. 6 Conclusion Causal retention asks whether a frozen learned state preserves an interventional answer map beyond what task success requires. Probe Bayes risk, posterior retest coverage, and the exact edit decomposition connect interface identification to selective adaptation without assuming a particular decoder. Finite SCMs, continuous control, a pretrained world model, and frozen Qwen states expose target, context, value, and delay errors despite useful task or source-decoding performance. The continuous results concern a fixed distribution of state, intervention, and horizon queries rather than recovery of an unrestricted latent SCM. Appendix A Auxiliary Theoretical Results The finite constructions below establish the interface separations and the extensions used by the selective-adaptation analysis. All logarithms are natural. A.1 Coarse Behavioral and Predictive Interfaces Proposition 9. There are finite deterministic structural causal model classes in which reward, optimal-policy, intervention-ranking, graph, synchronized-readout, and passive hidden-context interfaces are strictly coarser than the exact mechanism-probe interface. For each interface, a point-mass probe distribution has minimax mechanism risk at least 1/21/2. Proof. Each construction consists of two models with the same coarse interface and different answers to one mechanism probe. The lower bound then follows from Theorem 2 with ρ=1ρ=1. For reward and optimal-policy interfaces, let U,Y∈0,1U,Y∈\0,1\ and let a be the only nontrivial action. In M0M_0, a directly sets Y←1Y← 1 and leaves U unchanged. In M1M_1, a directly sets U←1U← 1 and the structural equation is Y←UY← U. Initialize U=Y=0U=Y=0 and let reward be Y. Every policy induces the same reward sequence in the two models, and the optimal value and optimal-policy set agree. The direct target of a is Y in M0M_0 and U in M1M_1. For intervention ranking, add a null action b with utility zero. In both models above, a yields utility one and b yields utility zero. Hence every pairwise ranking label is identical, while the direct target of the preferred action still differs. For a graph interface, use the same graph A→YA→ Y in both models and let do(A=1)do(A=1) set Y←1Y← 1 in M0M_0 and Y←0Y← 0 in M1M_1 from a common baseline Y=1/2Y=1/2. The graph is identical, whereas the value coordinate of the mechanism answer differs. A delay or context coordinate can be varied in the same way without changing the graph. For a synchronized-readout interface, let R be an observed proxy. In M0M_0, a directly sets Y←1Y← 1 and R←YR← Y; in M1M_1, a directly sets R←1R← 1 and Y←RY← R. Both produce the observed pair (Y,R)=(1,1)(Y,R)=(1,1) after a, but the direct target is Y in M0M_0 and R in M1M_1. For passive hidden-context observation, fix p∈(0,1)p∈(0,1). In M0M_0, action a sets Y←1Y← 1 with probability p independently on each trial. In M1M_1, a latent H∼Bernoulli(p)H (p) is drawn before each trial and a deterministically sets Y←1Y← 1 if and only if H=1H=1. The passive law of (a,Y)(a,Y) is Bernoulli with parameter p in both models, while only M1M_1 has a hidden eligibility context. These pairs establish all claimed strict coarsenings. ∎ The proposition does not assert that reward, graph estimation, or ranking is uninformative. It states that these population objects need not determine all coordinates of the mechanism answer. Additional intervention contrasts can refine the interface and remove a particular ambiguity. A.2 Finite-Sample Contrast Gates The constructive results use paired intervention contrasts rather than raw action correlations. The following calculation supplies the common finite-sample term for the target, context, and temporal gates. Lemma 10. For u∈0,1u∈\0,1\, let Y1u,…,YnuuY_1^u,…,Y_n_u^u be independent random variables in [0,1][0,1] with means pup_u. Define Δ^=1n1∑r=1n1Yr1−1n0∑r=1n0Yr0,Δ=p1−p0. = 1n_1 _r=1^n_1Y_r^1- 1n_0 _r=1^n_0Y_r^0, =p_1-p_0. Then, for every x>0x>0, ℙ|Δ^−Δ|≥x≤2exp−2x2n1−1+n0−1.P\| - |≥ x\≤ 2 \! \- 2x^2n_1^-1+n_0^-1 \. If a gate accepts when Δ^≥λ ≥λ and every population contrast is at least λ+γλ+γ or at most λ−γλ-γ, the probability of any error among K candidates is at most 2Kexp−2γ2n1−1+n0−1.2K \! \- 2γ^2n_1^-1+n_0^-1 \. For a balanced design n0=n1=mn_0=n_1=m, this becomes 2Ke−mγ22Ke^-mγ^2. Proof. Center all observations and write Δ^−Δ=∑r=1n1Yr1−p1n1−∑r=1n0Yr0−p0n0. - = _r=1^n_1 Y_r^1-p_1n_1- _r=1^n_0 Y_r^0-p_0n_0. Each summand in the first sum has an interval of possible values of length 1/n11/n_1, and each summand in the second has interval length 1/n01/n_0. Hence the sum of squared interval lengths is n1(1n1)2+n0(1n0)2=1n1+1n0.n_1 ( 1n_1 )^2+n_0 ( 1n_0 )^2= 1n_1+ 1n_0. Hoeffding’s inequality (Hoeffding, 1963) gives the stated exponential bound for each tail; adding the two tails proves the first display. A population contrast on either side of the threshold can be misclassified only if |Δ^−Δ|≥γ| - |≥γ. Applying the first bound and then a union bound over K candidates proves the remaining claims. ∎ The independent-pair calculation extends to adaptive interaction when action probabilities are logged. Lemma 11. Let (ℱt)t=0N(F_t)_t=0^N be the interaction filtration. At round t, let Yt(1),Yt(0)∈[0,1]Y_t(1),Y_t(0)∈[0,1] be potential responses and suppose At⟂(Yt(1),Yt(0))|ℱt−1,et=ℙ(At=1∣ℱt−1)∈[ϵ,1−ϵ]A_t \!\!\! (Y_t(1),Y_t(0)) _t-1, e_t=P(A_t=1 _t-1)∈[ε,1-ε] almost surely for some 0<ϵ≤1/20<ε≤ 1/2. Observe Yt=AtYt(1)+(1−At)Yt(0)Y_t=A_tY_t(1)+(1-A_t)Y_t(0) and define Δ^N=1N∑t=1NAtYtet−(1−At)Yt1−et, _N= 1N _t=1^N \ A_tY_te_t- (1-A_t)Y_t1-e_t \, Δ¯N=1N∑t=1N[Yt(1)−Yt(0)∣ℱt−1]. _N= 1N _t=1^NE[Y_t(1)-Y_t(0) _t-1]. Then, for every x>0x>0, ℙ|Δ^N−Δ¯N|≥x≤2exp−Nϵ2x2/2.P\| _N- _N|≥ x\≤ 2 \-Nε^2x^2/2\. Consequently, K adaptive gates whose average contrasts are separated from their thresholds by γ have joint error probability at most 2Kexp−Nϵ2γ2/22K \-Nε^2γ^2/2\. Proof. Set Vt=AtYtet−(1−At)Yt1−et,dt=[Yt(1)−Yt(0)∣ℱt−1].V_t= A_tY_te_t- (1-A_t)Y_t1-e_t, d_t=E[Y_t(1)-Y_t(0) _t-1]. Consistency and sequential randomization imply [Vt∣ℱt−1] [V_t _t-1] =[AtYt(1)∣ℱt−1]et−[(1−At)Yt(0)∣ℱt−1]1−et = E[A_tY_t(1) _t-1]e_t- E[(1-A_t)Y_t(0) _t-1]1-e_t =[Yt(1)∣ℱt−1]−[Yt(0)∣ℱt−1]=dt. =E[Y_t(1) _t-1]-E[Y_t(0) _t-1]=d_t. Thus Dt=Vt−dtD_t=V_t-d_t is a martingale difference. Conditional on ℱt−1F_t-1, VtV_t lies between −1/(1−et)-1/(1-e_t) and 1/et1/e_t; subtracting dtd_t does not change the interval width, and 1et+11−et≤2ϵ. 1e_t+ 11-e_t≤ 2ε. Hoeffding’s conditional lemma therefore gives [eλDt∣ℱt−1]≤expλ2/(2ϵ2).E[e^λ D_t _t-1]≤ \λ^2/(2ε^2)\. Iterating conditional expectation yields expλ∑t=1NDt≤expNλ2/(2ϵ2).E \! \λ _t=1^ND_t \≤ \Nλ^2/(2ε^2)\. For λ>0λ>0, Markov’s inequality gives ℙ∑t=1NDt≥Nx≤exp−λNx+Nλ2/(2ϵ2).P\! \ _t=1^ND_t≥ Nx \≤ \-λ Nx+Nλ^2/(2ε^2)\. The minimizing value is λ=ϵ2xλ=ε^2x, which gives exp−Nϵ2x2/2 \-Nε^2x^2/2\. Applying the same argument to −Dt-D_t and adding the tails proves the first claim. The gate bound follows by setting x=γx=γ and taking a union bound over K candidates. ∎ Proposition 12. Let ℱF be a finite class of candidate readout formulas and let A(f)=ℙf(X)=RA(f)=P\f(X)=R\. From m independent observations define A^m(f)=m−1∑r=1mf(Xr)=Rr A_m(f)=m^-1 _r=1^m1\f(X_r)=R_r\. The rule declares R a formula readout when maxf∈ℱA^m(f)≥c _f A_m(f)≥ c. If some f⋆f satisfies A(f⋆)≥c+γA(f )≥ c+γ, its false-negative probability is at most e−2mγ2e^-2mγ^2. If every f satisfies A(f)≤c−γA(f)≤ c-γ, the false-positive probability is at most |ℱ|e−2mγ2|F|e^-2mγ^2. Proof. In the first case, a false negative implies A^m(f⋆)<c A_m(f )<c. Therefore ℙfalse negative \false negative\ ≤ℙA^m(f⋆)−A(f⋆)≤−γ \ A_m(f )-A(f )≤-γ\ ≤e−2mγ2. ≤ e^-2mγ^2. In the second case, a false positive implies that at least one f∈ℱf satisfies A^m(f)−A(f)≥γ A_m(f)-A(f)≥γ. Hoeffding’s inequality gives probability at most e−2mγ2e^-2mγ^2 for each fixed f; a union bound over ℱF proves the result. ∎ A.3 Hidden-Context Acquisition The no-signal bound is exact for a uniformly hidden label. If B∼Unif(0,1m)B (\0,1\^m) and I(B,S)=0I(B;S)=0, then B remains uniform conditional on S. Consequently every deterministic or randomized decoder B B based on S satisfies dH(B^,B)=m/2,ℙB^=B=2−m.Ed_H( B,B)=m/2, \ B=B\=2^-m. Active setup changes the rate at which informative eligible outcomes arrive. Proposition 13. Fix a hidden-gated candidate. Suppose a setup policy reaches an eligible state within at most h actions with probability at least β, independently across reset attempts. To collect m eligible process outcomes with probability at least 1−δ1-δ, it is sufficient to run N≥8β(m+log1δ)N≥ 8β (m+ 1δ ) setup attempts, for total action cost at most N(h+1)N(h+1). Passive access is the same calculation with the ambient eligibility probability q in place of β; its expected number of attempts is m/qm/q. Proof. Let KNK_N be the number of successful setup attempts. It stochastically dominates a Binomial(N,β)Binomial(N,β) variable with mean ν=Nβν=Nβ. The stated condition implies ν≥8(m+log1δ),m≤ν/2.ν≥ 8 (m+ 1δ ), m≤ν/2. The multiplicative Chernoff lower-tail inequality (Boucheron et al., 2013) therefore gives ℙ(KN<m) (K_N<m) ≤ℙ(KN≤ν/2)≤e−ν/8 (K_N≤ν/2)≤ e^-ν/8 ≤e−m−log(1/δ)=e−mδ≤δ. ≤ e^-m- (1/δ)=e^-mδ≤δ. Each successful attempt contributes one process outcome and uses at most h+1h+1 actions. Under passive access, eligible arrivals are Bernoulli with parameter q, so the waiting time for m arrivals is negative binomial with expectation m/qm/q; the same Chernoff calculation gives the high-probability 1/q1/q factor. ∎ A.4 Randomized Lists and Adaptive Stopping Let R be internal randomness independent of (B,Z)(B,Z). Conditional on Z=zZ=z, the coverage of ℒ(z,R)L(z,R) is a convex combination of the coverage values of deterministic lists and is therefore at most Γk(z) _k(z). An adaptive procedure that stops after retesting at most k distinct entries induces the list of all entries it examined. If its final memory is exact, this list must contain every shifted entry unless an untested entry was supplied by an external oracle. Hence its success probability is at most Γk(Z)E _k(Z) as well. Adaptive ordering can reduce the number of tests on favorable paths, but cannot increase the coverage permitted by a hard cap of k distinct entries. A.5 Diagnostic Localization Lower Bound Consider the symmetric one-shift family in Proposition 5. Let J be uniform on [n][n]. Conditional on J=jJ=j, the m samples at coordinate j are Bernoulli(1/2+Δ/2)Bernoulli(1/2+ /2) and those at every other coordinate are Bernoulli(1/2−Δ/2)Bernoulli(1/2- /2). Denote the joint law by PjP_j. For u=(1+Δ)/2u=(1+ )/2 and v=(1−Δ)/2v=(1- )/2, kl(u,v)=Δlog1+Δ1−Δ.kl(u,v)= 1+ 1- . When 0<Δ≤1/20< ≤ 1/2, log1+Δ1−Δ=log(1+Δ)−log(1−Δ)≤Δ+Δ1−Δ≤3Δ, 1+ 1- = (1+ )- (1- )≤ + 1- ≤ 3 , so both directed Bernoulli divergences are at most 4Δ24 ^2. For j≠kj≠ k, PjP_j and PkP_k differ only at coordinates j and k; product additivity therefore gives KL(Pj∥Pk)≤8mΔ2.KL(P_j\|P_k)≤ 8m ^2. If P∘=n−1∑k=1nPkP_ =n^-1 _k=1^nP_k, convexity in the second argument yields I(J,X) I(J;X) =1n∑j=1nKL(Pj∥P∘) = 1n _j=1^nKL(P_j\|P_ ) ≤1n2∑j,k=1nKL(Pj∥Pk)≤8mΔ2. ≤ 1n^2 _j,k=1^nKL(P_j\|P_k)≤ 8m ^2. Fano’s inequality (Cover and Thomas, 2006) now implies ℙ(J^≠J)≥1−8mΔ2+log2logn.P( J≠ J)≥ 1- 8m ^2+ 2 n. Requiring this error to be at most ε and rearranging proves m≥(1−ε)logn−log28Δ2.m≥ (1- ) n- 28 ^2. A.6 Top-s Diagnostic Selection Suppose |B|=s|B|=s, shifted entries have means at least p, stable entries have means at most q<pq<p, and Δ=p−q =p-q. Let B^s B_s contain the s largest empirical means. If minj∈BX¯j>p+q2andmaxj∉BX¯j<p+q2, _j∈ B X_j> p+q2 _j∉ B X_j< p+q2, then B^s=B B_s=B. Hoeffding’s inequality and a union bound give ℙB^s≠B≤nexp(−mΔ2/2).P\ B_s≠ B\≤ n (-m ^2/2). Consequently m≥2Δ−2log(n/δ)m≥ 2 ^-2 (n/δ) is sufficient for exact top-s recovery with probability at least 1−δ1-δ. A.7 Coordinate Loss and End-to-End Sample Composition Let ej,ke_j,k indicate an error in coordinate k∈ctx,tar,val,delk∈\ctx,tar,val,del\ of entry j. Exact entry error satisfies maxkej,k≤θ^jt≠θjt≤∑kej,k. _ke_j,k 1\ θ_j^t≠ _j^t\≤ _ke_j,k. Summing first over S and B and then dividing by n yields Ladapt=|S|nLstab+|B|nLshift,L_adapt= |S|nL_stab+ |B|nL_shift, and bounds each term by its coordinate losses. Let Ftar,Fctx,Fro,FtimeF_tar,F_ctx,F_ro,F_time, and FhidF_hid denote, respectively, a target-contrast error, a visible context error, a readout-classification error, a delay error, and a hidden setup or hidden-label error in the source memory. Then Fsrc⊆Ftar∪Fctx∪Fro∪Ftime∪Fhid.F_src F_tar∪ F_ctx∪ F_ro∪ F_time∪ F_hid. Diagnostic selection controls whether the appropriate target entry is tested; local retesting controls its replacement coordinates. This is the component factorization used by Theorem 7. For balanced paired designs, Lemma 10 gives ℙ(Ftar) (F_tar) ≤2Ktare−mtarγtar2, ≤ 2K_tare^-m_tar _tar^2, ℙ(Fctx) (F_ctx) ≤2Kctxe−mctxγctx2, ≤ 2K_ctxe^-m_ctx _ctx^2, ℙ(Ftime) (F_time) ≤2Ktimee−mtimeγtime2. ≤ 2K_timee^-m_time _time^2. Proposition 12 and the hidden-label threshold test give the following bounds, where FmaxF_ is the largest readout formula-class size: ℙ(Fro) (F_ro) ≤Kro(Fmax+1)e−2mroγro2, ≤ K_ro(F_ +1)e^-2m_ro _ro^2, ℙ(Fhid) (F_hid) ≤δset+2Khide−mhidγhid2, ≤ _set+2K_hide^-m_hid _hid^2, where Proposition 13 supplies an attempted-action budget for δset _set. Diagnostic localization and local retesting obey ℙ(Fdiag) (F_diag) ≤ne−mdiagΔ2/2, ≤ ne^-m_diag ^2/2, ℙ(Fupd) (F_upd) ≤∑j∈B|j|e−rjηj2/2. ≤ _j∈ B|N_j|e^-r_j _j^2/2. For example, the sufficient allocations mg m_g ≥γg−2log2Kgδg, ≥ _g^-2 2K_g _g, g∈tar,ctx,time, g∈\tar,ctx,time\, mro m_ro ≥12γro2logKro(Fmax+1)δro, ≥ 12 _ro^2 K_ro(F_ +1) _ro, mhid m_hid ≥1γhid2log2Khidδlabel, ≥ 1 _hid^2 2K_hid _label, mdiag m_diag ≥2Δ2lognδdiag, ≥ 2 ^2 n _diag, rj r_j ≥2ηj2log|B||j|δupd ≥ 2 _j^2 |B|\,|N_j| _upd make the corresponding terms no larger than their assigned failure budgets. Set δsrc=δtar+δctx+δro+δtime+δhid,δhid=δset+δlabel. _src= _tar+ _ctx+ _ro+ _time+ _hid, _hid= _set+ _label. If δsrc+δϕ+δdiag+δupd≤δ _src+ _φ+ _diag+ _upd≤δ, substitution into Theorem 7 yields ℙ(Ladapt>0)≤δP(L_adapt>0)≤δ. Appendix B Experimental Protocols and Results B.1 Fixed-Memory Evaluation During interaction, each method may use its allowed internal state to choose actions. Learning then stops. The evaluator queries only the frozen answer map Tμ^T_ μ on held-out probes and compares it with the SCM answer GMG_M. Ground-truth mechanisms are used to construct probes and scores after the run; they are never supplied to the writer during interaction. Exact mechanism accuracy uses tuple matches. The finite discovery and language tables report target-edge precision, recall, and F1 from Eq. (22), while context, value, delay, and readout coordinates are probed separately. The FP column counts |E(μ^)∖E(μ⋆)||E( μ) E(μ )|. Readout false positives are the subset whose direct-target field is a designated sensor or deterministic proxy. Hidden recall counts true hidden-gated mechanisms whose context type is retained. AdaptScore equals one only when all shifted entries are corrected and all stable entries are preserved. B.2 Baselines The reward learner stores effects associated with high-return actions. Passive correlation stores observed changes without action contrasts. Conditional discovery thresholds conditional transition differences; its readout-oracle variant removes known readouts before scoring. Tuple lift converts discovered dependencies into the full output schema, and its oracle variant receives the same readout information. The latent world model and neural transition probe fit source transitions and are decoded only after their states are frozen. TD-MPC2 uses the released MT30 checkpoint; linear and nonlinear response decoders are fitted to source intervention pairs and frozen before the actuator remapping. Its global and selective updates receive identical target pairs. The ranking predictor minimizes pairwise intervention-ranking loss. Ungated writers admit every proposed tuple; gated variants use the same proposals but apply Eqs. (5) and (6). These controls separate objective and interface effects from output-format effects. Method Target F1 FP Readout FP Hidden recall Hidden FP Causal Core context search 0.7930.793 1.0631.063 0.0000.000 0.4580.458 0.0000.000 Conditional discovery, tuned 0.6400.640 8.4388.438 6.8136.813 0.4170.417 3.2503.250 Conditional discovery, readout oracle 0.7630.763 1.9381.938 0.0000.000 0.4170.417 1.5001.500 Tuple lift, tuned 0.6500.650 8.0008.000 7.4387.438 0.4790.479 10.87510.875 Tuple lift, readout oracle 0.7890.789 1.3131.313 0.0000.000 0.5000.500 6.1886.188 Latent world model, tuned 0.2390.239 48.68848.688 16.25016.250 0.2500.250 46.75046.750 Latent world model, readout oracle 0.3190.319 43.56343.563 0.0000.000 0.2500.250 39.87539.875 Transition probe, tuned 0.3960.396 38.62538.625 13.25013.250 0.3130.313 28.06328.063 Transition probe, readout oracle 0.4790.479 25.37525.375 0.0000.000 0.3130.313 21.18821.188 Table 1: Matched finite-SCM baselines. All rows in Table 1 receive the same trajectories. The latent world model attains next-bit accuracy 0.9050.905 (and 0.9070.907 with the readout oracle), so its probe error is not explained by failed source prediction. Treating each generated SCM family as the independent unit, the mean target-edge F1 and family-clustered 95% t intervals are 0.7930.793 [0.749,0.8370.749,0.837] for Causal Core, 0.6400.640 [0.606,0.6730.606,0.673] for conditional discovery, 0.7890.789 [0.740,0.8390.740,0.839] for oracle tuple lift, 0.2390.239 [0.185,0.2920.185,0.292] for latent modeling, and 0.3960.396 [0.387,0.4040.387,0.404] for the transition probe. The intervals are descriptive across the eight generated families; the exact family-seed values are in the source-data archive. Task Ranking signal Ranking predictor Causal Core Noisy-hidden ranking Source rank 0.5210.521 Target F1 0.0000.000; hidden 0.0000.000 Target F1 0.7640.764; hidden 0.6720.672 Renamed adaptation Source/target rank 0.865/0.5000.865/0.500 AdaptScore 0.0000.000 AdaptScore 1.0001.000 Semantic-readout adaptation Source/target rank 0.756/0.4280.756/0.428 AdaptScore 0.0000.000 AdaptScore 1.0001.000 Table 2: Ranking and selective-adaptation results. B.3 Continuous Metric Cells The continuous SCM has two physical state coordinates, a two-dimensional action, a deterministic pressure readout, and a context-specific vent effect. The evaluator fixes metric cells before training. A pulse is correct only when its direct target, sign, and context cell match the reference answer. Method Precision Recall F1 FP Readout FP Context error Random correlation 0.2520.252 1.0001.000 0.4020.402 17.8517.85 5.955.95 0.000.00 Global linear regression 0.5000.500 1.0001.000 0.6670.667 6.006.00 4.004.00 2.002.00 Metric Causal Core 1.0001.000 1.0001.000 1.0001.000 0.000.00 0.000.00 0.000.00 Table 3: Continuous metric-cell results. For noise levels 0.02,0.05,0.10,0.150.02,0.05,0.10,0.15, metric Causal Core retains F1 1.0001.000. The corresponding random-correlation F1 scores are 0.402,0.321,0.265,0.2520.402,0.321,0.265,0.252; global linear scores are 0.667,0.667,0.667,0.6950.667,0.667,0.667,0.695. B.4 Distributional World-Model Probes The public TD-MPC2 MT30 checkpoint contains 5,241,0665,241,066 parameters and is evaluated without weight updates. Its SHA-256 digest, checkpoint metadata, and all per-run outputs are included with the source data. The tasks are cheetah-run, walker-walk, and reacher-easy, with three state-sampling seeds per task. States are collected under the checkpoint policy with Gaussian action noise. The evaluator draws unit action directions and horizons h∈1,2,4h∈\1,2,4\, then branches the MuJoCo physics state to compute Eq. (23). Each source decoder receives 900 training queries; source and target evaluation use 240 held-out queries each. The linear decoder is ridge regression. The nonlinear decoder has two 256-unit GELU layers, uses a disjoint 15% validation split, and stops after 24 epochs without improvement. Both receive the encoded state, the difference of positive and negative latent rollouts, the action direction, and the horizon. The normalized response error is NRMSE=(∑i‖Δ^i−Δi‖22∑i‖Δi‖22)1/2.NRMSE= ( _i\| _i- _i\|_2^2 _i\| _i\|_2^2 )^1/2. Effect-sign accuracy evaluates the sign at the largest-magnitude coordinate of the true response. For seed s, actuator (s−1)modda(s-1) d_a changes sign. Adaptation uses five states per actuator. The selective update compares positive and negative actuator hypotheses and changes an entry only when the alternative halves its paired loss. The global update fits an unrestricted residual probe head to the same target pairs. Thresholds from 0.450.45 through 0.900.90 all localize the same single actuator in every run. Task Source NRMSE Frozen sign Global sign Selective sign Localized Cheetah run 0.7010.701 0.0280.028 0.7940.794 0.9750.975 3/33/3 Walker walk 0.8410.841 0.1420.142 0.7860.786 0.8690.869 3/33/3 Reacher easy 0.3430.343 0.0030.003 0.5000.500 1.0001.000 3/33/3 Mean 0.6290.629 0.0570.057 0.6940.694 0.9480.948 9/99/9 Table 4: Distributional intervention responses. On random target directions, NRMSE is 1.1381.138 for the frozen model, 1.5481.548 for the global update, and 0.6300.630 for the selective update. On directions that exclude the shifted actuator, the corresponding stable-response errors are 0.6730.673, 1.4601.460, and 0.6730.673. The selective update therefore matches the oracle actuator map at the reported precision without changing stable responses. The fixed finite query support together with a bounded truncation of Euclidean loss instantiates Eq. (2); untruncated NRMSE is reported to expose large errors. The experiment does not infer an unrestricted latent SCM. B.5 Language-Model States and Proposal Policies The hidden-state experiment presents Qwen2.5-7B-Instruct with a controlled intervention record followed by a binary direct-target or delay probe. Twelve procedural families supply 504 balanced training probes; six disjoint families supply 360 balanced test probes. Each family and condition receives a fresh random permutation of neutral action and state tokens, so a decoder cannot identify a mechanism from a recurring name. Every record contains two controlled repetitions per action. Prompt length is 697 tokens on average and 812 at maximum, with no truncation. The language model is frozen. At the answer position, hidden vectors are extracted from layers 0,7,14,21,0,7,14,21, and 2828. A logistic decoder is fit at each layer, and a two-layer multilayer perceptron and an RBF support-vector decoder are fit to the last layer. The controls are query-only TF–IDF, the model’s calibrated Yes/No logit margin, and a paired-evidence rule. The gated mechanism state admits a candidate only when the controlled pair supports its lag and unrelated interventions do not support the same target. All decoders and thresholds are frozen before evaluation on the six test families. Balanced accuracy is reported for mixed-label probes; readout FPR is the acceptance rate on synchronized-readout candidates. Standard errors in Figure 4 use the six test families as clusters. State or decoder Source BA Renamed BA Shifted-delay BA Readout FPR Query-only TF–IDF 0.7360.736 0.6110.611 0.0000.000 0.6110.611 Qwen Yes/No logits 0.6530.653 0.5830.583 0.4170.417 0.0000.000 Qwen last-layer linear 0.9580.958 0.8470.847 0.5830.583 0.0000.000 Qwen last-layer RBF 0.9440.944 0.8750.875 0.2500.250 0.1110.111 Paired evidence 1.0001.000 1.0001.000 1.0001.000 0.9440.944 Gated mechanism state 1.0001.000 1.0001.000 1.0001.000 0.0560.056 Table 5: Frozen language-model state probes. The source score of the last-layer linear decoder shows that substantial mechanism information is present in the frozen hidden state. Its shifted-delay score of 0.5830.583, together with the RBF decoder’s score of 0.2500.250, shows that source decodability does not yield a stable answer under a held-out mechanism change. Paired evidence recovers the changed delay but accepts 94.4%94.4\% of synchronized readouts. The gate retains its perfect shifted-delay score while reducing that false-positive rate to 5.6%5.6\%. Qwen2.5-7B-Instruct is used only to propose the next legal intervention from the current transcript. Runs use deterministic decoding, 100 interaction steps, and three seeds. The action is executed in the same complex noisy-hidden family used by the non-language comparison. Explanations and reward predictions receive no probe credit. Invalid outputs are mapped by the same deterministic action parser in every language condition. Agent Precision Recall Target F1 FP Readout FP Hidden recall Ungated Qwen explorer 0.1270.127 0.4300.430 0.1960.196 92.392.3 37.037.0 0.000.00 Qwen with causal prompt 0.1090.109 0.3230.323 0.1630.163 82.382.3 32.032.0 0.000.00 Qwen + context-search gate 0.7670.767 0.6130.613 0.6780.678 6.06.0 0.00.0 0.170.17 Qwen + control-planner gate 0.9010.901 0.6130.613 0.7270.727 2.02.0 0.00.0 0.080.08 Pure context-search planner 0.8230.823 0.5810.581 0.6800.680 4.04.0 0.00.0 0.170.17 Pure control planner 0.7900.790 0.6340.634 0.6970.697 5.75.7 0.00.0 0.170.17 Table 6: Language-model proposal policies. The causal prompt does not remove readout contamination. The gated variants use the same language model as a proposal source but apply the evidence gate after observing the transition. Their advantage therefore comes from the memory update, not from treating generated explanations as causal labels. B.6 Candidate Budgets and Implementation Candidate schedules cap computation but do not enter the probe metric. Small, default, and wide schedules yield the same target-edge F1 at 80 and 160 steps (0.6980.698 and 0.7630.763), zero readout false positives, and hidden recall rising from 0.1250.125 to 0.6250.625. All thresholds, target budgets, metric cells, saved MuJoCo states, language prompts, and parsers are fixed before held-out scoring; malformed language proposals count as failures. Finite-SCM actions are interventions. The writer estimates Eq. (4) from action-conditioned lift and repeated context visits. Because these trajectories are adaptive, the iid bounds in Appendix A.2 are not reported as confidence intervals for the empirical tables. Lemma 11 covers randomized adaptive logging; the continuous protocols use the paired design directly. Candidate caps determine which contrasts are attempted, not how frozen probes are scored. 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