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Equilibrium Causal Digital Twins: Validation, Transport, and Identification Limits
Faraz Dadgostari, Neda Nazemi
Intelligence
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Summary
This paper investigates the validation, transport, and identification limits of equilibrium causal digital twins in systems with feedback. It establishes conditions under which counterfactual predictions can be validated within a single domain and transported across domains where mechanisms change. Key contributions include the introduction of cyclic selection diagrams, criteria for direct reuse and hybrid models, and an impossibility result demonstrating that experimental agreement alone is insufficient for validation without structural assumptions. The study also provides statistical tests and characterizes identification ranges for linear models.
Entities (9)
Relation Signals (6)
Validation → requires → Structural Assumptions
confidence 90% · An impossibility result constructs systems that agree under every experiment in a finite design but disagree on the target counterfactual, showing that validation requires structural assumptions.
Digital Twin → usedfor → Counterfactual
confidence 90% · Digital twins are often used to predict how a system would respond to an intervention.
Cyclic Selection Diagram → enables → Transport
confidence 85% · We then introduce cyclic selection diagrams and derive criteria for direct reuse and for hybrid models that combine invariant source mechanisms with target information.
Equilibrium Causal Game → requires → Validation
confidence 85% · For equilibrium causal games, we give conditions on the mechanisms, equilibrium selection, and intervention design under which agreement with experimental distributions identifies the counterfactual of interest.
Hybrid Model → combines → Invariant Source Mechanisms
confidence 80% · hybrid models that combine invariant source mechanisms with target information.
Linear Model → has → Identification Requirements
confidence 80% · For linear models, we derive intervention requirements that depend on the mechanisms that changed, the observation model, and graph support.
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Abstract
Abstract:Digital twins are often used to predict how a system would respond to an intervention. In systems with feedback, a twin must reproduce an equilibrium counterfactual, and a twin developed in one domain may fail after mechanisms change. We study when these predictions can be validated and transported. For equilibrium causal games, we give conditions on the mechanisms, equilibrium selection, and intervention design under which agreement with experimental distributions identifies the counterfactual of interest. We show why agreement of means and covariances is insufficient for distributional queries. We then introduce cyclic selection diagrams and derive criteria for direct reuse and for hybrid models that combine invariant source mechanisms with target information. An impossibility result constructs systems that agree under every experiment in a finite design but disagree on the target counterfactual, showing that validation requires structural assumptions. For linear models, we derive intervention requirements that depend on the mechanisms that changed, the observation model, and graph support. When point identification fails, we characterize the remaining range of query values. We also provide statistical tests for reconstructed means and covariances and illustrate the theory in synthetic feedback systems.
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- Source: https://arxiv.org/abs/2607.21667v1
- Canonical: https://arxiv.org/abs/2607.21667v1
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Equilibrium Causal Digital Twins: Validation, Transport, and Identification Limits Faraz Dadgostari Department of Mechanical & Industrial Engineering, Montana State University Neda Nazemi Gianforte School of Computing, Montana State University Abstract Digital twins are often used to predict how a system would respond to an intervention. In systems with feedback, a twin must reproduce an equilibrium counterfactual, and a twin developed in one domain may fail after mechanisms change. We study when these predictions can be validated and transported. For equilibrium causal games, we give conditions on the mechanisms, equilibrium selection, and intervention design under which agreement with experimental distributions identifies the counterfactual of interest. We show why agreement of means and covariances is insufficient for distributional queries. We then introduce cyclic selection diagrams and derive criteria for direct reuse and for hybrid models that combine invariant source mechanisms with target information. An impossibility result constructs systems that agree under every experiment in a finite design but disagree on the target counterfactual, showing that validation requires structural assumptions. For linear models, we derive intervention requirements that depend on the mechanisms that changed, the observation model, and graph support. When point identification fails, we characterize the remaining range of query values. We also provide statistical tests for reconstructed means and covariances and illustrate the theory in synthetic feedback systems. 1 Introduction Digital twins are used to answer intervention questions: what would the system do if a mechanism were changed? For a system with feedback, the answer is an equilibrium rather than a one-way propagation through an acyclic graph. A simple two-agent feedback loop already shows the difference. Removing one feedback arrow can erase the path that carries both the intervention effect and its sensitivity to a change of domain (Chin et al., 2021). This paper separates three questions. Validation asks whether a twin gives the correct counterfactual in the domain where it was developed. Transport asks whether that answer remains correct after some mechanisms change. Identification asks whether the available observational and experimental distributions determine the relevant mechanisms or, more modestly, the query itself. These questions are related, but none can be replaced by the others. The distinction is especially important for counterfactuals. Interventions in cyclic latent models can be identified under graphical and solvability conditions (Forré and Mooij, 2019; Bongers et al., 2021), but a counterfactual links factual evidence to a hypothetical intervention and therefore lies at a higher level of the causal hierarchy (Bareinboim et al., 2022; Bongers et al., 2021). In acyclic models, prior work characterizes testable counterfactuals (Shpitser and Pearl, 2007; Tian and Pearl, 2002) and develops transportability through selection diagrams and do-calculus (Pearl and Bareinboim, 2014; Bareinboim and Pearl, 2016, 2013, 2012; Correa et al., 2022). Existing digital-twin approaches likewise use acyclic potential-outcome models (Laudy, 2026), while time-unrolled approaches remain acyclic within each time slice (Blondel et al., 2016). The equilibrium setting requires a different treatment because the feedback solution itself carries the causal response. Main results. First, we give conditions under which agreement with a collection of interventional distributions validates a counterfactual within one domain. The conditions separate monotonicity of the mechanisms, stability of equilibrium selection, and identification of the query-relevant response. They also distinguish full factual information from partial factual information and distributional agreement from agreement of only means and covariances. Second, we develop transport rules for cyclic selection diagrams. Direct reuse is valid when the relevant post-intervention ancestors are invariant and aligned across domains. When changed mechanisms are relevant, a hybrid model combines invariant source mechanisms with mechanisms re-identified in the target. Third, we show why structural assumptions cannot be avoided. Under explicit witness conditions, two systems can agree under every experiment in a finite design while assigning different values to the same counterfactual. Thus experimental agreement alone does not validate a cross-world prediction. Fourth, for linear models we derive target-intervention requirements that depend on the set of changed mechanisms, the observation model, and graph support. We characterize the remaining set of query values when point identification fails, show when query-specific experiments can require less information than full model recovery, and provide statistical procedures for comparing reconstructed means and covariances. Scope. The positive results require the structural and design conditions stated with each theorem. Intervention counts are not universal: they depend on what is observed, which mechanisms may differ, and which support restrictions are known. The finite-sample and asymptotic procedures likewise apply only under their stated sampling, rank, and regularity assumptions. The numerical examples illustrate the constructions but do not establish general identification or empirical performance. Organization. Section 2 introduces equilibrium causal twins, counterfactual queries, and domain changes. Sections 3 and 4 develop validation within one domain. Section 5 gives direct and hybrid transport results, and Section 7 explains why acyclic reductions can fail. Section 8 establishes the finite-design impossibility result. Sections 6, 9, and 10 study target interventions, partial identification, alignment, and query-specific design. Section 11 presents the statistical procedures, followed by numerical illustrations and limitations. 2 Equilibrium causal twins and counterfactual queries 2.1 Model and equilibrium selection An equilibrium causal game (ECG) consists of structural equations Vi=fi(Vpa(i),Ui)V_i=f_i(V_pa(i),U_i). The scalar noises are mutually independent and have continuous, strictly increasing distribution functions. Each strongly connected component has a unique solution under a declared measurable selection rule Sel, and that rule is stable across the interventions under study. In the linear specialization, V=BV+c+UV=BV+c+U, ρ(B)<1ρ(B)<1, and the observed variables satisfy X=HVX=HV, where H has full column rank and is invariant across environments. 2.2 Counterfactual queries and interventions A per-unit equilibrium counterfactual χ(;,I)χ(S; f,I) is obtained by recovering the exogenous state from the factual observation, applying intervention I, solving the modified system under Sel, and reading eY⊤Vcfe_Y V_cf. The query must be invariant under the residual gauge of the identified object (the sign-absorbed orbit ≃R _R). Otherwise the gauge width in Section 9 remains irreducible. Unless stated otherwise, I changes a mechanism. A payoff or curvature intervention is covered only after the reward layer has been transported and re-evaluated at target moments. In particular, target omitted-variable bias can reintroduce bias into a reward that was unbiased in the source, and the transported reward’s λ-scale gauge must be restored once per domain. These obligations are established in the companion reward paper (Dadgostari, 2026). The results below extend to payoff interventions only when those target-domain conditions hold. 2.3 Conditions for validation Validation proceeds through four logically separate steps. The first recovers factual noise ranks, the second determines what partial factual information implies, the third fixes the equilibrium branch, and the fourth asks whether the experiment identifies the mechanisms relevant to the query. Table 1 summarizes their roles; the formal conditions follow immediately below. Condition Object controlled Consequence for the query T1 strict noise monotonicity factual values recover the private-noise ranks T2 complete legal fibre every legal member induces the same post-intervention query kernel T3 SCC-local selection ranks and the solution set select the same equilibrium branch T4 declared response object response equality forces query-kernel equality throughout the fibre Table 1: The four steps in the validation argument. None substitutes for another: T1 concerns abduction, T2 partial factual information, T3 equilibrium choice, and T4 the information supplied by the experiment. T1: noise ranks. For every node and parent value, fi(v,⋅)f_i(v,·) is strictly increasing. A factual value therefore recovers the rank of its private noise. Definition 1 (Partial-factual query condition (T2)). For retained evidence y, let ℱ(y)F_D(y) be the complete legal fibre, let W be a common labelled, gauge-invariant standard-Borel factual variable with evidence-implied law PWyP_W^y, and let the query output lie in a common aligned or quotiented space YIY_I. Every θ∈ℱ(y)θ _D(y) supplies a regular conditional law Λwθ _w^θ of all private exogenous blocks and selection seeds used by the query and a jointly measurable, well-posed post-intervention solve SIθS_I^θ; write KIθ(w,⋅)=(SIθ)#ΛwθK_I^θ(w,·)=(S_I^θ)_\# _w^θ. A partial-factual query is point-identified exactly when [KIθ]PWy:θ∈ℱ(y)\[K_I^θ]_P_W^y:θ _D(y)\ is a singleton, and only PWyP_W^y-almost everywhere. For a fixed finite block partition, conditional independence instead requires Λwθ=⨂bΛw,bθ _w^θ= _b _w,b^θ almost everywhere. Membership in that null and calibration of a particular test are separate questions. Loop-gain measurability is only an optional Jacobian diagnostic: by itself it is neither necessary nor sufficient for either conclusion. T3: equilibrium selection. Within each strongly connected component, the shared selection rule is a measurable function of the private-noise rank vector and the solution set. It is invariant under strictly increasing reparameterizations of each noise coordinate. Thus the same ranks and the same solution set select the same branch. Definition 2 (Design sufficiency for a query (T4)). Let R(θ)R_D(θ) be the full declared response object supplied by the experimental design, with all labels, intervention operators, signed doses, units, and response laws retained. The design is sufficient for the query if, for every θ,θ′θ,θ in the complete legal fibre, R(θ)=R(θ′)⟹[KIθ]PWy=[KIθ′]PWy.R_D(θ)=R_D(θ ) [K_I^θ]_P_W^y=[K_I^θ ]_P_W^y. For full factuals the same definition uses the corresponding point-mass factual law. This is an observable response condition, not an assumption that the mechanisms have already been identified. The empirical evidence has three components. (C-a) compares predicted and observed responses under staged interventions. Agreement of means and free-block covariances is a moment condition; agreement of the complete interventional kernels is a distributional condition. The former supports the latter only for the specific query classes described in Theorem 5. Even in a linear model, matched first and second moments do not identify the noise distribution. (C-b) examines invariance of structural residuals across regimes, using declared separators and disciplined conditional tests. (C-c) checks the structural class itself, including monotonicity, noise assignment, and the selection rule. For a candidate twin T, write () Accept_D(T) for the event that every predeclared, nonempty block procedure associated with design D accepts. This notation records only the conjunction of those procedures; it does not assert that their implications identify an arbitrary query. 2.4 Domain changes and alignment A source πS^π and target τS^τ are ECGs on a shared graph G over V (|V|=d|V|=d), each uniquely solvable per SCC under a shared Sel. Define the discrepancy set by =i:(fi,PUi) differ across domains. D=\i:(f_i,P_U_i) differ across domains\. For i∉i∉ D, both the mechanism and its noise law are invariant. Exogenous independence (standing): in both domains the UiU_i are mutually independent (diagonal Ω ), so all cross-domain difference is localized to the per-node (fi,PUi)\(f_i,P_U_i)\; this excludes a shared-marginal copula shift (an invariant D—even =∅ D= —with a distinct exogenous joint law, which would reproduce every node-marginal yet move the equilibrium law). The cyclic selection diagram is D=G+i→i:i∈D=G+\S_i→ i:i∈ D\ with mechanism-level semantics (Definition 3). Linear/LQ instance: V=BωV+cω+UωV=B^ωV+c^ω+U^ω, zero-diagonal BωB^ω, ρ(Bω)<1ρ(B^ω)<1; rows i∉i∉ D share (Bi,⋅,ci,Ωii)(B_i,·,c_i, _i); sensor map Xω=HωVωX^ω=H^ωV^ω, HωH^ω full column rank. Alignment A: latent frames are matched either by shared sensors (Hτ=Hπ=:H^τ=H^π=:H, “anchoring”) or by the invariance-labeled frame; Proposition 41 quantifies the residual ambiguity when neither suffices. Definition 3 (Cyclic selection diagram; mechanism-level switches). iS_i annotates the mechanism of i, never a solution map. The augmented model is an ioSCM whose per-regime unique solvability holds because each fixed pure S-value reproduces one domain’s model (under the exogenous-independence assumption above, so all cross-domain difference is S-annotated); for mixed S-profiles (some coordinates π, others τ) we carry the Forré and Mooij (2019) SCC/loop-solvability hypothesis on the augmented model — or, w.l.o.g. for Rule-D, take S as a single global switch node, which admits only the two pure profiles. The generalized directed global Markov property applies to it. 3 Counterfactual agreement under monotone mechanisms The next lemma connects interventional distributions to unit-level counterfactuals. It provides the common argument for validation within a domain (Theorem 5) and transport across domains (Theorem 15). Lemma 4 (Quantile abduction under feedback). Let ,′∈+S,S ^+ share the graph, Sel, the intervention semantics of I, and the query-relevant interventional kernels P(Vi∣do(Vpa(i)=vpa(i)))P\! (V_i (V_pa(i)=v_pa(i)) ) for every i on the post-surgery ancestral SCC-set of the query (with Sel rank-measurable per (T3)). Then every full-factual per-unit counterfactual coincides: χ(;v,I)=χ(′;v,I)χ(S;v,I)=χ(S ;v,I) for a.e. v (w.r.t. the factual law). Proof. By (T1) each mechanism is fi(v,u)=Qido(v,Fi(u))f_i(v,u)=Q_i^do(v,F_i(u)), where QidoQ_i^do is the conditional quantile of the interventional kernel P(Vi∣do(Vpa(i)))P(V_i (V_pa(i))) — the structural response with the parents held fixed — and not the observational P(Vi∣Vpa(i))P(V_i V_pa(i)), which under feedback confounds UiU_i with its own realized parents and is therefore not the abduction object. Abduction fixes the rank τi=FVi∣do(pa)(vi) _i=F_V_i (pa)(v_i); the post-surgery rank-coupled system is re-solved by the shared Sel, which by the strengthened (T3) is rank-measurable and so reads only the shared ranks, not the per-node noise gauge. Every ingredient is a functional of shared interventional objects. □ The argument is the equilibrium instance of BGM monotone-determinism (Nasr-Esfahany et al., 2023), which is stated for structural/interventional mechanisms rather than cyclic observational conditionals. Interpretation. Strict monotonicity lets the factual observation recover a rank for each noise coordinate. Once the relevant interventional kernels and the equilibrium selection rule agree, the same ranks generate the same post-intervention outcome. Observational conditionals alone do not supply this information in a feedback system. Sharpness (observational conditionals do not suffice). Two systems in +T^+ sharing every observational conditional but differing on a query-relevant interventional kernel can disagree on χ: the linear two-node witness V1=aV2+U1,V2=bV1+U2V_1=aV_2+U_1,\ V_2=bV_1+U_2 with common Σ=(10.50.51) = ( smallmatrix1&0.5\\ 0.5&1 smallmatrix ) realises do(V2)do(V_2)-slopes a=0.2a=0.2 and a=0.4a=0.4, while observation-only abduction returns the wrong common slope 0.50.5; hence the hypothesis cannot be weakened to observational conditionals. When only observational laws and invariance (C-a,C-b without staged interventions) are available, χ is set-identified with the sharp interval of §9, not a point. 4 Validation within one domain We now state the conditions under which experimental agreement determines the counterfactual of interest. The result separates the population identification requirement from the finite-sample procedures used to compare a fixed candidate twin with the reference system. Theorem 5 (When experimental agreement identifies the query). If ,∈+()S,T ^+(D) (verified by C-c) with shared graph/ Sel and the staged set meets a sufficiency design of Theorem 42 for χ — one achieving Δχ()=0 _χ(D)=0 (e.g. any (T4) design, or the conditional-affine Vminqry(χ)V_ qry(χ) block cover; not the bare first-order VminFOV_ FO, which the y+x2y+x^2 counterexample shows insufficient off the affine case; §10), and T passes distributional (C-a) and (C-b) population-exactly, then χ(;v,I)=χ(;v,I)χ(T;v,I)=χ(S;v,I) a.e. for full factuals. For a partial factual, equality instead requires the exact (T2) boundary: the complete legal fibre has a singleton PWyP_W^y-a.e. class of query pushforward kernels [KIθ]PWy[K_I^θ]_P_W^y. Identification of the full (Λθ,SIθ)( ^θ,S_I^θ) pair is sufficient but not necessary; product factorization supplies the declared conditional independence, not equality of partial-factual query posteriors. The moment (C-a) back-tests deliver distributional (C-a) only query-by-query, never blanketly: even in the linear identified class, matching means and covariances does not identify the noise distribution (B=0,H=1B=0,H=1: U∼N(0,1)U\! \!N(0,1) and U∼Exp(1)−1U\! \!Exp(1)-1 share every additive-shift moment yet P(U>2)=0.023≠0.050P(U>2)=0.023≠ 0.050). Under identified linear structure the transfer is therefore query-specific: full-factual state-functional queries transfer from structure and the full factual state alone (no noise-distribution identification); interventional moment queries transfer when they factor through the identified first two moments; interventional distributional queries transfer only under a declared moment-determined noise family (or the distributional (C-b)/PIT route); and partial-factual queries only when the complete-fibre pushforward class is a singleton as in (T2). A general nonlinear mechanism fails even the interventional-moment step (moment-matched twins differ on the interventional kernel and on χ). So distributional (C-a) is an added, query-specific hypothesis, not delivered by moment matching alone. Proof. See Appendix A. □ Proposition 6 (Selected-summary modulus). For finite samples, keep the population object and the finite-sample summary distinct. Let ℒ(θ)L_D(θ) be the complete indexed observational and staged laws used by Theorem 42. Separately, let mc(θ) M_D mc(θ) be the fixed, predeclared mean–covariance summary actually tested: for each of the E environments it has one reconstructed-mean block and one Frobenius-isometric svecfree(ΣFF(e))svec_ free( _F^(e)) block. Zero-dimensional blocks with Fe=∅F_e= are omitted; let ℬB_D index the remaining blocks and put B=|ℬ|≤2EB=|B_D|≤ 2E. After fixed, data-independent block scalings (identity here), write ‖Rmc(θ)‖⊕2:=‖mc(θ)−mc(θ0)‖⊕2=∑e=1E(‖rem(θ)‖22+‖reΣ(θ)‖22).\|R_D mc(θ)\|_ ^2:=\| M_D mc(θ)- M_D mc( _0)\|_ ^2= _e=1^E (\|r_e^m(θ)\|_2^2+\|r_e (θ)\|_2^2 ). For a fixed, validation-data-independent twin in a deterministic localization Kη,RK_η,R, assume the selected-summary modulus ‖χ(θ)−χ(θ0)‖≤CL‖Rmc(θ)‖⊕s∀θ∈Kη,R,\|χ(θ)-χ( _0)\|_ Q≤ C_L\|R_D mc(θ)\|_ ^s ∀θ∈ K_η,R, where ∥⋅∥\|·\|_ Q is the declared query norm and CL,s>0C_L,s>0 are truth-specific instance constants. This is a moment-to-query sufficiency and separation premise; it is not implied by constancy on the complete-law fibre. A data-selected twin needs a uniform power statement for its selection rule, and a twin outside Kη,RK_η,R contributes Pr[localizationfails] [localization\ fails] unless another modulus is supplied. The modulus may be derived by Łojasiewicz only when Kη,R/∼K_η,R/ is a compact legal quotient that is closed and Hausdorff, and is definable in a common polynomially bounded o-minimal expansion. Moreover, mc M_D mc, χ, and the declared norms descend to continuous definable maps on the entire quotient (or on a finite closed-branch decomposition with common exponent s=minisis= _is_i and an adjusted maximum constant); and the separate truth-centred moment-fibre condition mc(θ)=mc(θ0)⟹χ(θ)=χ(θ0)(θ∈Kη,R) M_D mc(θ)= M_D mc( _0) χ(θ)=χ( _0) (θ∈ K_η,R) holds. These hypotheses give existential, not numerically computed, instance constants by the definable Łojasiewicz inequality (van den Dries and Miller, 1996). Merely o-minimal is insufficient: by Miller’s dichotomy (Miller, 1994), χ(t)=tχ(t)=t and M(t)=e−1/tM(t)=e^-1/t on [0,1][0,1] defeat every power modulus. The linear/LQ class is semialgebraic (with restricted-analytic tail queries) on Kη,R=∥B∥F≤R,ηI⪯Ω⪯RI,σmin(H)≥η,∥H∥≤R,∥c∥≤R,η≤cΣ≤R,ϕ∈Ξ,|λi(B(e))|≤1−η∀e∈∪patterns(χ)K_η,R=\\|B\|_F≤ R,\ η I RI,\ _ (H)≥η,\ \|H\|≤ R,\ \|c\|≤ R,\ η≤ c_ ≤ R,\ φ∈ ,\ | _i(B^(e))|≤ 1-η\ ∀ e (χ)\, where Ξ is closed, bounded and definable. The per-pattern spectral margins keep every relevant resolvent finite; a ρ(B)ρ(B)-only truncation does not. The constants can diverge at an unprobed-pattern pole, covariance floor or rank boundary. The exponent s is the exponent of (χ,Rmc)(χ,R_D mc); it equals a ratio of vanishing orders only after a uniform arc-wise proof. The equal-moment Gaussian/centred-exponential nonlinear pair violates (Suff-M), so this moment route is unavailable; complete-law separation cannot be substituted while retaining a moment-based rate. Proof. See Appendix A. □ Corollary 7 (Finite-sample query tolerance). Fix the candidate independently of the validation sample, and fix all block scales before seeing that sample. Use the Euclidean norm for reconstructed means and the Frobenius-isometric svecsvec norm for covariance blocks; their direct-sum norm is the norm in Proposition 6. For each actual block b∈ℬb _D assume the block-specific power condition (BPb):∥rb∥b>qb,n⟹Prblock b accepts≤βb.(BP_b): \|r_b\|_b>q_b,n \block $b$ accepts\≤ _b. The aggregate decision accepts only when every block accepts. Put Qn:=(∑b∈ℬqb,n2)1/2,εn:=CLQns.Q_n:= ( _b _Dq_b,n^2 )^1/2, _n:=C_LQ_n^s. For B≥1B≥ 1, εn≤CL[Bmaxbqb,n]s _n≤ C_L[ B _bq_b,n]^s. If B=0B=0, use the empty-sum convention Qn=εn=0Q_n= _n=0 and omit the maximum display; (Suff-M) then permits no query-changing candidate in the localization, so the wrong-query event below is empty. For B≥1B≥ 1 and a fixed predeclared localized twin, ‖χ−χ‖>εn\| _T- _S\|_ Q> _n forces some block ‖rb∗‖2>qb∗,n\|r_b \|_2>q_b ,n, and therefore Pr[()∧‖χ−χ‖>εn] [ Accept_D(T)\ \ \| _T- _S\|_ Q> _n] ≤βb∗+Pr[localizationfails]≤maxbβb+Pr[localizationfails]. ≤ _b + [localization\ fails]≤ _b _b+ [localization\ fails]. This is single-test containment, not a union bound; test levels control false rejection of the correct twin. For the nonasymptotic sub-Gaussian route with known H, Proposition 47 gives the mean block qe,nm=te,nm(αe)+te,nm(βe)q_e,n^m=t_e,n^m( _e)+t_e,n^m( _e); with estimated H, use q~e,nm=t~e,nm(αe)+t~e,nm(βe) q_e,n^m= t_e,n^m( _e)+ t_e,n^m( _e) instead. The covariance block order is qe,nΣ=Oκ¯Σ,e[keue/ne+keue/ne]q_e,n =O\ κ_ ,e[ k_eu_e/n_e+ k_e\,u_e/n_e]\, ue=log[4ke/(αe∧βe)]u_e= [4k_e/( _e _e)], with κ¯Σ,e=CBKZ,e2λ¯e κ_ ,e=C_BK_Z,e^2 λ_e and λ¯e≥λmax(ΣFF(e)) λ_e≥ _ ( _F^(e)), as in Proposition 49. Thus both Bernstein branches and the 2ke 2k_e conversion from the raw-entry threshold to the svecsvec norm are retained. The bound is existential/order-level: neither (CL,s)(C_L,s) nor a numerical εn _n is estimated in the numerical illustrations, which do not establish (⋆mc _ mc) or the near-threshold theorem. The fixed-d ADF-Wald route is separate: it gives pointwise asymptotic power for a fixed nonzero residual in the declared estimable range. It enters no finite-n sum in († ). A singular branch additionally needs fixed-rank/eigengap control and the proved kernel companion (including deterministic coordinates), or it refuses; an estimated random Wald seminorm does not replace the deterministic norm above. The displayed sub-Gaussian orders use independent probe observations nen_e; clustered data require their own cluster-sum concentration theorem, rather than a raw-row or generic effective-size substitution. This aggregate does not bound the d×d×d parameter error. The identification backbone below is the linear/LQ class (together with the nonlinear source-block positive result of the companion representation paper, Dadgostari and Nazemi (2026)); general diffeomorphic mixing inherits that paper’s retraction and is not claimed here. Proof. See Appendix A. □ Interpretation. Validation is query-specific. Full interventional distributions support the general counterfactual conclusion, whereas mean and covariance agreement is enough only when the query factors through those moments or when the noise family is determined by them. With partial factual information, the entire legal fibre must induce the same query distribution. Theorem 8 (Why the boundary conditions are needed). The four conditions exclude different failures. Without T1, a parent-dependent non-monotone rank-band reflection Ti(pa,u)=Fi−1∘ρ(pa,⋅)∘Fi(u)T_i(pa,u)=F_i^-1\! ρ(pa,·) F_i(u) at a node whose band-driving parent is not a descendant of i is measure-preserving for every papa, so all laws in every regime coincide while the counterfactual changes on the positive-measure symmetric difference of the factual and counterfactual bands. In the symmetric-noise linear specialization this is u↦−u -u on the band and the gap is 2|eY⊤(I−BI)−1ei||ui|>02\,|e_Y (I-B^I)^-1e_i|\,|u_i|>0; without symmetry the gap remains positive but has no such closed form. (A parent-independent annulus reflection is instead a measure-preserving involution — an SCM isomorphism with counterfactual gap 0 — so the parent dependence is essential to the T1 argument.) Without T2, the failure is already present without a cycle. Let V1=U1V_1=U_1, V2=U2V_2=U_2, and V3=V1+V2+U3V_3=V_1+V_2+U_3 with independent standard Gaussian noises, and condition on W=V3W=V_3. Every loop-gain condition is vacuous, yet Cov(U1,U2∣V3)=−1/3Cov(U_1,U_2 V_3)=-1/3, so the tensor-product identity fails. In the explicit Gaussian difference test, calibrating D=U1−U2D=U_1-U_2 with the false product variance 4/34/3 instead of its true conditional variance 22 makes nominal level 0.050.05 equal 0.1095310.109531. Conversely, whenever the declared regular conditional law equals the tensor product of its block marginals, the conditional independence follows regardless of loop gain; level validity of any named test still requires its own sampling and calibration theorem. The two-node feedback example gives a second analytic control: its conditional log-density has mixed partial −1/(1−v0v1)2≠0-1/(1-v_0v_1)^2≠ 0. The same calculation has an acyclic-collider negative control and an independent-noise positive control. Thus T2 is exactly the singleton query-kernel condition, whereas its tensor-product clause is exactly the declared conditional-independence null; full conditional-law identification is only sufficient. Without T3, take the primitive Z∼N(0,1)Z N(0,1), f0(Z)=tanhZ=:uf_0(Z)= Z=:u, state space [−2,2][-2,2], and best-response update Φ(v,u)=v−0.1v3−3v−u. (v,u)=v-0.1\v^3-3v-u\. Its equilibria solve v3−3v−u=0v^3-3v-u=0. The two outer roots are stable and are separated by more than 17/517/5 for every u∈(−1,1)u∈(-1,1). Two models with the same mechanism and noise, one using the minimum-stable-root rule and the other the maximum-stable-root rule, therefore differ only in Sel and disagree by more than 17/517/5 on the corresponding query. Each model uses its declared rule in every regime. This witness is selection-specific; it does not claim that a design which directly observes the selected branch is uninformative. Without T4, let B0=(03/51/20)B_0= ( smallmatrix0&3/5\\ 1/2&0 smallmatrix ), Ω0=I _0=I, H0=IH_0=I, A0=(I−B0)−1A_0=(I-B_0)^-1, and let QθQ_θ be the planar rotation at θ=0.7θ=0.7. Put Aθ=A0QθA_θ=A_0Q_θ, Wθ=Aθ−1W_θ=A_θ^-1, Dθ=diag((Wθ)11,(Wθ)22)D_θ=diag((W_θ)_11,(W_θ)_22), Bθ=I−Dθ−1WθB_θ=I-D_θ^-1W_θ, Ωθ=Dθ−2 _θ=D_θ^-2, and Hθ=IH_θ=I. Both models are stable Gaussian ECGs and have the same passive law because AθDθΩθDθAθ⊤=A0A0⊤A_θD_θ _θD_θA_θ =A_0A_0 . For factual state (1,1/2)(1,1/2) and the query obtained by setting V0=2V_0=2 and reading V1V_1, their values are 11 and 1.39166495961.3916649596. No probing-stage equality is asserted: the example shows precisely why passive Gaussian evidence alone does not satisfy T4. Proof. See Appendix A. □ Interpretation. The assumptions rule out distinct failure modes rather than repeating the same restriction. Monotonicity fixes the noise ranks, the conditional condition controls partial-factual abduction, the selection condition controls which equilibrium is chosen, and the design condition removes query-relevant observational equivalences. 5 Transport across domains Write AnGI(⋅)An_G_I(·) for the post-surgery ancestor set. The engine is that ancestral sub-models are autonomous. Lemma 9 (SCC-closed ancestral autonomy). For any surgery I and target set Y∪ZY∪ Z, A:=AnGI(Y∪Z)A:=An_G_I(Y∪ Z) is closed under parents and under SCC-membership, the restricted SCM on A is autonomous and inherits unique solvability per SCC, and the post-surgery law of (VY,VZ)(V_Y,V_Z) is a functional of fjω,PUjω:j∈A\f_j^ω,P_U_j^ω:j∈ A\ alone. In the linear class, every entry ew⊤(I−BI)−1e_w (I-B_I)^-1 is a walk-sum over ancestors of w; the identity persists by rational continuation to all det(I−BI)≠0 (I-B_I)≠ 0 (the trek/⟂t _t argument). □ The lemma isolates the part of the system that can affect the query after intervention. Mechanisms outside this ancestral subsystem do not enter the post-intervention law. Theorem 10 (Direct transport under ancestral separation). Assume source identification, alignment secured on A=AnGI(Y∪Z)A=An_G_I(Y∪ Z), and the cross-domain invariance condition (the (C-b) comparison, run with source∪\,∪\,target regimes and conditional discipline) holds on the mechanisms of A. If (sep):AnGI(Y∪Z)∩=∅(no -node is an ancestor of Y∪Z in DI), (sep): _G_I(Y∪ Z)∩ D= (no S-node is an ancestor of Y∪ Z in D_I), then Pτ(Y∣do(I),Z)=Pπ(Y∣do(I),Z)P^τ(Y (I),Z)=P^π(Y (I),Z); in the linear class this is an all-parameter identity (Lemma 9). Proof. By (sep) and Lemma 9 the target law of (Y,Z)(Y,Z) under do(I)do(I) is a functional of A-indexed mechanisms, all invariant by (sep)+\,+\,the cross-domain invariance condition; the same functional at the same mechanisms is the source law, computable from [π][S^π] since identification makes the class a point on A (or χ is gauge-invariant). □ Interpretation. Direct reuse requires four ingredients: the source mechanisms must be identified, the relevant latent frames must be aligned, the retained mechanisms must satisfy the cross-domain invariance condition, and (sep) must exclude every changed mechanism from the post-intervention ancestral subsystem of the query. These conditions are sufficient; failure of (sep) does not by itself show that reuse is impossible for a particular query. Lemma 11 (Domain changes within a feedback component). If i∈i∈ D lies in an SCC C with |C|≥2|C|≥ 2, then every j∈Cj∈ C is graphically downstream of iS_i. A generic distributional conclusion needs an additional regularity premise. On any connected finite-dimensional real-analytic stratum, suppose a finite distribution-separating signature sC(θ)s_C(θ) is analytic and there is an admissible witness at which the domain change moves sCs_C. Then the parameter values for which the law of C does not change lie in a proper analytic zero set. In the linear model, changing intercept cic_i by t moves E(Vj)E(V_j) by t[(I−B)−1]jit[(I-B)^-1]_ji whenever that entry is nonzero, which supplies such a witness. For an unrestricted nonlinear stratum no analytic-generic claim is made; the corresponding conclusion requires an explicit distributional-faithfulness assumption. In every case, nodes in C∖C D retain their mechanisms even when their equilibrium laws change. Proof. See Appendix B. □ Theorem 12 (Transport by replacing changed mechanisms). Fix the surgery, labelled alignment, mutually independent private noises, SCC-local selection rule, and well-posedness conditions. Form a hybrid by taking a complete pair (fi,PUi)(f_i,P_U_i) either from the aligned source or from target re-identification. Full-law transport. If the hybrid pair equals the target pair for every nonintervened node, then the hybrid and target have the same post-surgery joint law. Query-law transport. For a query on Y, it is enough that those complete pairs agree on the post-surgery ancestral SCC closure AnGI(Y)An_G_I(Y). Then the hybrid and target have the same law of Y. Structural row recovery by itself identifies only the corresponding linear coefficients and intercepts; it does not identify an unrestricted noise law. Proof. See Appendix B. □ Corollary 13 (Moment-class hybrid transport). If only the target structural rows and first two noise moments are re-identified, the hybrid conclusion holds for queries that factor through those moments. A full distributional conclusion additionally requires a moment-determined noise family or direct identification of the target noise laws on the relevant closure. Interpretation. When a changed mechanism lies on a relevant feedback path, source reuse can be repaired by estimating that mechanism in the target while retaining invariant mechanisms from the source. The resulting hybrid is evaluated as one equilibrium system rather than as a patchwork of acyclic effects. Theorem 14 (A graphical transport rule). Treating S as exogenous roots, Rule-D: if ⟂σY∣Z∪XS _σY Z∪ X in DID_I then Pπ(y∣do(I),z,x)=Pτ(y∣do(I),z,x)P^π(y (I),z,x)=P^τ(y (I),z,x) for the source separator law. To use this kernel at target, assume the regime-specific domination μτZX≪μπZX _τ^ZX _π^ZX for the post-surgery law of the full separator. The target integral then uses the common kernel with the target separator law. The conditioning set of the conclusion must contain the full separating set Z∪XZ∪ X, not Z alone. Dropping X is unsound: in →X→YS\!→\!X\!→\!Y we have ⟂σY∣XS _σY X yet Pπ(y∣do(I),z)≠Pτ(y∣do(I),z)P^π(y (I),z)≠ P^τ(y (I),z) once S shifts X’s law; to transport the X-marginalized P(y∣do(I),z)P(y (I),z) one needs X intervened, the target law Pτ(x∣do(I),z)P^τ(x (I),z), equality of the source and target X∣ZX Z laws, or an x-constant kernel. Composed with the Forré and Mooij (2019) rules on DID_I this licenses selection-diagram transport formulas. Soundness holds under that paper’s SCC/loop solvability hypothesis (carried verbatim); the linear route (Lemma 9) needs no such hypothesis. Completeness is open. Proof: the augmented model’s generalized directed global Markov property gives Y⟂∣Z,XY \!\!\! Z,X; conditioning on the two S point masses gives equality μπZX _π^ZX-a.e.; domination transports that equality to the target separator law. □ Interpretation. The graphical rule transports a conditional distribution only with the complete separating set retained. If a separator is subsequently marginalized, its distribution must itself be transported or fixed by intervention. Theorem 15 (Transporting unit-level counterfactuals). For a per-unit counterfactual at target factuals, suppose the target hybrid lies in +T^+: it has monotone noise, the complete-fibre pushforward-kernel singleton of (T2), and shared Sel. Then a partial-factual χ transports PWyP_W^y-a.e. because every member of the complete target fibre induces the same query-pushforward kernel by (T2); for full-factual queries, Lemma 4 applies to the hybrid, whose query-relevant interventional kernels are pinned by Theorem 10 or the query-law clause of Theorem 12 through equality of the complete mechanism–noise pairs on the post-surgery ancestral SCC closure. Below the boundary only bounded transfer holds, with W=diamχ()W=diamχ(A) and, under recombination closure and a shared Sel (else the selection-branch gap of Theorem 36 is added), the cross-world width bound W≤W¯comp+W¯mono+W¯rotW≤ W^comp+ W^mono+ W^rot (Theorem 36; additively exact only in the decoupled orbit-invariant-coefficient subclass; without closure only the joint diameter) at target parameters. □ (For partial-factual abduction and transport, the query-pushforward class over the complete fibre is the relevant object; loop gain alone does not determine it.) Interpretation. Transport of an interventional distribution and transport of a unit-level counterfactual are not the same claim. The latter additionally requires the target factual evidence to select a unique query distribution over the complete legal fibre; otherwise the appropriate answer is a range of possible values. Definition 16 (Complete transport fibre). Let Θ be a complete admissible joint source–target class. It includes every profiled alignment and nuisance parameter, all mechanisms, the joint noise law, and the full equilibrium-selection kernel for every evidence and query regime. The underlying kernel spaces are standard Borel, and the common query-output space has already been quotiented by the declared alignment and gauge equivalence. Fix a pre-query design d, a reference θ⋆∈Θ _ ∈ , and the exact evidence map EdE_d, comprising all permitted source and target design laws, including the observed target factual law but excluding the query-bearing law and unavailable experiments. Define ℱd(θ⋆)=θ∈Θ:Ed(θ)=Ed(θ⋆).F_d( _ )=\θ∈ :E_d(θ)=E_d( _ )\. Let μ⋆ _ be the fixed factual-input law contained in Ed(θ⋆)E_d( _ ) and hence common to this fibre. For query surgery I, put QI(θ)=[KIθ]μ⋆Q_I(θ)=[K_I^θ]_ _ , or use the corresponding unconditional law, in a proper quotient equipped with a separating metric ϱ . Proposition 17 (Pointwise transport identification). The query is pointwise transport-identified at (d,θ⋆)(d, _ ) if and only if QI(ℱd(θ⋆))Q_I(F_d( _ )) is a singleton. It fails to be pointwise transport-identified if and only if the actual fibre contains θ0,θ1 _0, _1 with ϱQI(θ0),QI(θ1)>0 \Q_I( _0),Q_I( _1)\>0. Proof. This is the definition of identification on the complete evidence fibre; no global measurable factorization through EdE_d is required. □ Lemma 18 (Exact-evidence curves). Let γ:(−ε,ε)→Θγ:(- , )→ , ε>0 >0, satisfy γ(0)=θ⋆γ(0)= _ and Edγ(t)=Ed(θ⋆)E_d\γ(t)\=E_d( _ ) throughout its domain. If some admissible t≠0t≠ 0 satisfies ϱ[QIγ(t),QI(θ⋆)]>0, [Q_I\γ(t)\,Q_I( _ )]>0, then the query is not pointwise transport-identified. Proof. Both endpoints belong to the same actual evidence fibre and are separated by the query metric; apply Proposition 17. □ Theorem 19 (Query-active complete-block collision). Let ε>0 >0, let ℬB be a complete parameter block, and consider θ(t)=ιℬ(cℬ⋆+tv;θ−ℬ⋆)θ(t)= _B(c_B +tv; _-B ) for |t|<ε|t|< , where ιℬ _B replaces the indicated block and leaves its complement fixed. Assume: (i) every θ(t)θ(t) belongs to Θ for every evidence and query regime, including cross-block noise compatibility, positive definiteness, and the full selection specification; (i) Edθ(t)=Ed(θ⋆)E_d\θ(t)\=E_d( _ ) exactly; and (i) a quotient-well-defined scalar functional ℓ that is insensitive to null representatives—for example, integration against a bounded test function—satisfies ℓQI(θ(t))=α+βt,β≠0. \Q_I(θ(t))\=α+β t, β≠ 0. Then every admissible nonzero t∈(−ε,ε)t∈(- , ) gives an actual-fibre collision separated by the query, and the query is not pointwise transport-identified. Proof. Conditions (i)–(i) put θ(t)θ(t) and θ⋆ _ in the same complete admissible fibre. Condition (i) gives distinct quotient query laws for every admissible nonzero t. Proposition 17 then gives the conclusion. □ Corollary 20 (Instantiation by the existing collision constructions). The stable complete-block family in Theorem 26 instantiates Theorem 19 whenever its primitive construction establishes membership in the complete joint class for a nondegenerate interval, exact equality of the complete design evidence, and nonzero query motion. Rotational or split-frame families, a finite-budget family, a non-monotone reshuffle, or a T2/T4 ambiguity is an instantiation only when its own primitive argument supplies the same three facts together with quotient-well-defined query separation. In particular, a reshuffle outside the declared class does not prove an in-class collision, and a graphical label or a positive-dimensional stabilizer does not by itself establish query motion. Proof. Each qualifying construction supplies the curve and separating functional required by Theorem 19. □ Interpretation. Non-identification requires a design-null direction that is also query-active. Fixed, cancelled, or surgery-excised queries fail the condition β≠0β≠ 0. The intersection AnGI(Y)∩An_G_I(Y)∩ D locates a possible direct post-surgery response channel, but it is not a general necessity condition for conditional counterfactual or selection-mediated failure. Neither raw ancestry nor membership outside +T^+ is, by itself, a sufficient non-transportability claim. 6 Identifying changed mechanisms in linear models We now specialize to linear models and ask how many target interventions are needed to identify the changed mechanisms. Assume that invariance and alignment have established the rows indexed by c D^c. We consider single-target interventions. In the CFFC_F-nonsingular ambient branch, active probing of all discrepancy rows identifies the target. Smaller support-aware designs and every singular branch are governed by the complete legal fibre, not by a universal numerical fallback. Definition 21 (Complete stratified linear fibre). For a fixed design, let the complete legal fibre be the union of all admissible support, rank, normalization, alignment, stability, and selection strata. On an affine stratum s, profile the fixed nuisances and write its relative-interior patch as ℱs∘=θs+Qst:Mst=0∩Θs∘,Qs∈ℝns×ps,rank(Qs)=ps.F_s =\ _s+Q_st:M_st=0\∩ _s , Q_s ^n_s× p_s, (Q_s)=p_s. Here MsM_s has psp_s columns. Its local dimension is ps−rank(Ms)p_s-rank(M_s). Local identification is constancy on this patch; global identification is constancy over the union of every admissible stratum and every disconnected component. Lemma 22 (Free-block ambiguity at a singularity). Order the coordinates as F,F, D. Fix an invertible C with unit diagonal, its known row block CF,⋅C_F,·, and the pinned response rows Z=P,⋅Z=P_ D,· of P=C−1P=C^-1, and write Z=[ZFZ]Z=[Z_F\ Z_ D]. If q=nullity(CFF)=nullity(Z)q=nullity(C_F)=nullity(Z_ D) and N∈ℝ||×qN ^| D|× q spans kerZ Z_ D, then every compatible discrepancy-row completion Y=C,⋅Y=C_ D,· is uniquely Y=Y0+NT,T∈ℝq×d.Y=Y_0+NT, T ^q× d. For a fixed support branch whose discrepancy-row equalities are affine, let ℒG(Y)=bGL_G(Y)=b_G collect the unit-diagonal equations and every declared exact zero in those rows, and put rG=rankT↦ℒG(NT).r_G=rank\T _G(NT)\. The legal completion fibre is an affine set of dimension qd−rGqd-r_G, intersected with the open conditions that C is invertible, ρ(I−C)<1ρ(I-C)<1, all staged systems are well posed, and declared present edges remain nonzero. Any additional nonlinear model equality requires its own constant-rank analysis. Thus every relative-interior patch has dimension qd−rGqd-r_G. In the ambient unknown-support class, where only the diagonal is imposed, rGr_G is the number of nonzero rows of N and is generically ||| D|; fixed support can increase rGr_G and can identify a completion even when CFFC_F is singular. Local query identification requires constancy on the local patch; global query identification requires constancy on the entire legal fibre, including all of its components and admissible branches. Proof. The pinned-row identity ZC=EZC=E_ D is ZY=E−ZFCF,⋅Z_ DY=E_ D-Z_FC_F,·. Its complete solution set is Y0+NTY_0+NT. The displayed structural equalities are linear in Y, so rank–nullity gives qd−rGqd-r_G; intersecting strict inequalities leaves the local dimension unchanged. The equality of the two nullities is the complementary nullity theorem for the invertible pair (C,P)(C,P). For diagonal-only constraints, different discrepancy nodes select different columns of T, so the rank is exactly the number of nonzero rows of N. □ Theorem 23 (Intervention requirements in linear models). Consider active single-target probes on distinct D-nodes. Intervening on every discrepancy node pins the labelled discrepancy rows P,⋅τP^τ_ D,· of Pτ=(I−Bτ)−1P^τ=(I-B^τ)^-1. Together with the invariant rows, these probes identify Bτ=I−(Pτ)−1B^τ=I-(P^τ)^-1 only when the remaining free-row completion is unique. In the ambient unknown-support branch, that completion is unique exactly when CFFPF=EF−CFP,C=I−Bτ,det(CFF)≠0,C_FP_F=E_F-C_F DP_ D, C=I-B^τ, (C_F)≠ 0, or, equivalently, det(Pτ)≠0 (P^τ_ D D)≠ 0. Nonsingularity of I−BτI-B^τ alone does not imply this condition. On a singular or support-restricted branch, identification is governed instead by the complete stratified fibre above. A scalar query is locally identified iff it is constant on every relevant relative-interior patch and globally identified iff it is constant across all strata and components. No numerical intervention count is inferred from singularity or local dimension alone. Proof. See Appendix C. □ Proposition 24 (Joint-influence free-block inference). Fix F,F, D, the dimensions, environments, labels, support/rank, model branch, and ηF>0 _F>0 independently of the inference sample. A declared calibrated source experiment π supplies the complete structural row block BF,⋅πB^π_F,· before the target free-row solve, and the fixed common structural frame satisfies the explicit interface assumption BF,⋅τ=BF,⋅π.B^τ_F,·=B^π_F,·. The source design has complete labelled coverage of the observational response map and every source single-node intervention/P-row object used to reconstruct all rows of PπP^π that form BF,⋅πB^π_F,·; a partial collection is outside this result. For each labelled response-map environment e, with a shared baseline arm 0, probe arms 1:q1:q, and fixed full-row-rank design D, form R^e=[X¯e1−X¯e0,…,X¯eq−X¯e0],M^e=R^e⊤(⊤)−1. R_e=[ X_e1- X_e0,…, X_eq- X_e0], M_e= R_e D ( D D )^-1. The labelled structural row j is reconstructed, without a target PFτP_F^τ solve, by mj=M^0ej,Dj=M^j−M^0,wj⊤=−mj⊤Djmj⊤mj,P^jj=11−wjj,P^j,:=ej⊤+P^jjwj⊤.m_j= M_0e_j, D_j= M_j- M_0, w_j =- m_j D_jm_j m_j, P_j= 11-w_j, P_j,:=e_j + P_jw_j . Stacking the source rows gives B^π=I−P^π−1 B^π=I- P_π^-1. A fixed, predeclared structural stochastic-coordinate mask retains the estimated entries and projects the known zero diagonal and every declared exact structural zero to zero; those deterministic coordinates never enter the Wald basis. The target reconstruction supplies only the labelled rows P^,:τ P^τ_ D,:. Thus the pure pre-solve estimator is C^=IF−B^FFπ,G^=−B^Fπ,Z^=P^,:τ. C=I_F- B^π_F, G=- B^π_F D, Z= P^τ_ D,:. Let the joint primitive vector contain every source and target arm mean, each shared baseline only once, and every nuisance actually plugged into these maps. The sampling unit is an independent cluster: within-cluster multi-arm and source–target dependence is retained in one influence vector. If N is the number of clusters, Na/N→πa∈(0,1)N_a/N→ _a∈(0,1) for every required fixed stratum. Assume the displayed reconstruction denominators and PπP_π are nonsingular; the joint cluster array obeys a Lindeberg CLT and finite covariance (a fixed 2+ϵ2+ε moment bound suffices); and the complete cluster sandwich is consistent and positive definite on the retained stochastic coordinates. The invariant full-column-rank sensor/no-direct-effect model remains in force. The estimator does not plug in an estimated sensor. A variant that introduces an estimated sensor would require its influence function and all source–target and nuisance cross-covariances; that extension is not proved here. The result is not a selective-inference statement: the branch is fixed before the inference sample, or selected on an independent split. For conditioning alone, use the reduced source-only primitive system and let uC=vecoff(C)u_C=vecoff(C) contain exactly the coordinates selected by the fixed stochastic map. Known diagonal ones and declared structural zeros are deterministic. Let TCT_C insert uCu_C into vec(C)vec(C), let JCJ_C be the reconstruction Jacobian, and set V^C=J^CΩ^μJ^C⊤,ΔN(αC)=χrC,1−αC2Nλmax(TCV^CTC⊤),rC=dim(uC)>0. V_C= J_C _μ J_C , _N( _C)= χ^2_r_C,1- _CN _ (T_C V_CT_C ), r_C= (u_C)>0. Pointwise at every fixed data-generating process satisfying these assumptions, lim infN→∞Pr‖C^−C‖2≤ΔN(αC)≥1−αC. _N→∞ \\| C-C\|_2≤ _N( _C)\≥ 1- _C. With s^=σmin(C^) s= _ ( C), LN=s^−ΔNL_N= s- _N, and UN=s^+ΔNU_N= s+ _N, accept the ηF _F-conditioning claim iff LN≥ηFL_N≥ _F, reject it iff UN<ηFU_N< _F, and refuse otherwise. On the confidence event, acceptance gives σmin(C)≥ηF _ (C)≥ _F and ‖C−1‖2≤ηF−1\|C^-1\|_2≤ _F^-1; its false-accept probability and the false-reject probability of the stated threshold claim each have asymptotic limsup at most αC _C. These are unconditional error statements, not coverage conditional on acceptance. The solve bound uses a separate, single joint source–target system uS=(uC,vecG,vecZ)u_S=(u_C,vecG,vecZ) with V^S=J^SΩ^μJ^S⊤ V_S= J_S _μ J_S , dimension rSr_S, and one χrS,1−αS2χ^2_r_S,1- _S ellipsoid. Each vectorization retains only its declared stochastic coordinates. If SjS_j selects block j∈C,G,Zj∈\C,G,Z\ and TjT_j reconstructs its matrix, set δj=χrS,1−αS2Nλmax(TjSjV^SSj⊤Tj⊤). _j= χ^2_r_S,1- _SN _ (T_jS_j V_SS_j T_j ). Writing Q=EF−GZQ=E_F-GZ and Q^=EF−G^Z Q=E_F- G Z, retain the full bilinear remainder δQ=δG‖Z^‖2+(‖G^‖2+δG)δZ. _Q= _G\| Z\|_2+(\| G\|_2+ _G) _Z. The full solve uses the guard from this same joint event, LS=σmin(C^)−δC≥ηF.L_S= _ ( C)- _C≥ _F. This is distinct from the reduced αC _C diagnostic above. On the single joint Wald event, and only after this full-system guard accepts, ‖X^−X‖2≤δQ+δC‖X^‖2LS,X=C−1Q,X^=C^−1Q^.\| X-X\|_2≤ _Q+ _C\| X\|_2L_S, X=C^-1Q, X= C^-1 Q. Equivalently, writing the displayed right-hand side as bNb_N, the pointwise unconditional error statement is lim supN→∞Prfull accept and ‖X^−X‖2>bN≤αS; _N→∞ \full accept and \| X-X\|_2>b_N\≤ _S; the deterministic inequality is not asserted outside the joint Wald event. Separately calibrated marginal intervals may not be spliced into this solve claim. The result requires the source-to-target invariance interface, nonvanishing allocation, a prespecified branch, valid reconstruction denominators, compatible dimensions, and a finite positive-definite covariance on the retained coordinates; without them the radius is undefined. Operationally, missing source provenance or invariance, invalid allocation, post-selection on the inference sample, a failed reconstruction denominator, incompatible dimensions, nonfinite output, or an unusable retained-coordinate covariance yields neither a radius nor acceptance. If F=∅F= there is no free-block guard; if |F|=1|F|=1, CFF=[1]C_F=[1] exactly and conditioning is deterministic. If a target structural row block BF,⋅τB^τ_F,· is supplied externally and is exact in the declared target coordinate frame, then ΔC=ΔG=0 _C= _G=0; this conclusion is conditional on the external block and its frame. This result is pointwise, fixed-dimensional, fixed-environment, fixed-branch, and asymptotic—not finite-sample, uniform, growing-dimensional, selective, oracle, or bootstrap inference. Finally, the conditioning check is applied to CFFC_F, not PτP^τ_ D D: BFFτ=[0M00]B^τ_F=[ smallmatrix0&M\\ 0&0 smallmatrix] (remaining blocks zero) has ρ(Bτ)=0ρ(B^τ)=0 and Pτ=IP^τ_ D D=I yet σmin(CFF)→0 _ (C_F)→ 0 as M→∞M→∞. Proof. See Appendix C. □ Corollary 25 (Support-aware identification criterion). Let g(B)=det(I−BFF)g(B)= (I-B_F) on a fixed admissible affine support stratum. At an admissible zero where Dg≠0Dg≠ 0, the singular locus is locally codimension one. This statement is local and regime-specific: the zero set may be empty (in particular, if |F|=1|F|=1 then CFF=[1]C_F=[1]), and fixed support can force a different rank geometry. On a fixed corank-q affine branch, Lemma 22 gives local dimension qd−rGqd-r_G. The support-aware intervention minimum is therefore the smallest design for which the query is constant on the entire complete legal fibre; full identification requires every stratum and component to be a singleton modulo the declared equivalence. Numeric counts from the CFFC_F-nonsingular ambient branch, from an orthogonal orbit, or from a known-support example apply only after those regimes have been established. They are not fallbacks for an unresolved singular branch. Proof. See Appendix C. □ Interpretation. The intervention requirement depends on what is already aligned and on how the target is observed. Active probing with an unknown sensor map generally requires every changed row, whereas a shared sensor map can save one intervention when the free-block completion is unique. Known support can reduce the requirement further. Theorem 26 (A lower bound for complete discrepancy blocks). Assume the discrepancy set is a complete source block of size m≥2m≥ 2, CF=0C_ DF=0, and the normalized local fibre is A-locally complete for the SO(m)SO(m) action of Proposition 41. For each design below, also assume the proposition’s normalized equivariance and local-faithfulness condition, so equality of the anchor laws is equivalent, locally, to fixing the signed labelled anchor frame. For every design of m−2m-2 linearly independent signed labelled anchors, the local survivor is SO(2)SO(2) and therefore contains a nontrivial stable curve through the truth. Every query with nonzero derivative along that curve has a genuine local collision; an invariant query does not. Hence m−1m-1 anchors are necessary and sufficient to remove the connected rotational ambiguity. This is a theorem about that normalized complete-block fibre, not a universal intervention count. Global model or query identification still requires exclusion of transverse and disconnected branches. Separately, consider the ambient unknown-support Gaussian branch with distinct indices r,sr,s, zero-mean Gaussian sources having a common covariance, identical stochastic-intervention source laws in the two members, and every retained named stage disjoint from r,s\r,s\. Suppose 1−CrsCsr≠01-C_rsC_sr≠ 0 and the small split-frame perturbation in Appendix C remains admissible and stable. Then two members agree on all complete retained Gaussian laws, while the labelled held-out s-stage response row moves; the structural response query (Bt)sr(B_t)_sr also moves off the stated exceptional locus. Fixed known H generically excludes this collision. This is a scoped collision example, not a positive identification theorem and not a complete-law claim for arbitrary support classes. Proof. See Appendix C. □ Interpretation. The sufficiency count cannot be read as a universal property of ||| D| alone. Below the stated boundary, distinct stable target systems can agree on the available responses. The particular collision depends on the observation regime, support, and completion conditions. Remark 27 (Numerical evaluation of εO_ ). Near instability, ‖Σ‖\| \| can be large enough that an absolute tolerance on ‖Σ′−Σ‖\| - \| falls below the corresponding floating-point scale. Numerical comparisons must therefore use relative error. The angle should likewise be selected by tracing the connected component of the identity, rather than by a fixed global grid, because the admissible component can shrink as ρ(B)ρ(B) approaches one. Corollary 28 (Query-specific intervention savings). In the ambient active-probe branch with det(CFF)≠0 (C_F)≠ 0, probing all ||| D| discrepancy rows identifies the target. In the normalized complete source-block SO(m)SO(m) branch, m−1m-1 independent signed labelled anchors remove the connected rotation. Outside these proved regimes there is no universal numerical formula: known support, singular strata, alignments, and disconnected branches are summarized by the complete-fibre minimum KminIDK_ ID. For a single query the cost refines to the complete-fibre block cover of Section 10; it can be smaller when every remaining direction is query-irrelevant. A bare tangent rank gives only the first-order lower bound. □ Remark 29 (Design-relativity — collapse under richer target designs). The count is stated for the two regimes above. Two enrichments collapse it: (α) shared & known H with full active probing (M0τM_0^τ measured on all nodes): then M0τ(I−Bτ)=HM_0^τ(I-B^τ)=H is a full-rank linear system and BτB^τ is identified at Kminτ=0K^τ_ =0 without using invariance (which becomes cross-domain testable — a feature); (β) shared H with a per-component LiNG target: per-component non-Gaussian ICA pins MτM^τ only up to the LiNG signed-permutation equivalence class (Lacerda et al., 2008) — for cyclic models uniqueness is not generic and needs the additional LiNG stability/support conditions — and, given that resolution, BτB^τ follows at Kminτ=0K^τ_ =0. A rotation that is invisible to second moments can create fourth-order dependence under non-Gaussian sources; the algebraic recovery begins only after the mixing has been resolved. Every KminτK^τ_ claim therefore states its target observation model. 7 Why acyclic transport can fail under feedback Theorem 30 (Acyclification can give the wrong transport conclusion). There is a selection diagram and query for which the acyclic (intended-DAG) Pearl–Bareinboim reading concludes that the effect transports and computes effect 0, while equilibrium semantics give domain-dependent, nonzero answers. For the two-agent cycle qi=θi−bωqj+εiq_i= _i-b^ωq_j+ _i with b∈b∈ D, take bπ=0.5b^π=0.5 and bτ=0.3b^τ=0.3. The DAG reading θi→qi _i→ q_i d-separates S from the query and predicts a null/invariant effect, whereas the equilibrium do-effect is −b/(1−b2)-b/(1-b^2): source −2/3≈−0.667-2/3≈-0.667 versus target −30/91≈−0.330-30/91≈-0.330. Acyclifying a cyclic selection diagram is unsound. Proof: direct computation; the DAG omits the feedback walk carrying both the effect and the S-dependence (Lemma 9 with ∩An(Y)≠∅ D (Y)≠ ). □ This is why the transport theory cannot be a relabeling of the acyclic theory (Pearl and Bareinboim, 2014): the equilibrium solution map carries the domain dependence, and mechanism-level switch semantics (Definition 3) are the correct lift. Interpretation. Acyclic reduction can remove the feedback walk that transmits both an intervention and a domain change. A separation conclusion on that reduced graph therefore need not describe the equilibrium counterfactual. 8 Limits of validation from finite experiments Under the witness-admissibility conditions below, validation and transport share the same indistinguishability obstruction. We state that obstruction once. Theorem 31 (No unconditional cross-world validation from a finite design). Fix any finite design D, any query surgery I, and any +∈+S^+ ^+. Suppose +S^+ admits a node r that is witness-admissible for (,I)(D,I), i.e. (a) D never probes r’s mechanism, and I does not intervene on r (a surgery on r excises the reshuffled mechanism, so uru_r never enters); (b) r has a parent π that is not a descendant of r and whose value is moved by I with positive probability. Because π is a non-descendant it necessarily lies in a strictly earlier component of the condensation, so π is exogenous to UrU_r even when r itself sits inside a cycle; this is what keeps the reshuffled twin a well-posed SCM (the equilibrium system I−BI-B is untouched). If no such π exists — in particular if r is a root — then only parent-independent reshuffles are available at r, and those are measure-preserving involutions, i.e. SCM isomorphisms whose counterfactual gap is identically zero; the conclusion below then genuinely fails. (c) r is a post-surgery ancestor of the readout with non-vanishing coefficient: eY⊤(I−BI)−1er≠0.e_Y (I-B^I)^-1e_r≠ 0. Then there is an −∉+S^- ^+, obtained from +S^+ by a parent-dependent non-monotone measure-preserving reshuffle Tr(π,⋅)T_r(π,·) at r — a reflection of a band of UrU_r whose bounds are driven by π, taken in rank space, Tr=Fr−1∘ρ(π,⋅)∘FrT_r=F_r^-1\! ρ(π,·) F_r, so that it is exactly measure-preserving for every continuous strictly-increasing FrF_r — agreeing with +S^+ on every law of every regime of D, while |χ(+)−χ(−)|≥Δ>0on a positive-measure factual set,|χ(S^+)-χ(S^-)|\;≥\; >0 a positive-measure factual set, namely the symmetric difference of the factual- and counterfactual-parent bands. The bound map π↦br(π)π b_r(π) is ours to choose and need only separate the factual from the counterfactual parent law on a set of positive probability, and satisfy br(π)>ab_r(π)>a there (otherwise the band is empty and Δ≡0 ≡ 0); any strictly monotone brb_r bounded below by a does — the construction uses br(π)=a+softplus(π)>ab_r(π)=a+softplus(π)>a. (A bound depending on |π||π| alone does not suffice in general: a sign-flipping I with πcf=−πFπ^cf=-π^F moves π with probability one yet leaves the bands coincident, and the gap collapses to zero.) In the linear class with noise at r symmetric about the origin — so that the rank-space reflection is the annulus negation u↦−u -u on a≤|u|≤br(π)a≤|u|≤ b_r(π) — the gap has the closed form Δ=2|eY⊤(I−BI)−1er||ur| =2\, |e_Y (I-B^I)^-1e_r |\,|u_r|. Without symmetry the reflection is still exactly measure-preserving in rank space and Δ>0 >0 persists, but it is no longer 2|⋅||ur|2|·||u_r|. Hence any test accepting the correct twin under +S^+ with probability ≥1−α≥ 1-α accepts the same (now wrong-by-Δ ) twin under −S^-: cross-world claims are necessarily conditional on class membership. Interventional-layer claims remain unconditionally testable (the Forré and Mooij, 2019 baseline). Proof. See Appendix A. □ Interpretation. A finite experiment collection cannot by itself rule out every change in the cross-world coupling. The theorem constructs a system outside the monotone class that matches every observed experimental law but changes the counterfactual. Validation is therefore conditional on structural class membership. Conditions (a)–(c) identify the query regimes in which the impossibility construction applies; they are not vacuous restrictions but the exact statement of where the impossibility bites. A parentless (root) node, for instance, admits only parent-independent reshuffles — which are SCM isomorphisms with gap 0 (below) — so a root is never witness-admissible. Why parent-dependence is essential. A parent-independent annulus reflection T(u)=−uT(u)=-u on |u|∈[a,b]|u|∈[a,b] is a measure-preserving involution: (,id)(S,id) and (′,T)(S ,T) are then the same structural model up to a noise-gauge relabelling (an SCM isomorphism), the cross-world reshuffle cancels, and the intrinsic counterfactual gap is identically 0. It is the dependence of the reshuffle on a parent that moves between the factual and counterfactual worlds which breaks cross-world rank invariance — and this is exactly why (T1) is required. The node must be non-descendant-parented: inside a feedback cycle the parent seen by TrT_r is the equilibrium value, itself a function of UrU_r, and the twin exits the well-posed class (existence/uniqueness of the equilibrium fail on a positive-measure set). Corollary 32 (Cross-domain validation is conditional). Apply Theorem 31 to the switch-augmented model of Definition 3, with D = any finite source design together with any finite target design, the reshuffle placed on the rS_r-indexed mechanism of r, and r witness-admissible at target parameters — conditions (a)–(c) of Theorem 31 read with Bτ,IB^τ,I and the target factual law, so in particular eY⊤(I−Bτ,I)−1er≠0e_Y (I-B^τ,I)^-1e_r≠ 0 and the target law is non-degenerate on r’s bound-driving parent. Then all source laws and all target-design laws coincide across the pair, while the value reported by a procedure would transport is wrong under −S^- by a gap Δ>0 >0 on positive measure; in the symmetric-noise specialization this closes to Δ=2|eY⊤(I−Bτ,I)−1er||ur| =2|e_Y (I-B^τ,I)^-1e_r|\,|u_r|, whereas under asymmetric noise Δ>0 >0 still holds but with a different value. Hence any transport procedure sound without target-class assumptions is trivial: cross-world transport conclusions are necessarily conditional on target +T^+-membership, while interventional transport claims remain testable. □ Remark 33 (Why the target-side clause is needed). Witness-admissibility must be checked at target parameters, and this is a genuinely new requirement rather than a restatement of the within-domain one: the discrepancy rows D are exactly the parameters that need not be generic relative to the source. If a target discrepancy sets B⋅rτB^τ_· r so that eY⊤(I−Bτ,I)−1er=0e_Y (I-B^τ,I)^-1e_r=0, the transported gap is 0 even though the source-side witness is perfectly admissible — the impossibility simply does not bite for that query in that target. The witness may be placed at an unprobed discrepancy node whenever the declared design leaves one available. Lemma 4 uses monotonicity as the sufficient full-factual boundary in this paper, while Theorem 31 shows it is minimax-necessary under that theorem’s witness-admissibility conditions, within a domain and across domains alike. This is not a necessity claim for every individual query or system. For conditional instruments the exact null boundary is the tensor-product identity for the declared finite block partition; for partial-factual queries it is a singleton query-pushforward class over the complete legal fibre. Full conditional-law/map identification is sufficient but not necessary. Loop gain is only a local diagnostic. Together, the positive and negative results locate the boundary between identification and non-identification. 9 Partial identification and alignment When point identification fails, the appropriate object is the complete set of query values compatible with the retained laws and structural restrictions. We first describe the ambiguity created by rotations, completions, and cross-world couplings, and then examine how alignment information can reduce it. Theorem 34 (Identified sets under structural ambiguity). For an un-instrumented full-rotational source block, let ℱεF_ be the complete rotation–completion fibre consistent with the design, declared support/selection stratum, denominator guards, and ρ(B)≤1−ερ(B)≤ 1- . Under the second-moment observation model the identified set is exactly χ(ℱε)χ(F_ ); its rotation and transverse-completion projections are the floors WgaugerotW_gauge^rot and WgaugecompW_gauge^comp. In the finite-dimensional LQ branch, if the complete fibre is encoded by finitely many semialgebraic support/selection strata and χ is a scalar semialgebraic function on the denominator-valid locus, then χ(ℱε)χ(F_ ) is a finite union of points and intervals whose endpoints may be open or closed, finite or infinite. Every included value is attained; a finite extremum is attained exactly when the corresponding endpoint belongs to the image. A closed, bounded, endpoint-attained identified set is guaranteed when the full fibre is compact and χ extends continuously across all stratum closures with every query denominator bounded away from zero. These are sufficient conditions for that conclusion, not a necessity claim. A spectral margin alone does not guarantee finiteness or attainment: the population pinned-response algebra admits a noncompact completion-escape direction, but that illustration is not a complete-law noncompactness proof unless its members are shown to remain in the complete fibre. On the exact two-node normalized-rotation branch, rationalized by t=tanθt= θ and restricted only by strict stability ρ(B)<1ρ(B)<1 (not by a fixed positive margin ε ), the connected component of the identity has scalar query image equal to the open interval (12−517068,12+517068). ( 12- 5 17068, 12+ 5 17068 ). The endpoints are excluded because stability is strict. Per-component non-Gaussian source assumptions plus a valid acquisition/alignment result reduce the rotation ambiguity to signed permutations; they do not collapse transverse completions. Proof. See Appendix D. □ Interpretation. The sharp identified set is the image of the complete legal fibre, not merely the image of a local tangent or a single group orbit. Compactness and continuity guarantee endpoint attainment; a spectral stability margin alone does not. Proposition 35 (Why the selection condition matters). On the fixed-support rational rotation branch above, the stable query image is exactly the displayed open interval. Removing the strict stability restriction allows the readout to diverge as the diagonal normalization approaches a regular zero. Thus T3 is load-bearing for this branch. For a complete rotation–completion fibre, stability alone does not imply compactness, coercivity, or a nonzero query denominator. The separate completion-escape construction concerns the pinned-response algebra only and is not used as a complete-law counterexample. Proof. See Appendix D. □ Theorem 36 (Decomposition of cross-world ambiguity). Without T1, the scalar partial-ID width is the query diameter over the joint ambiguity orbit, W=diamχ()W=diamχ(A); in the additive linear scalar-noise subclass its two cross-world parts are (no global nonlinear direct-sum claim): (a) for a node i satisfying the witness-admissibility conditions of Theorem 31 for the two worlds under comparison — in particular, a non-descendant parent moves across those worlds on a set of positive probability and the legal ambiguity class contains the required parent-indexed law-preserving rearrangements — the per-node reshuffle part WcwmonoW_cw^mono has sharp extended essential supremum |eY⊤(I−BI)−1ei|esssupUi−essinfUi.|e_Y (I-B^I)^-1e_i|\ *ess\,supU_i- *ess\,infU_i\. For an atomless noise law, measure-preserving parent-dependent rearrangements approach this value; a pointwise maximum need not exist. Positive-mass endpoint attainment requires compatible endpoint atoms. Randomization may split compatible atomic mass, but it cannot create endpoint atoms while preserving atomless marginals. A parent-independent involution is an SCM isomorphism contributing 0. Without witness-admissibility, the essential-range display is only an upper bound for an enlarged coupling class, not the sharp width of the actual legal fibre; the actual contribution can be smaller and is 0 when only parent-independent relabellings are admissible. Under witness-admissibility this ambiguity is invisible to every design law, removed only by assuming T1 (for nonlinear monotone mechanisms the same role is played by a Lipschitz-modulus / transport-diameter bound); (b) the within-block rotation part WcwrotW_cw^rot is the cross-world reading of the same rotation floor WgaugerotW_gauge^rot that Theorem 34 measures at the identification layer — the identical floor, not an independent one: on an un-instrumented block Wcwrot=WgaugerotW_cw^rot=W_gauge^rot, and once the design identifies the rotation (Wgaugerot=0W_gauge^rot=0 on χ’s block) Wcwrot=0W_cw^rot=0 too. The primary object is the joint-orbit diameter W=diamχ()W=diamχ(A). Proof. See Appendix D. □ Corollary 37 (A uniform upper bound under recombination). Under recombination closure — A the product hull comp×mono×rotA^comp\!×\!A^mono\!×\!A^rot, with hull ∩+∩\,T^+ admissible and any two points joined by single-block moves each block moved at most once, and under a shared Sel (else the selection-branch gap below is added) — the base-uniform upper bound W≤W¯comp+W¯mono+W¯rotW\;≤\; W^comp+ W^mono+ W^rot holds, each W¯∙ W being χ’s diameter over the ∙ -orbit uniformly over the other floors’ admissible orbits (corner-path telescoping; every floor worst-cased). Without closure only the joint diameter is guaranteed: for a non-product A the decomposition fails (the diagonal =(0,0),(1,1)A=\(0,0),(1,1)\, χ=c+rχ=c+r — every one-block slice a singleton, both floors 0, yet W=2W=2), and revisiting a block gives only W≤∑∙k∙W¯∙W≤ _ k_ W (multiplicities k∙k_ ; staircase W¯=1+1 W=1+1 but W=4W=4). Proof. See Appendix D. □ Corollary 38 (Exact additivity in the decoupled subclass). The charged-once additive form W=Wgaugecomp+Wcwmono+WcwrotW=W_gauge^comp+W_cw^mono+W_cw^rot (rotation entered once as the identification-layer WgaugerotW_gauge^rot, mono at the truth) holds in the decoupled regime — floors on transverse coordinate blocks with a query separable across them, in particular the off-block mono node below (orbit-invariant coefficient), or when T4 pins the rotation/frame (Wgaugecomp=Wcwrot=0W_gauge^comp=W_cw^rot=0). Truth-anchored additivity fails when the floors co-activate: a Q-dependent (especially sign-changing) readout coefficient makes the joint width exceed the truth-anchored sum — the bilinear χ=q⋅uχ=q\!·\!u is superadditive — which is why the general bound worst-cases every floor over the others. The rotation orbit is charged once within the rotation group (closure under composition); the completion floor WgaugecompW_gauge^comp acts on the support/frame directions (the rotation and GL(d)GL(d)-frame gauge constructions are transverse; completion-vs-rotation transversality in general d is an additional stated condition, and the completion×mono cross-term is bounded conservatively), and the mono reshuffle acts on an off-block node i∉Si∉ S whose reshuffle-driving parent lies in a strictly earlier SCC, so walks i→Yi\!→\!Y never enter the source block S and its coefficient |eY⊤(I−BI)−1ei||e_Y (I-B^I)^-1e_i| is orbit-invariant. It is collapsed to signed permutations by per-component LiNG (only). A fourth floor, the selection-branch gap (a multiplicity of admissible selections), is charged separately and vanishes under a shared Sel (Corollary 39 and Appendix D track all four). Proof. See Appendix D. □ Interpretation. The width below the monotone boundary combines ambiguity in the within-world model with ambiguity in how factual and counterfactual noises are coupled. The additive display is exact only in the stated decoupled subclass. Corollary 39 (Moment agreement can miss gauge ambiguity). Any validator that is a functional of second-moment laws assigns the truth’s pass/fail to every Q∈εQ _ . Consequently, whenever the validator passes the truth, it also passes every legal representative on which χ differs, thereby false-validating those representatives. Restriction to the in-class orbit is essential: an unstable representative is not an ECG and therefore does not define a false validation. Proof. See Appendix D. □ Lemma 40 (Gauge transformation of transported queries). A transported χ is well-defined exactly when it is constant on the complete representative fibre of all legal source representatives, target representatives, alignments, and discrete branches matching the retained labelled laws. On a component that has separately been proved to be a single group orbit, this reduces to invariance under that group action; without such a transitivity proof, an orbit image need not be the complete ambiguity set. □ Proposition 41 (Query identification under partial alignment). Fix the complete retained baseline evidence y0y_0. Let 0(y0) A_0(y_0) be the set of all legal source representatives, target representatives, alignments, nuisance completions, and discrete support, normalization, sign, permutation, and selection branches matching all retained labelled laws. For an intervention design D, let aDa_D return the complete labelled interventional laws, including the operator, target, signed dose, units, and response law, and put D=α∈0(y0):aD(α)=u0,ℐD(χ)=χ(α):α∈D. A_D=\α∈ A_0(y_0):a_D(α)=u_0\, _D(χ)=\χ(α):α∈ A_D\. Then ℐD(χ)I_D(χ) is the sharp population identified set and χ is point identified iff its extended diameter is zero. Uniform identification requires this constancy on every admissible fibre. A positive-width claim therefore requires two surviving legal members that move χ; positive stabilizer dimension alone is insufficient. Sharp endpoints are guaranteed to be attained when the representative fibre is compact and χ is continuous; other mechanisms may also yield attainment. One may quotient by a declared reparameterization only after proving that both aDa_D and χ descend to that quotient; the representative-fibre formulation needs no such compatibility assumption. The following anchor count is deliberately confined to a normalized special case. Let S=S= D be a complete source block of size m, with CF=0C_ DF=0, and suppose the normalized local residual action is SO(m)SO(m). Assume 0(y0) A_0(y_0) is A-locally complete: every local legal fibre direction is generated by that action. Let k anchors be signed, labelled, linearly independent directions in shared units, collected as A=[v1,…,vk]A=[v_1,…,v_k], and suppose their response law is equivariant and locally faithful in the sense that, for Q near the identity, aD(TQα)=aD(α)⟺Q⊤A(α)=A(α).a_D(T_Qα)=a_D(α) Q A(α)=A(α). Generically, as witnessed by a nonzero minor of the anchor Jacobian, the local survivor stabilizer is SO(m−k),dimSO(m−k)=(m−k2),SO(m-k), SO(m-k)= m-k2, and the removed rank is (m2)−(m−k2)=k(2m−k−1)/2 m2- m-k2=k(2m-k-1)/2. Thus, for m≥2m≥ 2, m−1m-1 such anchors remove the connected SO(m)SO(m) ambiguity. This identifies the full local fibre only under A-local completeness. Global identification additionally requires the complete fibre to contain no transverse completion, disconnected support, reflection, permutation, normalization, or selection branch on which χ moves. Unsigned, zero-dose, duplicate, or unlabelled anchors do not satisfy this count. Proof. The identified-set statement is the image of the complete survivor level set. In the normalized special case, local faithfulness makes equality of anchor laws equivalent to fixing the k-frame. Its orientation-preserving stabilizer is SO(m−k)SO(m-k); subtraction of the two Lie dimensions gives the displayed rank. The nonzero-minor premise makes this the generic local rank. The remaining qualifications follow because the group action describes the complete local fibre only under A-local completeness and does not enumerate other components. □ Interpretation. Alignment is query-specific. Interventions identify the query exactly when every legal alignment and nuisance completion that matches the labelled laws gives the same query value. Familiar anchor counts apply only after the residual ambiguity has been shown to have the corresponding group action. 10 Query-specific experimental design Both halves share a design principle: the relevant cost is a minimum over environment blocks, and is set by the query and the complete retained-law fiber rather than by a nominal scalar rank. Theorem 42 (Query-specific experiment design). Fix an admissible query χ, differentiable at the truth θ0 _0, and a finite unit-cost environment library. For D in that library, let =θ′:ℒ(θ′)=ℒ(θ0)K_D=\θ :L_D(θ )=L_D( _0)\ be the global complete legal fiber: all models in the declared class that match every retained observational and staged law, after profiling all nuisance parameters, including disconnected and discrete branches. Here ℒL_D denotes those complete indexed laws, not the finite moment stack used by the tests. For a scalar query ∥⋅∥\|·\|_ Q is absolute value; for a vector query it is a declared norm. Put Δχ()=supθ′,θ′∈‖χ(θ′)−χ(θ′)‖,Vminqry(χ)=min||:Δχ()=0, _χ(D)= _θ ,θ _D\|χ(θ )-χ(θ )\|_ Q, V_ qry(χ)= \|D|: _χ(D)=0\, If the displayed set is empty, its minimum is defined as +∞+∞. Thus Vminqry(χ)V_ qry(χ) is the minimum query-identifying intervention-block cost, and population query identification is exactly complete-fibre constancy. If T_K_D denotes the directions realised by C1C^1 feasible curves through θ0 _0, define VminFOV_ FO by replacing fiber constancy with Dχ(θ0)T=0Dχ( _0)T_K_D=0. For the same environment library, baseline tests, class, and cost convention, VminFO(χ)≤Vminqry(χ)≤KminID,V_ FO(χ)≤ V_ qry(χ)≤ K_ ID, where KminIDK_ ID is the minimum block count giving a singleton complete legal fiber. The first inequality follows by differentiating constancy along feasible curves; the second because full identification identifies every query. Neither inequality asserts universal strictness. Proof. See Appendix E. □ Corollary 43 (Affine row-space criterion). For an affine query χ(θ)=c+Gθχ(θ)=c+Gθ, the boundary-safe exact criterion is Gspan(−)=0G\,span(K_D-K_D)=0. A computable row-space equivalence requires a stronger, independently verified premise. Suppose all baseline laws, nuisances, support restrictions, and affine class equalities have been profiled and the complete baseline fiber is exactly (θ0+imQ)∩Θ( _0+imQ)∩ , where Q has full column rank and θ0 _0 is in the relative interior of the remaining open inequalities. Suppose further that environment j adds exactly the fixed linear block AjQA_jQ, so for D M=stackj∈(AjQ),=(θ0+QkerM)∩Θ.M_D=stack_j (A_jQ), _D=( _0+Q M_D)∩ . Then Δχ()=0⇔kerM⊆ker(GQ)⇔row(GQ)⊆row(M), _χ(D)=0 M_D (GQ) (GQ) (M_D), and VminFO=VminqryV_ FO=V_ qry is the minimum number of environment blocks satisfying the displayed row-space condition, not the scalar rank of one matrix. More generally a vector-valued full-B query with selector GBG_B requires GBQkerM=0G_BQ M_D=\0\. This reduces to QkerM=0Q M_D=\0\ only when the profiled parameter is exactly vecBvecB; one random scalar gradient cannot identify full B. In the population pinned-response algebra, P0=(I−B0)−1P_0=(I-B_0)^-1 and a target j fixes zj=ej⊤P0z_j=e_j P_0. For B=B0+EB=B_0+E, zj(I−B)=ej⊤⇔zjE=0,z_j(I-B)=e_j z_jE=0, so Aj(E)=zjEA_j(E)=z_jE is an exact affine block. This corollary applies only after the complete profiled-fiber premise above has been established: pinned-response equations alone do not characterize the complete observational, interventional, or distributional ECG law. The illustrative calculation below therefore uses a declared pinned-response-only affine observation algebra, not a universal complete-law claim. In its zero-diagonal, unknown-support three-node chain, B01B_01 needs two unit-cost blocks while full B needs three, and exact stable twins exist below both minima; a dense four-node control gives equality 3=33=3. Hence query identification can be strictly cheaper, but a local query need not be cheaper. Changing the support changes the count. Proof. See Appendix E. □ Remark 44 (Limits of first-order design criteria). The row-space shortcut does not apply at an active boundary, with estimated zjz_j, for a nonlinear query, with an unprofiled law/class/support restriction, or a near-rank block. Off the affine case, first order is only necessary: χ(θ)=θ2χ(θ)=θ^2 at 0 and χ(x,y)=y+x2χ(x,y)=y+x^2 on y=0\y=0\ satisfy the pointwise tangent condition but are nonconstant. Complete-fiber constancy remains the criterion; local tangent or finite-jet searches are refutation-only across singular or incompletely enumerated branches. A finite-tolerance moment suite accesses a localized set τmc=‖mc(θ)−mc(θ0)‖⊕≤τK_τ mc=\\| M_D mc(θ)- M_D mc( _0)\|_ ≤τ\, not the complete-law fibre; it therefore needs the separate selected-summary modulus (and moment-fibre sufficiency) of Proposition 6. Interpretation. An experiment need not identify the whole model to identify a particular query. It is sufficient that the query be constant on the complete legal fibre remaining after the experiment. A first-order rank condition gives only a local lower bound unless the stated affine assumptions hold. Remark 45 (The transport instance of the same principle). Theorem 26 and Corollary 28 are the cross-domain reading of Theorem 42: where validation replaces full identification by the query-relative block cost Vminqry(χ)V_ qry(χ), transport replaces it by the discrepancy-relative cost. The active-probe CFFC_F-nonsingular ambient branch of Theorem 23 is identified by probing all ||| D| rows; the normalized complete-block SO(m)SO(m) branch has the separate anchor count of Proposition 41. Support-specific and singular classes use their own complete-fibre KminIDK_ ID and need not obey either count. The query-relative refinement Kminτ(χ)K^τ_ (χ) composes the two. In every stated regime the query-identifying design costs no more than full identification and can be cheaper, but need not be; the collapse regimes (Remark 29) show the count is design-relative rather than intrinsic. 11 Statistical procedures The preceding results are population statements. This section gives procedures for comparing the reconstructed mean and covariance responses of a fixed candidate twin with those of the reference system, under explicit sampling, rank, and tail assumptions. Theorem 46 (Statistical checks and their limits). (i) The staged mean back-test is run on the reconstructed latent free-block mean m^F(e)=SFH^+m^X(e)∈ℝ|Fe| m_F^(e)=S_F H^+ m_X^(e) ^|F_e|: SFS_F selects the environment-e free block FeF_e, so the dimension is |Fe|=d−|e||F_e|=d-| D_e| (not d — e.g. d=2d=2 with node 0 hard-clamped gives Fe=1F_e=\1\, a one-dimensional statistic). It is the mean analogue of the covariance test’s Z=SFH+XZ=S_FH^+X; the observed mean map M(e)=HPM_T^(e)=HP is p-dimensional but the free block carries only |Fe||F_e| random directions, so the observed-mean covariance has rank≤|Fe|<prank≤|F_e|<p and the raw p-dimensional inverse does not exist. Write m,F(e):=E[SFH+X(e)]m_T,F^(e):=E_T[S_FH^+X^(e)] for the candidate’s full environment mean, including the structural intercept and the declared hard/soft intervention semantics; it reduces to a response-map times dose only in a baseline-centred zero-intercept specialization. Two mean routes are distinct. Fixed-dimensional ADF Wald: Tem=n(m^F(e)−m,F(e))⊤Σ^m,F+(m^F(e)−m,F(e))→dχrm,e2T_e^m=n\,( m_F^(e)-m_T,F^(e)) _m,F^+( m_F^(e)-m_T,F^(e)) _dχ^2_r_m,e under H0H_0, with rm,e=rank(Σm,F(e))r_m,e=rank( _m,F^(e)) (generically |Fe||F_e|) and Σ^m,F _m,F the ADF estimate of the per-observation covariance. For externally known H this is the fixed-frame covariance. For an independent sample-split H H, add JHΣH^JH⊤J_H _ HJ_H , where JH=∂mF/∂vec(H)J_H=∂ m_F/ (H). For a same-sample influence-function estimate, use the full joint sandwich Σm+JHΣHJH⊤+ΣmHJH⊤+JHΣHm _m+J_H _HJ_H + _mHJ_H +J_H _Hm; omitting the cross terms is invalid. This route is pointwise asymptotic; its finite-n law is not replaced by χ2χ^2. Proof. See Appendix E.1. □ Proposition 47 (Concentration for reconstructed means). Suppose H is externally known and the probe observations are independent. Assume that the reconstructed free response ZF(e)Z_F^(e) obeys ‖⟨ZF(e)−EZF(e),u⟩‖ψ2≤Km,e(u⊤Σm,F(e)u)1/2,\| Z_F^(e)-EZ_F^(e),u \|_ _2≤ K_m,e(u _m,F^(e)u)^1/2, together with a declared envelope λ¯m,e≥λmax(Σm,F(e)) λ_m,e≥ _ ( _m,F^(e)). For a universal CmC_m, put te,nm(η):=CmKm,eλ¯m,e1/2|Fe|+log(2/η)ne,qe,nm:=te,nm(αe)+te,nm(βe).t_e,n^m(η):=C_mK_m,e λ_m,e^1/2 |F_e|+ (2/η)n_e, q_e,n^m:=t_e,n^m( _e)+t_e,n^m( _e). The test accepting when ‖m^F(e)−m,F(e)‖2≤te,nm(αe)\| m_F^(e)-m_T,F^(e)\|_2≤ t_e,n^m( _e) has level at most αe _e and power at least 1−βe1- _e whenever the population mean gap exceeds qe,nmq_e,n^m. For a sample-split estimated H, suppose the reconstructed mean residual has error at most he,n(η)h_e,n(η) except on an event of probability η. Define t~e,nm(η):=te,nm(η/2)+he,n(η/2) t_e,n^m(η):=t_e,n^m(η/2)+h_e,n(η/2). The estimated-H test accepts at t~e,nm(αe) t_e,n^m( _e) and uses q~e,nm=t~e,nm(αe)+t~e,nm(βe) q_e,n^m= t_e,n^m( _e)+ t_e,n^m( _e); the union allocation preserves level αe _e and Type-I error βe _e. Without such a bound this finite-n route is unavailable. No scale-free bound is possible without the envelope, and no (1−ρ(B))−1(1-ρ(B))^-1 substitution is made. Proof. See Appendix E.1. □ Proposition 48 (Covariance Wald procedure). This procedure is stated only in a fixed latent frame in which H is externally known exactly. A shared but unidentified H does not suffice, because its common residual gauge is generally GL(m)GL(m) rather than O(m)O(m). An estimated-H extension would require the complete influence function, including all nuisance and same-sample cross terms; that extension is not proved here. Let Z=SFH+XZ=S_FH^+X, let se=svecfree(Σ^(e)−Σ(e))s_e=svec_free( ^(e)- _T^(e)), and let We=Covsvecfree(Z~Z~⊤),ke=|Fe|(|Fe|+1)/2.W_e=Cov\svec_free( Z Z )\, k_e=|F_e|(|F_e|+1)/2. With independent probes, finite (4+δ)(4+δ) moments, an invertible free subsystem, and a consistent ADF estimate W^e W_e, if We≻0W_e 0 then TeΣ=nse⊤W^e−1se⇒χke2.T_e =ns_e W_e^-1s_e\ \ χ^2_k_e. This gives pointwise asymptotic level α. Against every fixed nonzero covariance alternative its power tends to one, and hence is eventually at least 1−β1-β; no finite uniform sample size is asserted. If the population rank re<ker_e<k_e is externally declared and separated by a positive eigenvalue gap, the range Wald and the hard-kernel companion of Proposition 50 are separate pieces: the range statistic has limit χre2χ^2_r_e, while the companion detects fixed kernel moves. A naive pseudoinverse over sample-created small eigenvalues is not this procedure. Raw vechvech and svecsvec are equivalent invertible coordinates in the full-rank case; svecsvec fixes the direct-sum norm used above. Proof. See Appendix E.1. □ Proposition 49 (Nonasymptotic covariance concentration). For an actual covariance block ke≥1k_e≥ 1, let W:=Z−EZW:=Z-EZ and suppose the free-block responses are iid and obey the centered whitened-shape condition ‖⟨W,x⟩‖ψ2≤KZ(x⊤ΣFF(e)x)1/2\| W,x \|_ _2≤ K_Z(x _F^(e)x)^1/2 for every x, with KZ≥1K_Z≥ 1 (translation- and scale-invariant — part of the stated assumptions), together with a declared covariance envelope λ¯e≥λmax(ΣFF(e)) λ_e≥ _ ( _F^(e)) (also an explicit assumption, since λmax(ΣFF(e)) _ ( _F^(e)) is an unknown population quantity — alternatively a sample-split upper confidence bound with error allocation, or the self-normalized empirical Bernstein form; the test considered here is the ADF Wald above, studentized by the empirical W^e W_e and hence oracle-free). For this analytic route use the divisor-n sample-mean-centred covariance Σ^n=n−1∑ℓ=1n(Zℓ−Z¯)(Zℓ−Z¯)⊤ _n=n^-1 _ =1^n(Z_ - Z)(Z_ - Z) . The identity Σ^n−Σ=n−1∑ℓ(WℓWℓ⊤−Σ)−W¯W¯⊤ _n- =n^-1 _ (W_ W_ - )- W W shows that the first term is entrywise sub-exponential and the empirical-centering term is absorbed by the same u/nu/n branch. For uη=log(4ke/η)u_η= (4k_e/η), the entrywise-Bernstein sup-norm test has threshold τn(η)=C1κ¯Σ(uη/n+uη/n) _n(η)=C_1 κ_ ( u_η/n+u_η/n), κ¯Σ:=CBKZ2λ¯e κ_ :=C_B\,K_Z^2 λ_e the operational (envelope) constant, and level ≤α≤α for all n. Against a fixed, validation-data-independent candidate it has power ≥1−β≥ 1-β whenever ‖svecfree(Σ−Σ)‖2>qe,nΣ:=2ke[τn(α)+τn(β)]\|svec_free( _S- _T)\|_2>q_e,n := 2k_e[ _n(α)+ _n(β)]; in particular it suffices that n≳κ¯Σ2(2ke)γ−2log4keα∧β∨κ¯Σ2keγ−1log4keα∧βn κ_ ^2\,(2k_e)\,γ^-2 4k_eα β\ \ κ_ \, 2k_e\,γ^-1 4k_eα β (the same κ¯Σ κ_ as in τn _n; it equals the oracle κΣ _ below when the envelope is tight, λ¯e≈λmax(ΣFF(e)) λ_e\!≈\! _ ( _F^(e)), and otherwise inflates the guaranteed sample size by a factor in [λ¯e/λmax,(λ¯e/λmax)2][ λ_e/ _ ,( λ_e/ _ )^2] — the max of the quadratic variance-branch term and the linear far-branch term, following whichever dominates and possibly switching branch) — two Bernstein branches, the sub-Gaussian variance branch carrying 2ke2k_e and the sub-exponential far branch 2ke 2k_e (ke≍d2/2k_e\! \!d^2/2) — with the oracle κΣ=CBKZ2λmax(ΣFF(e)) _ =C_B\,K_Z^2 _ ( _F^(e)); when ΣFF(e)≻0 _F^(e) 0, this also equals CBKZ2λmin(Pfree(e))−1C_B\,K_Z^2 _ (P_free^(e))^-1 for Pfree(e):=(ΣFF(e))−1P_free^(e):=( _F^(e))^-1, the actual marginal free-block precision (the centred-product norm ‖WiWj−Σij‖ψ1≤CBKZ2ΣiiΣjj≤CBKZ2λmax(ΣFF(e))\|W_iW_j- _ij\|_ _1≤ C_BK_Z^2 _i _j≤ C_BK_Z^2 _ ( _F^(e)); the structural (I−BFF(e))⊤ΩFF−1(I−BFF(e))(I-B_F^(e)) _F^-1(I-B_F^(e)) equals it only under a hard clamp — a soft probe leaves the complementary block random and inflates ΣFF _F), hence, on that positive-definite branch, κΣ2≍KZ4λmin(Pfree(e))−2 _ ^2 K_Z^4 _ (P_free^(e))^-2 (universal constant CB2C_B^2). The λmax _ form remains valid for singular Σ . KZ4K_Z^4 is a valid Bernstein upper-bound scale (not exact/minimax-necessary), not (2+κ¯4)(2+ κ_4): the excess-kurtosis form is false — for the continuous near-Rademacher U=(R+εG)/1+ε2U=(R+ G)/ 1+ ^2, 2+κ¯4→02+ κ_4→ 0 as ε→0 → 0 while Var(UiUj)=1Var(U_iU_j)=1 stays fixed, so off-diagonal detection still needs n=Ω(γ−2)n= (γ^-2). Here KZK_Z is a declared assumption of the result; a diagnostic ψ2 _2 pretest can reject gross violations but cannot establish an upper bound on KZK_Z. Proof. See Appendix E.1. □ Proposition 50 (Rank-degenerate covariance inference). In the fixed identified frame of Proposition 48, this branch applies exactly when the environment’s WeW_e is degenerate, with rank/kernel read off WeW_e — a population selection rule, taken here as an externally supplied fixed rank/kernel declaration (part of the assumptions), not a data-driven finite-sample rank estimate, since exact rank is not selectable uniformly near degeneracy. This branch lies outside the standing continuous-noise class; it records the limiting structural failure rather than enlarging the class used by the identification theorems. This branch inherits the full standing hypothesis set of the fixed-dimensional case — mutually independent free-block noises, each of positive variance σj2>0 _j^2>0 and finite (4+δ)(4+δ) moments, zero mean Uj=0EU_j=0, and an invertible free subsystem A=(I−BFF)−1A=(I-B_F)^-1 (i.e. det(I−BFF)≠0 (I-B_F)≠ 0) — not merely the zero-mean convention: degeneracy of WeW_e is a fourth-moment statement (on Var(Uj2)Var(U_j^2)) layered on top of these standing assumptions, never a consequence of centering alone. Since WeW_e is built on mean-centered variables, the exact degeneracy condition on coordinate j is Var((Uj−Uj)2)=0Var((U_j-EU_j)^2)=0, which under Uj=0EU_j=0 reads Var(Uj2)=0Var(U_j^2)=0. Only under a hard clamp (where Z=AFFUFZ=A_FU_F, so such a UjU_j degenerates WeW_e along QjQ_j) — or when F contains every still-random coordinate — the kernel is the r-dimensional span of the congruence pullbacks Qj=A−⊤EjjA−1Q_j=A^- E_jA^-1 of the degenerating diagonal directions, A=(I−BFF)−1A=(I-B_F)^-1 (in the fixed Frobenius-isometric coordinates the dual vectors are qj=svec(Qj)q_j=svec(Q_j), and both projectors use the same svecsvec convention), with r=#j:Var((Uj−Uj)2)=0r=\#\j:Var((U_j-EU_j)^2)=0\ counting the exogenous free-block noises of centered constant magnitude (not the reconstructed coordinates ZiZ_i: mixing gives Var(Zi2)>0Var(Z_i^2)>0 from the UjUkU_jU_k cross-term even when the underlying noise has centered constant magnitude — e.g. Z1=U1+aU2Z_1=U_1+aU_2, Z2=U2Z_2=U_2 has Var(Z12)=4a2>0Var(Z_1^2)=4a^2>0 yet rankWe=1rank\,W_e=1, r=2=mer=2=m_e), so r=mer=m_e iff every exogenous free-block noise is symmetric two-point/Rademacher up to scale. Each coordinate is independent: a coordinate j with Var((Uj−Uj)2)>0Var((U_j-EU_j)^2)>0 is non-degenerate and drops only its own QjQ_j from the kernel, so in general kerWe=spanQj:Var((Uj−Uj)2)=0 W_e=span\Q_j:Var((U_j-EU_j)^2)=0\ and We≻0W_e 0 (kernel dim 0) iff Var((Uj−Uj)2)>0Var((U_j-EU_j)^2)>0 for every j; one failing coordinate leaves the other QjQ_j intact (e.g. a 22-D clamp with U1∈1,−2U_1∈\1,-2\, U2U_2 Rademacher gives We=diag(2,4,0)W_e=diag(2,4,0) in svecsvec coordinates, kernel dim 11 not 0). Two ways a single coordinate fails: an asymmetric zero-mean two-point U∈1,−2U∈\1,-2\ has Var(U2)=2>0Var(U^2)=2>0; and a non-zero-mean constant-magnitude U (ℙ(U=1)=34P(U=1)= 34, ℙ(U=−1)=14P(U=-1)= 14: |U|≡1|U|≡1 but U=12EU= 12) has Var((U−U)2)=34>0Var((U-EU)^2)= 34>0. Under a soft probe Z=VFZ=V_F carries the complementary block’s noise, and this may remove some or all kernel directions: writing Z=LUZ=LU over the independent still-random Rademacher noises, Var(Z~i2)=4∑k<lLik2Lil2>0Var( Z_i^2)=4 _k<lL_ik^2L_il^2>0 iff free coordinate i genuinely mixes ≥2≥ 2 noises (row i of L has ≥2≥ 2 nonzeros); a lone-noise coordinate keeps its diagonal kernel direction. This per-coordinate mixing is necessary but not sufficient (e.g. Z1=U1+U2Z_1=U_1+U_2, Z2=U1−U2Z_2=U_1-U_2 mixes both yet Z~1Z~2≡U12−U22=0 Z_1 Z_2≡U_1^2-U_2^2=0, rank 11; in general rankWe≤p(p−1)/2rank\,W_e≤ p(p-1)/2 for p still-random noises). Full rank therefore requires the full mixing/spanning non-degeneracy of the congruence/moment map, not genericity alone — e.g. Z1=U1+aU3Z_1=U_1+aU_3, Z2=U2Z_2=U_2 leaves Z~22≡1 Z_2^2≡1 and WeW_e degenerate for every a (eigenvalues 0,2(1+a2),4a2\0,2(1+a^2),4a^2\), whereas the scalar free block (me=1m_e=1, F=1F=\1\) Z1=U1+aU2Z_1=U_1+aU_2 mixes both and We=4a2≻0W_e=4a^2 0 is full rank. The branch is selected from the actual WeW_e: when a soft probe restores full rank the full-rank Wald applies; when it leaves a residual kernel (declared r, which need not be mem_e and whose kernel need not be spanQjspan\Q_j\) the two-piece test applies. When degenerate, the test has two pieces. (i) Range Wald: the rank-truncated Wald (spectral cut at the declared rank ke−rk_e-r; not the naive Moore–Penrose inverse of W^e W_e, whose Op(1/n)O_p(1/n) sample eigenvalues re-inflate each kernel direction to a non-negligible, convention-dependent nonstandard contribution — e.g. (1−X2)2/(4X2)(1-X^2)^2/(4X^2), X∼N(0,1)X N(0,1), under the unbiased sample-covariance convention, not a χ2χ^2) is χke−r2χ^2_k_e-r on range(We)range(W_e), with level α and power against range-supported alternatives. (i) Companion kernel test: the r kernel functionals equal their population values up to Op(1/n)O_p(1/n) under H0H_0 (population-exact; the residual is only the Op(1/n)O_p(1/n) centering fluctuation) but shift by Θ(1) (1) under a twin moving a kernel direction, so a threshold tnt_n with 1/n≪tn≪11/n\! \!t_n\! \!1 (e.g. tn≍n−1/2t_n\! \!n^-1/2) has level →0→ 0 and power →1→ 1 against any Ω(1) (1) kernel move. Purely-kernel alternatives are caught only by (i), range alternatives by (i). This is not a “sub-Gaussian threshold” (two-point noise is sub-Gaussian; the failure is degeneracy, not tails, and the full-rank Wald does not apply — the sup-norm test still applies with KZ≍1K_Z\! \!1). Proof. See Appendix E.1. □ Corollary 51 (Aggregate query tolerance). For the fixed validation-independent candidate, fixed block scales, direct-sum Euclidean/svecsvec norm, and block-specific conditions (BPb)(BP_b) of Corollary 7, the finite-n sub-Gaussian tolerance is exactly the block expression in Corollary 7, εn=CL[∑b∈ℬqb,n2]s/2 _n=C_L[ _b _Dq_b,n^2]^s/2. For B≥1B≥ 1, the bound by CL[Bmaxbqb,n]sC_L[ B _bq_b,n]^s is only a conservative display; it does not replace the block sum. Here a known-H mean block uses qe,nmq_e,n^m, an estimated-H mean block uses q~e,nm q_e,n^m, and the operational covariance radius retains both Bernstein branches and its declared-envelope κ¯Σ κ_ ; the covariance rate cannot be read off a mean constant. Against (B,cΣΩ)(B,c_ ) every mean test passes. Let S be the source covariance with unbiased divisor n−1n-1 and suppose Σ=cΣΣ _T=c_ _S. Then the exact Gaussian pivot is cΣ(n−1)tr(Σ−1S)∼χm(n−1)2.c_ (n-1)tr( _T^-1S) χ^2_m(n-1). For D=tr(Σ−1S)−mD=tr( _T^-1S)-m, E(D)=m(1−cΣ)/cΣE(D)=m(1-c_ )/c_ , Var(D)=2m/[cΣ2(n−1)]Var(D)=2m/[c_ ^2(n-1)], and the exact squared signal-to-noise ratio is (n−1)m(1−cΣ)2/2(n-1)m(1-c_ )^2/2. In a separate orientation check, take source covariance Ω′=diag(1+a,1−a,1) =diag(1+a,1-a,1) and comparator I. Then the trace statistic has zero mean gap and is blind to this trace-free change, whereas the full Wald TeΣT_e rejects it. Reversing source and comparator is not blind because tr(Ω′−1)−3=2a2/(1−a2)>0tr( -1)-3=2a^2/(1-a^2)>0. The numerical illustrations exercise both block types of RmcR_D mc (means and free-block svecsvec), but do not establish the selected-summary modulus; W^e+ W_e^+ needs n>rankWen>rank\,W_e, and noise below the fourth moment is outside the result. Only spanned directions are validated (counterpart: a span-deficient wrong twin — the split-gauge completion family on the untested directions; finite level and power calculations do not address this identification failure). Proof. See Appendix E.1. □ Proposition 52 (Invariance diagnostics and their ceiling). Invariance is necessary; under the (T2) factorization null a named conditional test is level-valid only with its separate sampling/calibration theorem. It also has an existential ceiling. There are exact-law, quotient-valid pairs with invariant residual laws but different admissible queries. One is the passive Gaussian counterfactual pair in Theorem 8; another is the parent-dependent measure-preserving reshuffle in Theorem 31. These examples use query functionals that descend to the declared equivalence class, not entries of a particular representative H. Thus (C-b) tests environment invariance and can detect drift, but it does not by itself identify a cross-world query. The same ceiling applies cross-domain, as Corollary 32 shows. Proof. See Appendix E.1. □ Interpretation. The mean and covariance procedures test selected implications of a candidate twin; they do not establish the structural class or identify an arbitrary counterfactual. Their conclusions are limited to the reconstructed directions, sampling regimes, and rank conditions stated in the theorem. 12 Numerical illustrations Design and scope. The figures use small synthetic linear ECG systems with planted source–target discrepancies and fixed seeds. They illustrate selected constructions and finite-instance behavior; they do not establish theorem validity, identification, completeness, coverage, or operational performance. Figure 1: Interventions reduce rotational ambiguity in the displayed source block. Each additional query-relevant row removes rotational degrees of freedom; the residual dimension reaches zero after |D|−1|D|-1 rows are pinned. This calculation concerns the stated rotational orbit and does not give a universal intervention count. Figure 2: Different model discrepancies require different diagnostics. The cells summarize the displayed constructions. A staged intervention distinguishes the moment-matched latent rotation, whereas structural-class analysis distinguishes the parent-dependent reshuffle and the selection-branch change. The loop-gain row is marked “not established” because no matching construction is claimed here. Figure 3: Hybrid reconstruction versus direct source reuse. In the displayed synthetic systems, the hybrid model of Theorem 12 reproduces the target response to numerical precision, whereas direct reuse of the source model retains the domain discrepancy. Figure 4: Design relativity and the acyclification hazard. Panel A: in these examples, the target-intervention budget scales with the discrepancy set rather than the full system; no universal count is implied. Panel B: an acyclic reading misses the nonzero equilibrium effect in the example of Theorem 30. Interpretation. Figures 1 and 2 distinguish experimental information from structural restrictions. Figures 3 and 4 illustrate when direct reuse fails and why the observation model and support conditions must accompany every intervention count. 13 Discussion and limitations Relation to existing causal theory. The hedge constructions of Shpitser and Pearl concern identification in the acyclic causal hierarchy. The finite-design obstruction here serves a related logical purpose, but it concerns the cross-world coupling and the equilibrium response of a feedback system. Likewise, cyclic separation and intervention calculus describe relations among interventional distributions, while a transported per-unit counterfactual additionally requires factual abduction, equilibrium selection, and alignment across domains. These distinctions explain why neither acyclic hedges nor cyclic separation alone settles the questions studied here. Cross-world scope. The positive results require the declared boundary class. Monotonicity is an assumption, supported by the constructed necessity result and the additive linear specialization. For partial factual information, membership in a fixed-partition tensor-product null, calibration of a named test, and constancy of the query distribution over the complete legal fibre are distinct claims; loop gain alone establishes none of them. The completeness of the graphical transport rule remains open. Design-relative identification. Intervention counts depend on the observation model, support, environment library, incoming-block information, and complete legal fibre. The affine expression qd−rGqd-r_G applies only at fixed corank and support, within a fixed-branch relative-interior patch. The query-targeted comparison does not imply universal strict savings, and the alignment result does not imply a universal one-anchor rule. Statistical scope. Finite-sample query guarantees based on moments require both the selected-summary modulus and the stated moment-fibre sufficiency condition. Constancy on a complete-law fibre cannot replace either condition. The joint-influence free-block Wald procedure is pointwise and asymptotic, with fixed dimension, environment, and branch. The covariance rank-degenerate branch requires its declared rank, eigengap, and kernel conditions. An estimated-sensor extension of the free-block procedure is not proved here. Evidence and external validity. The numerical material consists of synthetic or analytic illustrations. It does not establish general identification, completeness, coverage, or performance in an operational system. Evaluation on substantive application domains remains future work. Open problems. Three boundaries remain unresolved. First, a complete graphical characterization of transport under feedback is not yet available. Second, the exact intervention requirement for incomplete discrepancy blocks with live relay mechanisms remains structure-dependent. Third, finite-sample procedures with an estimated sensor map and data-dependent covariance rank require additional analysis. The conditional cumulant hierarchy discussed in the supporting theory should be regarded as a direction for future work, not as a result established here. 14 Conclusion For feedback-capable causal twins, validation and transport are governed by the same basic decision logic. The mechanism-level separation criterion gives a sufficient condition for direct reuse within the boundary class; when it fails, direct reuse may still hold for a particular query. When separation fails but the source and target frames are aligned, the shared selection rule is retained, and the target discrepancy mechanisms and their noise laws are exactly re-identified, hybrid reconstruction recovers target post-surgery laws. Per-unit counterfactual transport additionally requires the target boundary-class conditions; otherwise the complete legal fibre provides the partial-identification answer within the declared class. This separation between direct reuse, target re-identification, and irreducible ambiguity is the main practical consequence of the theory. Availability Code and data for the numerical illustrations are available from the author upon reasonable request and will be released publicly upon journal publication. Appendix A Technical arguments for validation This appendix collects the arguments used in the main text. Definitions and remarks do not require separate proofs. When a result concerns a complete legal fibre, all nuisance, support, alignment, normalization, and selection branches named in the statement are part of that fibre. A.1 Rank abduction and partial factual information Lemma 4. Strict monotonicity gives fi(vpa(i),u)=Qido(vpa(i),Fi(u))f_i(v_pa(i),u)=Q_i^do(v_pa(i),F_i(u)), where QidoQ_i^do is the conditional quantile of the structural intervention that fixes the parents. Thus a full factual observation fixes τi=Fi(ui) _i=F_i(u_i) almost surely. An increasing reparameterization of uiu_i preserves τi _i. The post-surgery system may therefore be written entirely in terms of the shared ranks, the shared interventional kernels, and the selection rule. Rank-measurability of Sel makes the two selected solutions equal. This is why observational conditionals cannot replace the parent-fixing interventional kernels in a feedback model. For partial factual information, standard-Borel disintegration gives a regular conditional law Λwθ _w^θ, unique PWyP_W^y-almost everywhere. Its pushforward through the well-posed solve is KIθK_I^θ. Hence all data-consistent models give the same posterior query law exactly when [KIθ]PWy\[K_I^θ]_P_W^y\ is a singleton. Conditional independence of a declared finite partition is instead equivalent to the tensor-product identity for Λwθ _w^θ; it neither implies nor is implied by equality of arbitrary query pushforwards. A.2 Population validation Theorem 5. At a query-sufficient design, equality of the complete staged interventional kernels places the candidate and reference systems in the same complete law fibre. Constancy of the query on that fibre identifies the query-relevant kernels. Lemma 4 then gives the full-factual conclusion, and the singleton pushforward condition above gives the partial-factual conclusion. The invariance comparison extends the kernel equality from the staged regimes to every regime used by the query. The moment-only alternatives require a separate argument. Under identified linear structure, a full-factual state-functional query is fixed by the factual state and the structural map; an interventional moment query is fixed when it factors through the identified first two moments. A distributional query additionally requires a family determined by those moments or equality of the complete interventional kernels. The Gaussian and centred-exponential example in the theorem has equal first two moments but unequal tail probabilities, proving that this additional premise cannot be omitted. Proposition 6. On the compact, closed, Hausdorff definable quotient, define g(θ)=‖χ(θ)−χ(θ0)‖g(θ)=\|χ(θ)-χ( _0)\|_ Q and h(θ)=‖Rmc(θ)‖⊕h(θ)=\|R_D mc(θ)\|_ . The condition (Suff-M) gives h−1(0)⊆g−1(0)h^-1(0) g^-1(0). The definable Łojasiewicz inequality in a common polynomially bounded o-minimal expansion therefore supplies g≤CLhsg≤ C_Lh^s for instance-dependent CL,s>0C_L,s>0. A finite closed-branch decomposition uses the minimum exponent and the maximum adjusted constant. The example g(t)=tg(t)=t, h(t)=e−1/th(t)=e^-1/t shows why polynomial boundedness is needed. Per-pattern spectral margins and the remaining compactness conditions make every relevant rational solve continuous; a margin on the observational matrix alone does not control an unobserved surgery. Corollary 7. If the query error exceeds CLQnsC_LQ_n^s, the modulus gives ‖Rmc‖⊕>Qn\|R_D mc\|_ >Q_n. Since Qn2=∑bqb,n2Q_n^2= _bq_b,n^2, at least one block has residual norm greater than its own qb,nq_b,n. Acceptance of all blocks is then contained in acceptance by that one separated block, giving its Type-I bound. This is a single-block containment argument, not a union bound. The empty-block convention makes the wrong-query event empty under (Suff-M). A.3 Necessity constructions Theorem 8. For T1, apply a parent-dependent rank-band reflection at a node whose band-driving parent is not its descendant. For every fixed parent value the map preserves the noise law, so every designed regime has the same law. When the intervention moves that parent, the factual and counterfactual bands have a positive-measure symmetric difference and the readout changes. A parent-independent reflection composes with itself and is only a noise relabeling, which explains the parent-dependence requirement. For T2, take independent standard Gaussian U1,U2,U3U_1,U_2,U_3 and V3=U1+U2+U3V_3=U_1+U_2+U_3. Conditioning on V3V_3 gives Cov(U1,U2∣V3)=−1/3Cov(U_1,U_2 V_3)=-1/3. Thus the factorization null fails even without a cycle; using the product variance 4/34/3 in place of the true variance 22 gives the size stated in the theorem. The converse is the defining tensor-product identity for conditional independence. For T3, let Z∼N(0,1)Z N(0,1), let the surgery indicator be s∈0,1s∈\0,1\, and use the state space [−2,2][-2,2] with f0(v,Z)=tanhZ,f1(v,Z)=v−110v3−3v−tanhZ.f_0(v,Z)= Z, f_1(v,Z)=v- 110\v^3-3v- Z\. At s=0s=0 the equilibrium is uniquely v=tanhZv= Z. At s=1s=1, writing a=tanhZ∈(−1,1)a= Z∈(-1,1), the equilibrium equation v3−3v−a=0v^3-3v-a=0 has roots r−(a)∈(−2,−1)r_-(a)∈(-2,-1), r0(a)∈(−1,1)r_0(a)∈(-1,1), and r+(a)∈(1,2)r_+(a)∈(1,2). Since ∂vf1=1.3−0.3v2 _vf_1=1.3-0.3v^2, the outer roots are stable and the middle root is unstable. The two models use, consistently across all their regimes, either the minimum-stable or the maximum-stable SCC-local selector. They therefore agree in every design that does not stage s=1s=1, but disagree after do(s=1)do(s=1). Indeed r+(a)−r−(a)>17/5r_+(a)-r_-(a)>17/5 for all a∈(−1,1)a∈(-1,1): the gap is even, decreases for a>0a>0, and at the endpoint a=1a=1 its square is 12−3x2>741/64>289/2512-3x^2>741/64>289/25, where the middle root is −x-x and 0<x<3/80<x<3/8. The example uses the paper’s declared-selection notion of well posedness; it would not satisfy a requirement of raw uniqueness of every fixed point. For T4, put B0=(03/51/20),C0=I−B0,Ω0=I,B_0= pmatrix0&3/5\\ 1/2&0 pmatrix, C_0=I-B_0, _0=I, and, for the planar rotation QθQ_θ, define d1=cosθ−12sinθ,d2=cosθ+35sinθ,Λθ=diag(d1−1,d2−1),d_1= θ- 12 θ, d_2= θ+ 35 θ, _θ=diag(d_1^-1,d_2^-1), Cθ=ΛθQθ⊤C0,Bθ=I−Cθ,Ωθ=Λθ2.C_θ= _θQ_θ C_0, B_θ=I-C_θ, _θ= _θ^2. Whenever d1d2≠0d_1d_2≠ 0, Cθ−1ΩθCθ−⊤=C0−1C0−⊤C_θ^-1 _θC_θ^- =C_0^-1C_0^- , so the passive zero-mean Gaussian law is unchanged. At θ=.7θ=.7 both models are stable, whereas for factual v=(1,1/2)v=(1,1/2), intervention do(V0=2)do(V_0=2), and response Y=V1Y=V_1, the counterfactual gap is Bθ,10−12=7tan(.7)23tan(.7)+5>0.B_θ,10- 12= 7 (.7)2\3 (.7)+5\>0. This collision applies only to passive stages or stages lying in the exact stabilizer of the displayed law; a generic labelled structural probe separates the pair. The two necessity constructions are independent. Theorem 31 and Corollary 32. Choose the rank-space reflection so its upper band boundary separates the factual and counterfactual laws of the non-descendant parent on a set of positive probability. It preserves every conditional mechanism law and hence every regime in the finite design, but its two cross-world applications do not cancel on the symmetric difference of the bands. Multiplication by the nonzero resolvent coefficient gives a positive query gap. Under symmetric noise the reflection is u↦−u -u on the band and yields the displayed 2|c||u|2|c||u| formula; without symmetry only positivity of the gap is retained. The cross-domain result applies the same construction to the target switch-indexed mechanism and uses the target resolvent and target factual law. Appendix B Technical arguments for transport under feedback Lemma 9. In the condensation graph, the ancestor set of Y∪ZY∪ Z after surgery is closed under parents and under strongly connected components. Solving components in condensation order shows that its post-surgery law uses only mechanisms and noise laws in that set. In a linear model, expanding (I−BI)−1(I-B_I)^-1 in its convergent walk series proves the claim under ρ(BI)<1ρ(B_I)<1; multiplying by the determinant extends the resulting rational identity to the entire nonsingular locus. Theorems 10 and 12. Under (sep), Lemma 9 restricts the query law to mechanisms that are invariant and aligned, so the source and target functionals coincide. For the hybrid result, write κiω=(fiω,PUiω) _i^ω=(f_i^ω,P_U_i^ω) for the complete aligned mechanism–noise pair. Full target-law transport requires κih=κiτ _i^h= _i^τ for every nonintervened row, identical surgery and clamps, independent noises, identical variable/noise alignment, the same SCC-local selection semantics, and a well-posed selected solve in every post-surgery SCC. Induction along the condensation graph then gives the same law block by block. For a query on coordinates Q, equality is needed only on the post-surgery ancestral SCC closure AnGIeSCC(Q)An SCC_G_I_e(Q), yielding equality of the Q-law. Equality of only a specified class of moments transports only queries determined by that class; it does not imply equality of the complete law unless a separate moment-determinacy premise is supplied. Lemma 11. Fix a connected real-analytic support stratum. If a single measurable functional h makes Δ(θ)=∫hPθ,j1−∫hPθ,j0 (θ)= h\,dP^1_θ,j- h\,dP^0_θ,j finite and analytic, and an explicit point on that stratum has Δ≠0 ≠ 0, then equality of the two laws is contained in the proper analytic zero set Δ=0\ =0\. It consequently has empty interior and chart Lebesgue measure zero. The argument must be repeated on every connected stratum; where no separator and witness are available, the conclusion requires an explicit distributional-faithfulness assumption. For the linear-intercept specialization, replacing cic_i by ci+tc_i+t gives V(t)−V(0)=t(I−B)−1eiV^(t)-V^(0)=t(I-B)^-1e_i under the common-noise coupling, and hence E[Vj(t)]−E[Vj(0)]=t[(I−B)−1]ji.E[V_j^(t)]-E[V_j^(0)]=t[(I-B)^-1]_ji. The exact exceptional set is t=0t=0 together with the zero set of the corresponding cofactor of I−BI-B. On a connected analytic support stratum, one nonzero cofactor witness makes this a proper analytic set. Measure-preserving reparameterizations, or strata on which the cofactor vanishes identically, show why no unqualified global genericity claim is valid. Theorem 14. The generalized directed global Markov property of the augmented model gives Y⟂∣Z,XY \!\!\! Z,X after surgery. Evaluating the conditional law at the source and target point masses of S yields equality conditional on the full separator Z∪XZ∪ X. For this source version of the conditional kernel to define the target integral, the regime-specific target separator law must satisfy μτ,eZX≪μπ,eZX. _τ,e^ZX _π,e^ZX. Then, for the common target-a.e. version Kπ,eK_π,e, Pτe(Y∈A)=∫Kπ,e(A∣z,x)μτ,eZX(dz,dx).P_τ^e(Y∈ A)= K_π,e(A z,x)\, _τ,e^ZX(dz,dx). Marginalizing X requires the target conditional law of X given Z. Reusing the source-marginalized kernel is valid only when that conditional law agrees in the two domains target-a.e., or when the kernel is x-constant on the relevant target support. Matched marginals alone do not suffice: source mass on X=ZX=Z and target mass on X=1−ZX=1-Z give a direct counterexample. Theorem 15. The hybrid fixes the query-relevant interventional kernels. Lemma 4 proves the full-factual claim, while the complete-fibre singleton proves the partial-factual claim. Below point identification, the answer is the image of the target legal fibre; the width statements follow from Corollaries 37 and 38. Definition 16, Proposition 17, Lemma 18, Theorem 19, and Corollary 20. The standard-Borel premise makes every QI(θ)Q_I(θ) a well-defined equivalence class under the common factual law. Identification is constancy on the actual fibre, so a separated pair is both necessary and sufficient for pointwise failure. An exact-evidence curve lies in that fibre. In Theorem 19, the affine scalar functional changes by βtβ t, so every admissible nonzero point on the declared interval is separated from the reference. Existing constructions instantiate this argument only after establishing global membership, exact evidence equality, the full selection/noise requirements, and query motion. This explains why ancestry, a graph label, or stabilizer dimension alone is insufficient. Theorem 30. Solving the two-node equilibrium gives the effect −b/(1−b2)-b/(1-b^2). Substitution of b=.5b=.5 and .3.3 gives −2/3-2/3 and −30/91-30/91. The acyclic graph deletes the feedback walk and returns zero, so it cannot represent this domain dependence. Appendix C Technical arguments for linear target identification Lemma 22. The pinned-row identity is ZY=E−ZFCF,⋅Z_ DY=E_ D-Z_FC_F,·. Its complete solution is Y0+NTY_0+NT. Applying the affine structural equalities to NTNT gives rank rGr_G and hence dimension qd−rGqd-r_G. Strict stability, invertibility, and present-edge inequalities do not alter the relative-interior dimension. The complementary-nullity theorem for the inverse pair (C,P)(C,P) gives the equality of nullities. The associated determinant-zero locus is locally codimension one only at an admissible regular zero, that is, where the derivative of detCFF C_F is nonzero along an admissible tangent direction. A model-compatible witness is obtained from V=(10011001),W=(01100−1−10),B=VW⊤.V= pmatrix1&0\\ 0&1\\ 1&0\\ 0&1 pmatrix, W= pmatrix0&1\\ 1&0\\ 0&-1\\ -1&0 pmatrix, B=VW . Here B2=0B^2=0 and a two-by-two free block of C=I−BC=I-B is (1−1−11) ( smallmatrix1&-1\\ -1&1 smallmatrix ); its determinant has a nonzero admissible tangent derivative. At rank deficiency at least two the adjugate vanishes, so the determinant locus may be singular and no global hypersurface claim follows. Theorem 23. Active probing pins the labelled discrepancy rows of P=(I−B)−1P=(I-B)^-1. The invariant rows supply the equations CFFPF=EF−CFPC_FP_F=E_F-C_F DP_ D. Thus CFFC_F nonsingular gives the unique ambient solve, and inversion gives B. On a fixed singular support branch, Lemma 22 supplies the exact remaining affine fibre. For the normalized rotational calculation on a complete source block of size m, let A be the matrix of k signed, labelled, linearly independent anchor directions. Equivariance makes the anchor-preserving stabilizer Q∈SO(m):Q⊤A=A≅SO(m−k).\Q∈ SO(m):Q A=A\ SO(m-k). Its dimension is (m−k2) m-k2, so the anchors remove k(2m−k−1)/2k(2m-k-1)/2 rotational directions and m−1m-1 generic anchors eliminate that continuous orbit. The generic rank assertion follows from a nonzero minor at the identity anchor frame and persists on an open dense set of the fixed-support chart. This orbit calculation becomes a statement about the complete local data fibre only under the theorem’s local-completeness hypothesis: in a neighborhood of the reference point, every legal data-equivalent model must lie on this normalized anchor-preserving orbit. Global identification further requires the complete legal fibre to have no transverse, disconnected, or discrete branches. Proposition 24. For each independent cluster, stack the source, target, shared-baseline, and nuisance influence contributions before taking the outer product. The multivariate cluster central limit theorem and delta method give one joint affine chart for (C^,G^,Z^)( C, G, Z). Construct a single (1−αS)(1- _S) ellipsoid in that chart; all three radii δC,δG,δZ _C, _G, _Z are projections of this same event, not separately calibrated marginal intervals. Their covariance includes every overlap and plug-in cross term named in the statement. Weyl’s inequality gives σmin(C)≥L:=σmin(C^)−δC _ (C)≥ L:= _ ( C)- _C, and the procedure refuses unless L≥ηL≥η. With Q=EF−GZQ=E_F-GZ, δQ=δG‖Z^‖+(‖G^‖+δG)δZ. _Q= _G\| Z\|+(\| G\|+ _G) _Z. Finally, C(X^−X)=(C−C^)X^+(Q^−Q)C( X-X)=(C- C) X+( Q-Q) and ‖Q^−Q‖≤δQ\| Q-Q\|≤ _Q give ‖X^−X‖≤δQ+δC‖X^‖L.\| X-X\|≤ _Q+ _C\| X\|L. Thus the unconditional probability of accepting while the displayed bound fails has limsup at most αS _S. A reduced guard using its own αC _C is a separate construction and cannot be combined with this conclusion without a new allocation. Every listed refusal corresponds to a missing premise. In particular, an estimated sensor would alter the complete joint influence vector and is not supplied here. Corollary 25, Theorem 26, and Corollary 28. Unknown incoming edges leave d−||d-| D| unconstrained directions in an unprobed row. With known incoming edges, the missing-row completion reduces to the universal-parent condition on the stated no-relay block. In the shared-sensor covariance regime the complete unpinned block has the orthogonal deficit above. For the lower bound, every choice of ||−2| D|-2 probes leaves two rows whose stable rotation remains inside a complete full-rotational component and preserves the exact designed laws. Choosing a point in the identity component keeps ρ(B)<1ρ(B)<1; the query-motion premise provides a separated member. In incomplete support the rotation may leave the class, which is why no uniform lower bound is asserted there. The count comparison in Corollary 28 is therefore restricted to the ambient class stated in that corollary; support-aware classes use KminIDK_ ID. For the unknown-sensor collision used in the lower-bound discussion, choose distinct indices r,sr,s, put u=es−Crseru=e_s-C_rse_r, and define Et=tuer⊤,Ct=C(I+Et),Pt=(I+Et)−1P,Ht=H(I+Et),Ωt=Ω.E_t=tue_r , C_t=C(I+E_t), P_t=(I+E_t)^-1P, H_t=H(I+E_t), _t= . Then diag(CEt)=0diag(CE_t)=0, HtPt=HPH_tP_t=HP, and Pt=P−t1−tCrs(es−Crser)er⊤P.P_t=P- t1-tC_rs(e_s-C_rse_r)e_r P. For a Gaussian zero-mean source with common covariance, identical stochastic-intervention sources, and every retained named stage disjoint from r,s\r,s\, the P1 intervention identity makes the complete retained loading matrix—and hence every recorded joint Gaussian law—identical. In the ambient support model the perturbation is legal for small t, and es⊤Pt=es⊤P−t1−tCrser⊤Pe_s P_t=e_s P- t1-tC_rse_r P, so the labelled held-out s-stage response row moves. Moreover, dt(Bt)sr|0=−(1−CrsCsr) ddt(B_t)_sr|_0=-(1-C_rsC_sr), so the structural response query moves away from the exceptional locus. This is a quotient-valid query separation. Fixed known H generically excludes the collision, and a stage involving r or s, a forbidden support entry, a nonidentical intervention source, or a nonzero source mean not transformed consistently invalidates the construction. Appendix D Technical arguments for partial identification and alignment Theorem 34 and Proposition 35. By definition, the sharp identified set is the image of the complete legal fibre. On finitely many semialgebraic strata, the Tarski–Seidenberg theorem makes the scalar image a finite union of points and intervals. Compactness plus continuity guarantees a compact image and attainment of its extrema. A spectral margin does not bound transverse completion coordinates, so it cannot provide that conclusion alone. Removing the stability restriction lets the rotation approach the diagonal-normalization pole. For example, with B0=(03/51/20),t=tanθ,B_0= pmatrix0&3/5\\ 1/2&0 pmatrix, t= θ, the normalized rotation gives B12(t)=2(5t−3)5(t−2),B21(t)=5(2t+1)2(3t+5),χ(t)=13t+102(3t+5).B_12(t)= 2(5t-3)5(t-2), B_21(t)= 5(2t+1)2(3t+5), χ(t)= 13t+102(3t+5). The stable component containing the identity is the open interval ((1−170)/13,(1+170)/13)((1- 170)/13,(1+ 170)/13), and its query image is exactly (12−517068,12+517068). ( 12- 5 17068, 12+ 5 17068 ). The endpoints are excluded stability-boundary models. An exact unbounded response-row family is Bt=(0t1/(4t)0)B_t= ( smallmatrix0&t\\ 1/(4t)&0 smallmatrix ), Ht=I−BtH_t=I-B_t, Ωt=I _t=I: then Ht(I−Bt)−1=IH_t(I-B_t)^-1=I, ρ(Bt)=1/2ρ(B_t)=1/2, and det(I−Bt)=3/4 (I-B_t)=3/4, while the labelled response B12=tB_12=t is unbounded. This latter calculation is a response-row illustration, not a claim that every staged law is equal. Theorem 36. The query diameter is taken over the joint ambiguity set, so it is the sharp primary object. A parent-dependent measure-preserving reshuffle attains, in the extended-essential-supremum sense, the scalar-noise width multiplied by the resolvent coefficient when the node is witness-admissible for the two worlds and the legal ambiguity class contains the corresponding parent-indexed rearrangements. More precisely, under those conditions, if U is atomless with essential endpoints a,ba,b and c is the relevant coefficient, then supT0,T1esssup|c||T1(U)−T0(U)|=|c|(b−a), _T_0,T_1 *ess\,sup|c|\,|T_1(U)-T_0(U)|=|c|(b-a), where the supremum is over law-preserving rearrangements and may be infinite. The antitone quantile coupling gives the lower bound. This is not generally a pointwise maximum: for U∼Unif(0,1)U (0,1) the width is one but the endpoint gap is not attained almost surely. Positive-mass endpoint attainment needs compatible endpoint atoms. Randomization may split existing compatible atomic mass, but cannot create endpoint atoms while preserving atomless marginals. A parent-independent involution is a relabeling and contributes zero. Without witness-admissibility, the same essential-range quantity is only an enlarged-coupling upper bound; it need not equal the actual legal-fibre width. The rotation term is the same orbit already present in the within-world identified set, so it is not counted twice. Corollary 37. Under recombination closure, join any two admissible points by a path that changes each ambiguity block once. The triangle inequality bounds each step by the corresponding diameter uniformly over the other blocks. Summing gives the upper bound. The diagonal and staircase examples show respectively why product closure and the one-change path are required. Corollary 38. In the decoupled subclass the query is a sum of functions on transverse coordinate blocks, so the supremum and infimum separate and the diameter is the sum of the three diameters. A shared selection rule removes the fourth, branch-selection term. A bilinear readout violates this separability and explains why the general result is only the uniform upper bound. Corollary 39 and Lemma 40. A second-moment functional is constant on every legal second-moment orbit. If the query is not constant there, the same decision is assigned to two different query values. More generally, a transported query descends to the retained object exactly when it is constant on the entire representative fibre; a group orbit suffices only after transitivity has been established. Proposition 41. The first claim follows by retaining every legal member of the labelled-law level set. For a complete source block of size m, k signed, labelled, independent anchor directions form a matrix A satisfying the equivariance relation A(TQθ)=Q⊤A(θ)A(T_Qθ)=Q A(θ). Assume additionally the local faithfulness condition aD(TQθ)=aD(θ)a_D(T_Qθ)=a_D(θ) if and only if Q⊤A(θ)=A(θ)Q A(θ)=A(θ) for Q near the identity. Hence the residual normalized rotational stabilizer is StabSO(m)(A)≅SO(m−k),Stab_SO(m)(A) SO(m-k), with dimension (m−k2) m-k2. Thus m−1m-1 generic anchors eliminate the normalized continuous SO(m)SO(m) orbit. A special query can require fewer anchors precisely when this residual stabilizer acts trivially on that query. These are local orbit statements. Only the explicitly assumed local-completeness condition identifies the orbit with the complete nearby data fibre; global point identification also requires the absence of transverse, discrete, and disconnected branches. Without equivariance the survivor is a general level set, and tangent rank alone cannot remove those branches. Appendix E Technical arguments for query design and inference Theorem 42. Query identification is constancy on K_D, so minimizing over the environment library gives VminqryV_ qry. Constancy differentiates to zero along every feasible curve, proving VminFO≤VminqryV_ FO≤ V_ qry. A singleton complete fibre identifies every query, proving the second inequality. When the legal fibre has finitely many relative-interior strata and, on stratum s, the reference-law equality is exactly θ0+QsZs _0+Q_sZ_s, where Qs∈ℝns×psQ_s ^n_s× p_s has full column rank, while the additional environments impose exactly Msz=0M_sz=0 with MsM_s having psp_s columns, the complete fibre is ⋃sθ0+Qsz:z∈Zs∩kerMs. _s\ _0+Q_sz:z∈ Z_s∩ M_s\. At a relative-interior point its local dimension is ps−rankMsp_s-rankM_s. A query is globally identified if and only if it is constant on every connected component of every stratum and those constants agree across all components and strata. For an affine query LθLθ, within-component constancy is LQskerMs=0LQ_s M_s=\0\; the cross-component comparisons remain necessary. Thus zero local dimension or full Jacobian rank on one branch does not by itself establish global identification. Corollary 43. Under the complete affine-fibre premise, the surviving displacements are exactly QkerMQ M_D. Constancy of c+Gθc+Gθ is therefore GQkerM=0GQ M_D=0. The finite-dimensional fundamental theorem of linear algebra makes this equivalent to row(GQ)⊆row(M)row(GQ) (M_D). The pinned-response identity follows by multiplying P0(I−B0)=IP_0(I-B_0)=I and subtracting. None of these equalities characterizes a larger nonlinear or unprofiled law fibre. E.1 Inference proofs Theorem 46. The reconstructed mean is a smooth function of the sample mean and, when applicable, the sensor estimate. The multivariate central limit theorem and delta method give the stated influence covariance, including the two same-sample cross terms. Studentization on the declared estimable range gives the chi-square limit. A plugged-in sensor without its influence contribution is outside this argument. Proposition 47. The stated whitened sub-Gaussian condition gives the Euclidean sample-mean bound with radius te,nm(η)t_e,n^m(η). Under the alternative, the triangle inequality leaves a deviation larger than the null radius whenever the population gap exceeds the sum of the null and power radii. A sample-split sensor error is added with the displayed probability allocation. Proposition 48. Writing the centered quadratic score in Frobenius-isometric symmetric coordinates gives an asymptotically linear statistic with covariance WeW_e. The influence function includes the centering correction and every estimated nuisance derivative and overlap block stated in the proposition. In the externally known exact frame, the full-rank studentized quadratic form is chi-square. Merely sharing an unidentified H across twins is insufficient: the free-block selection and singular subspaces need not be preserved by the residual gauge. If a separately proved whitening map is used, it must be common, identified, positive definite, block preserving, and its estimation influence must enter the same joint covariance. The proposition makes no such optional claim. Proposition 49. Write Σ^−Σ=n−1∑i(WiWi⊤−Σ)−W¯W¯⊤ - =n^-1 _i(W_iW_i - )- W W . Products of sub-Gaussian coordinates are sub-exponential, so entrywise Bernstein gives the two-branch threshold. A union bound over the kek_e symmetric coordinates and ‖svec(A)‖2≤2ke‖A‖max\|svec(A)\|_2≤ 2k_e\|A\|_ give qe,nΣq_e,n and the two displayed sample-size terms. The declared eigenvalue envelope is needed to make the radius operational. Proposition 50. On a declared fixed-rank stratum, spectral truncation restricts the Wald statistic to the estimable range and gives the corresponding chi-square limit. Consistent rank selection requires a fixed population eigengap and Op(n−1/2)O_p(n^-1/2) projector error. The hard-kernel companion is available only when the kernel is exact: sample centering contributes Op(n−1)O_p(n^-1), whereas a fixed kernel alternative moves the score by Θ(1) (1), so a threshold tn↓0t_n 0 with ntn→∞nt_n→∞ has pointwise eventual power one and vanishing null rejection. The hard-clamp formula additionally assumes independent zero-mean coordinates and a proved span equality for the noise-square directions. That discrete construction lies outside the paper’s standing continuous-noise class and is presented only as a separately scoped extension. Under a soft probe the actual WeW_e determines the rank, and neither the hard-clamp span nor uniform near-singular power may be imported. Corollary 51 and Proposition 52. For a validation-independent fixed candidate, the aggregate result substitutes the mean and covariance radii above into Corollary 7, using the fixed block scales and the same direct-sum Euclidean and svecsvec norms as the blockwise procedures. Acceptance is the intersection of blockwise events; the power guarantee is therefore controlled by the largest separated block Type-I bound, not by a union of those bounds. For the Gaussian scale illustration, if Στ=cΣπ _τ=c _π and the source covariance uses divisor n−1n-1, then c(n−1)tr(Στ−1S)∼χm(n−1)2.c(n-1)tr( _τ^-1S) χ^2_m(n-1). Thus for D=tr(Στ−1S)−mD=tr( _τ^-1S)-m, E[D]=m(1−c)c,Var(D)=2mc2(n−1),SNR2=(n−1)m(1−c)22.E[D]= m(1-c)c, (D)= 2mc^2(n-1), ^2= (n-1)m(1-c)^22. With divisor n, both the pivot and null centering change. The scale statistic is blind to a trace-free change only in the stated orientation: source covariance Ω′=diag(1+a,1−a,1) =diag(1+a,1-a,1) and comparator I. Reversing the two gives tr(Ω′−1)−3=2a2/(1−a2)>0tr( -1)-3=2a^2/(1-a^2)>0. The full covariance statistic sees either nonzero change. Finally, residual-law invariance is necessary for an invariant mechanism but cannot distinguish two quotient-valid models whose regime-specific exact laws agree. 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