{"id":"86b5945b-97aa-4ad4-82e0-a517afabb190","arxiv_id":"2504.19944","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper completes the complexity landscape for probabilistic and causal satisfiability under graph-structure and small-model constraints, proving new NEXP-completeness and ∃RΣ-completeness results.","lead":"Researchers mapped how hard it is to check whether a set of statements about probabilities and cause-effect questions can be satisfied, when the shape of the causal model is fixed or the model is forced to be small. They found that fixing the causal graph can raise the difficulty to NEXP-complete for interventional reasoning, and that requiring small models changes the difficulty at the observational layer.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 7's reduction places Y→X edges, permitting X to depend on Y; taken literally, the hardness proof is unsound (e.g., ψ = x=y), though a graph with no Y→X edges fixes it.","rationale":"The reader's weakest_assumption identified that the graph must enforce the X/Y independence and flagged the confusing 'Y the only predecessors of X' sentence. I agree that this is a load-bearing point, but I would sharpen it: the Markovian assumption is explicitly stated for graph-constrained problems (Section 2.3), so the real vulnerability is the edge orientation, not latent confounding. Moreover, the error is not merely confusing; if taken literally it invalidates the soundness direction, as shown by the X := Y counterexample. The reader's conditional verdict already requires fixing this, so my stress-test does not move the verdict. A secondary concern, not developed here, is that the graph-encoding constraint in Theorem 11, ∑_v P([t]X_i ≠ [v\\xi]X_i) = 0, appears to sum over all v including those that change the parents, which would force X_i to be constant; this too should be checked if Theorem 11 is used for the NEXP upper bound. The primary concern, however, is the Theorem 7 reduction's graph orientation, because the central NEXP-hardness claim depends on it directly.","tokens_in":26063,"tokens_out":16686,"duration_ms":166823,"concrete_test":"Analytically test the soundness of Theorem 7 with the graph G as stated: take the Schönfinkel–Bernays sentence ∃x∀y(x = y), which is false for n ≥ 1. Build an SCM with binary variables X, Y, setting F_X(Y) = Y and Y uniform; define the relation variables so that constraint (6) is vacuously or identically satisfied. Check that ∑_y P([y](X = Y)) = 2^n holds, so equation (7) is satisfied. If this model is admitted by the DAG described in the proof (with Y the predecessors of X), the reduction maps a false sentence to a satisfiable instance, demonstrating unsoundness. Then modify G so that X has no incoming edges from Y (e.g., X are roots) and re-verify the soundness argument: for any u, x(u) is independent of y because F_X has no Y arguments, and equation (7) yields ∀y ψ′(x(u), y).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The NEXP-hardness of Satbase⟨Σ⟩_{DAG,L2} and Satlin⟨Σ⟩_{DAG,L2} rests on the reduction in Appendix B.1 from Schönfinkel–Bernays sentences. The reduction must ensure that the existential witnesses X are independent of the universal variables Y; otherwise the soundness argument that a single x(u) satisfies ∀yψ fails. The only mechanism available to enforce this independence is the graph G: under the paper's Markovian assumption, if there is no edge Y→X, then no function F_X may take Y as an argument, so X's value is determined by the exogenous variables alone. However, the proof text states: 'We continue by ensuring that the values of X are not affected by the values of Y by making the Y the only predecessors of the X in G.' Taken literally, this places edges Y→X, i.e., the opposite of what is needed. The DAG representation only requires that every argument of a function is an edge; it permits functions to ignore extra edges, so an edge Y→X does not force dependence, but it does not forbid it either. Consequently, nothing in the stated graph prevents a model from having F_X depend on Y. Such a model can satisfy the reduction's constraints while no single existential witness exists. Concretely, for the false sentence ∃x∀y(x = y) with binary variables, set F_X(Y) = Y and let Y be uniform. Then for every intervention [Y = y], we have X = y, so P([y](X = Y)) = 1 and equation (7), ∑_y P([y]ψ′) = 2^n, holds; equation (6) is satisfied by choosing the relation variables appropriately. The reduction would therefore map a false sentence to a satisfiable Satinstance, breaking soundness. The intended fix is to set G so that X has no incoming edges from Y (e.g., making X roots and, if desired, X the predecessors of Y). With that correction, the step 'values of X depend only on u' is justified and the soundness proof goes through. This is likely a typographical error, but as written the central hardness proof is unsound and must be corrected.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the computational complexity of satisfiability and validity problems for probabilistic and causal languages over Pearl's Causal Hierarchy. It parameterizes the languages by arithmetic (basic, linear, polynomial), presence of a compact marginalization operator, and PCH level, and adds two new constraints: a specified causal DAG as part of the input, and a polynomial upper bound on model size. The headline results are: with a fixed DAG, Satbase⟨Σ⟩_{DAG,L2} and Satlin⟨Σ⟩_{DAG,L2} are NEXP-complete (Theorem 7), while Satpoly⟨Σ⟩_{DAG,L3} remains equivalent to the unconstrained succ-∃R-complete problem (Theorem 11); with small models, Satpoly⟨Σ⟩_{sm,L2} and Satpoly⟨Σ⟩_{sm,L3} are NEXP-complete (Theorem 14), and the probabilistic level drops to ∃RΣ via prior work [5]. The paper also proves that Bayesian-network model checking for polynomial arithmetic with summation is in P^#P and organizes the results into two landscape tables. The proofs are based mainly on reductions from Schönfinkel-Bernays satisfiability and on prior results by the same group.","tokens_in":26423,"tokens_out":15727,"duration_ms":159016,"significance":"If the proofs are completed, the paper makes a solid contribution. It identifies a genuine complexity jump from PSPACE to NEXP when the causal diagram is included in the input at the interventional layer, and it gives a nearly complete multi-parametric map across six arithmetic choices, three PCH levels, and two model constraints. The main hardness reductions are from scratch and are stated in a testable way, and the authors are candid about the sketch in Theorem 11 and about leaving Satlin⟨Σ⟩_{DAG,L1} open. These results would be of real interest to the probabilistic-logic and causality communities. However, as the manuscript currently stands, the central NEXP-completeness claim in Theorem 7 is not established because the graph construction in its proof is written in the wrong direction.","major_comments":[{"comment":"The graph-construction step in the reduction from Schönfinkel–Bernays sentences is, as written, backwards. The text says: “We continue by ensuring that the values of X are not affected by the values of Y by making the Y the only predecessors of the X in G.” Under the causal-diagram definition in §2.3, an edge Y→X permits F_X to take Y as an argument; it does not forbid it. To make X independent of Y one must instead specify that no edge Y→X is present. As stated, the construction allows a model with F_X(Y)=Y and Y uniform: for the false sentence ∃x∀y(x=y), equation (7) is satisfied because P([Y=0](X=0))=P([Y=1](X=1))=1, while no single x works for both values of y. The subsequent soundness argument that “the values x of the variables X deterministically depend only on the values u of the exogenous variables” is therefore not justified. The proof should specify the intended graph (X with no incoming edges from Y) and should state explicitly that the DAG constraint is exact for absent edges, not merely a lower bound on the edge set.","section":"§B.1, Proof of Theorem 7"},{"comment":"The proof of Theorem 11 is not a proof as it stands. The encoding formula is displayed as ∑_v P([t_1,...,t_k] X_i ≠ [v\\ x_i] X_i)=0 and then deferred with the remark that “this constraint uses a bit of notational sugar, explained in detail in the proof of Theorem 7.” The proof of Theorem 7 does not explain this notation or the encoding. The displayed expression mixes two different interventional contexts inside one primitive, and the counterfactual semantics of L3 permit such an expression only after a precise definition of the abbreviation (e.g., as a conjunction of two post-interventional formulas over the same exogenous variables). Since Theorem 11 is used for Proposition 6, Corollary 12, and the NEXP upper bound in Theorem 7, this missing derivation is load-bearing. Please provide a complete formal encoding into E_counterfact and prove the equivalence for each of the three arithmetic classes.","section":"§4.2, Theorem 11"},{"comment":"The proof of Proposition 6 appears to state the reduction directions backwards. It says “Since any probabilistic formula is a special case of a counterfactual one and Satpoly⟨Σ⟩_{L1} is succ-∃R-complete, we get Satpoly⟨Σ⟩_{DAG,L1} ≤_P Satpoly⟨Σ⟩_{L1}.” This would be the direction needed for membership only if DAG-constrained L1 formulas were a special case of unconstrained L1 formulas, which they are not; the special-case relation gives a reduction from DAG,L1 to DAG,L3, not to L1. Conversely, Proposition 2 gives Satpoly⟨Σ⟩_{L1} ≤_P Satpoly⟨Σ⟩_{DAG,L1}, which is the hardness direction, not the “opposite” relation named in the proof. The proposition may be true and the fix is straightforward (membership via DAG,L1 ≤_P DAG,L3 together with Theorem 11), but the proof as written should be corrected.","section":"§4.1, Proposition 6"}],"minor_comments":[{"comment":"The definition of “DAG G represents the graph structure” only says that every argument of a function appears as an incoming edge; it does not say that the edge set is exact. Since the proof of Theorem 7 repeatedly uses the phrase “exactly the edges,” please state the exact-parent convention explicitly.","section":"§2.3"},{"comment":"The addition-encoding step says “the value should not be larger than P(X_k=0),” but the original ETR-INV equation is x_i+x_j=x_k, which is an equality. Please clarify how both the ≤ and ≥ directions are enforced, or correct the sentence.","section":"§B.1, Proof of Lemma 5"},{"comment":"The fragment name is consistently written “Schönfinkel-Bernay” in the proof of Theorem 7; it should be “Schönfinkel-Bernays.”","section":"Throughout"},{"comment":"The class ∃RΣ is used without a definition; since it is introduced in the companion paper [5], a one-sentence definition or a precise pointer to the relevant definition would help the reader.","section":"§2.4"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the venue and the overall research program is coherent. The central theorem's proof as written is unsound at a load-bearing point, but the flaw is local (the graph direction in the Schönfinkel-Bernays reduction) and appears readily repairable, so I recommend major revision rather than rejection. The reliance on the authors' own prior work [5,9,39] is appropriate and does not raise concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the paper is a genuine advance: it completes most of the complexity landscape for probabilistic and causal satisfiability with graph-constrained and small models, and the headline result — graph-constrained interventional satisfiability jumps from PSPACE to NEXP (Theorem 7) — is the first case I know where pinning down the graph changes the complexity. Second, the written proof of that theorem has an orientation bug that, taken literally, is unsound, and it needs a one-line repair.\n\nThe new results are real. Theorem 7's NEXP-completeness for Satbase⟨Σ⟩_{DAG,L2} and Satlin⟨Σ⟩_{DAG,L2} is proven by reduction from Schönfinkel-Bernays, and Theorem 14 extends the small-model story to NEXP at the interventional and counterfactual layers with polynomial arithmetic. The small-model probabilistic layer landing in ∃RΣ is tied down cleanly, and Facts 2–4 are careful: the distinction between observed-support and exogenous-support definitions of 'small' matters, and the prime-support counterexample for Markovian models is neat. Tables 2 and 3 will be the reference point for the next few years. The self-citations to [5], [9], and [39] look legitimate; those are prior published results being built on, and the new hardness proofs are from scratch.\n\nNow the soft spots, in proportion. The big one is Theorem 7's graph construction. The text says the values of X are made independent of Y 'by making the Y the only predecessors of the X.' Under the paper's own semantics, an edge Y→X permits F_X to take Y as an argument, so the soundness argument ('X depends only on u') does not follow. The stress-test counterexample is valid: for the false sentence ∃x∀y(x=y), taking F_X(Y)=Y and Y uniform satisfies equation (7), so the reduction as written maps a false sentence to a satisfiable instance. The fix is obvious — give the X variables no incoming edges. I believe this is a typo, but it is in the central theorem and must be corrected, not papered over.\n\nTheorem 11 is the second issue. Its proof is a sketch: the counterfactual constraint ∑_v P([parents]Xi ≠ [v\\Xi]Xi) = 0 is not written in the formal grammar, and the promised details in the proof of Theorem 7 are not there. Since Theorem 7's NEXP upper bound leans on Theorem 11, this needs a real derivation. Proposition 6's proof also garbles its direction claims; the equivalence is correct and follows from Proposition 2, but the text is confusing.\n\nWho this is for: anyone working on the complexity of probabilistic or causal reasoning, Pearl's hierarchy, or the ∃R/succ-∃R family. The paper deserves a serious referee, and the main theorems are likely all correct. If it were in front of me, I would ask for a revision fixing the Theorem 7 orientation, the Theorem 11 details, and the Proposition 6 prose, then accept. It is marked as ICALP 2025; the fixes belong in the archival version regardless.","headline":"Solid landscape-completing paper, but the written proof of the headline PSPACE-to-NEXP theorem has a fixable orientation bug in its graph construction.","tokens_in":27061,"tokens_out":11144,"would_cite":true,"duration_ms":98832,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68Q17","68Q25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Adding a causal graph to interventional satisfiability raises it from PSPACE-complete to NEXP-complete.","keywords":["probabilistic satisfiability","Pearl's causal hierarchy","structural causal models","interventional reasoning","compact marginalization","NEXP-completeness","existential theory of the reals","small-model property"],"falsifier":"Exhibit one satisfiable Schönfinkel-Bernays sentence whose translation in the proof of Theorem 7 is not satisfied by any Markovian SCM with the prescribed DAG, or is satisfied when the original sentence is not. Concretely, translate a small sentence such as $\\exists x\\,\\forall y\\,((R(x)\\wedge R(y))\\to x=y)$: if some assignment of exogenous variables with positive probability makes two copies of the relation $R$ disagree on equal declared arguments, then the graph edges alone have not forced the dependency the proof relies on, and the reduction fails.","tokens_in":25884,"feed_emoji":"🎲","tokens_out":11769,"duration_ms":112807,"temperature":0.7,"pith_summary":"This paper asks how hard it is to decide whether a formula about probabilities, interventions, or counterfactuals has a satisfying causal model when the model is constrained. Its headline result is at the interventional level: if the causal graph is supplied as part of the input, satisfiability for basic or linear arithmetic with compact marginalization jumps from $\\mathrm{PSPACE}$-complete to $\\mathrm{NEXP}$-complete. The proof encodes satisfiability of Schönfinkel-Bernays sentences, a first-order fragment whose satisfiability is $\\mathrm{NEXP}$-complete, into the interventional language; the graph edges enforce functional dependencies and compact sums express universal quantification. The paper also maps the remaining combinations of arithmetic, hierarchy level, graph constraint, and small-model constraint, showing which combinations change the complexity and which do not. The value is a near-complete computational landscape for problems that underlie causal inference, do-calculus verification, and probabilistic explanation.","feed_headline":"A fixed causal DAG makes interventional satisfiability NEXP-complete","feed_subtitle":"A prescribed DAG pushes interventional probabilistic logic from PSPACE to exponential-time hardness.","key_machinery":"The load-bearing mechanism is the reduction from Schönfinkel-Bernays sentences, the effectively propositional fragment $\\exists x\\,\\forall y\\,\\psi$, whose satisfiability is $\\mathrm{NEXP}$-complete, into interventional-layer satisfiability over a prescribed DAG. For each relation $R_i$, the DAG declares its parents to be exactly its argument variables $Z_i^1,\\ldots,Z_i^k$; a constraint of the form $\\sum_{t_1,\\ldots,t_k} P([T_1=t_1,\\ldots,T_k=t_k,Z_i^1=t_1,\\ldots,Z_i^k=t_k](R_i^j \\neq R_i)) = 0$ forces copies of the same relation to agree on equal arguments. A compact summation $\\sum_y P([y]\\psi') = 2^n$ over the universal variables acts as a for-all quantifier. These ingredients let the graph replace the counterfactual machinery used in earlier hardness proofs, which is why the jump appears already at the interventional level with weak arithmetic.","core_discovery":"The central discovery is that adding the causal graph to the input changes the difficulty of satisfiability only at the interventional layer: the problems $\\mathrm{Sat}^{\\mathrm{base}\\langle\\Sigma\\rangle}_{\\mathrm{DAG},L_2}$ and $\\mathrm{Sat}^{\\mathrm{lin}\\langle\\Sigma\\rangle}_{\\mathrm{DAG},L_2}$ are $\\mathrm{NEXP}$-complete, whereas the same languages without a fixed graph are $\\mathrm{PSPACE}$-complete. The proof reduces satisfiability of Schönfinkel-Bernays sentences to these problems. Boolean variables become random variables, relation occurrences become random variables whose only declared graph parents are their arguments, and the compact marginalization operator expresses that a formula holds for all values of the universally quantified variables. The graph, together with the Markovian independence assumption, forces the relation variables to behave as deterministic functions of their arguments, so the reduction works even with basic or linear arithmetic.","pith_inferences":["Beyond the paper: the same encoding suggests do-calculus verification—deciding whether an interventional rule holds for all SCMs with a given graph—could be inherently exponential when the query language uses compact sums, not just the satisfiability side.","Beyond the paper: because the $\\mathrm{NEXP}$ reduction uses a fixed variable order and sparse parent sets, a natural test is whether polytrees or bounded-treewidth DAGs already give the jump; if they do, the hardness is carried by interventions plus summation rather than by graph width.","Beyond the paper: the authors conjecture that the missing cell $\\mathrm{Sat}^{\\mathrm{lin}\\langle\\Sigma\\rangle}_{\\mathrm{DAG},L_1}$ is $\\exists\\mathbb{R}^{\\Sigma}$-complete. Adapting their ETR-INV encoding through the summation operator would settle that cell and complete the table.","Beyond the paper: the small-model results separate 'small support' from 'small mechanism' in causal models; polynomial support on noise does not bound the complexity of the functions $F$, which is the reason interventional queries stay exponentially hard."],"forward_implications":["Interventional satisfiability with a prescribed DAG is $\\mathrm{NEXP}$-complete for basic and linear arithmetic, so deciding it in polynomial time would imply $\\mathrm{P} = \\mathrm{NEXP}$, and the corresponding validity problem is $\\mathrm{co}$-$\\mathrm{NEXP}$-complete.","Compact marginalization is what unlocks the jump: without the summation operator the small-model property keeps the problem in $\\mathrm{NP}$, and with summation but no graph the same interventional layer is only $\\mathrm{PSPACE}$-complete.","At the counterfactual layer and with polynomial arithmetic, the formula can encode the graph itself through exponential sums, so fixing the graph does not raise the complexity beyond the unconstrained case.","The small-model constraint does not tame interventional reasoning: with polynomial arithmetic, small-model interventional and counterfactual satisfiability remain $\\mathrm{NEXP}$-complete because the deterministic mechanisms $F$ can still encode exponential information even when the support of the noise distribution is polynomially bounded.","For a fully specified Bayesian network, model checking a formula with polynomial arithmetic and summation lies in $\\mathrm{P}^{\\#\\mathrm{P}}$, in sharp contrast to the $\\mathrm{NEXP}$-complete satisfiability problem when only the graph is fixed."],"supporting_citations":[{"why":"Supplies the PSPACE-completeness of the corresponding unconstrained interventional problems and the proof template for the NEXP-hard reduction from Schönfinkel-Bernays sentences.","marker":"[9]"},{"why":"Proves satisfiability of Schönfinkel-Bernays sentences is NEXP-complete, the source of the lower bound.","marker":"[20]"},{"why":"Shows the NEXP-complete satisfiability remains hard when all variables are binary, justifying the reduction's binary-domain assumption.","marker":"[2]"},{"why":"Establishes the ∃R-completeness of polynomial probabilistic satisfiability and that the PCH layer alone does not change complexity; provides the baseline the graph-constrained results extend.","marker":"[23]"},{"why":"Introduced the probabilistic satisfiability framework, proved NP-completeness for basic and linear arithmetic, and supplied the small-model linear-algebra argument used throughout.","marker":"[12]"},{"why":"Introduced compact marginalization and the class succ-∃R, showing that the summation operator increases complexity; the paper's landscape builds directly on this work.","marker":"[39]"},{"why":"Gives the PSPACE algorithm for the existential theory of the reals used in the exponential-time upper bound for small-model polynomial arithmetic.","marker":"[33]"},{"why":"Defines the class ∃RΣ and settles the small-model probabilistic-layer case, which the paper transfers to SCM-based small models via its Fact 3.","marker":"[5]"}],"fun_headline_variants":["Fixed DAG makes interventional satisfiability NEXP-complete","Causal graph inclusion bumps interventional logic to NEXP","Interventional satisfiability is NEXP-complete with fixed DAG","Graph-fixed interventional logic: NEXP-complete complexity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof that a prescribed DAG raises interventional satisfiability to $\\mathrm{NEXP}$ assumes a Markovian causal model, meaning each observed variable has its own independent exogenous noise and no unobserved confounding is allowed. If shared hidden causes are allowed, the same DAG no longer pins down the functional dependencies among the observed variables, and the reduction's soundness collapses.","fun_headline_variants_meta":{"raw":{"variants":["Fixed DAG makes interventional satisfiability NEXP-complete","Causal graph inclusion bumps interventional logic to NEXP","Interventional satisfiability is NEXP-complete with fixed DAG","Graph-fixed interventional logic: NEXP-complete complexity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000332,"raw_usage":{"total_tokens":1910,"prompt_tokens":1075,"completion_tokens":835,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":691,"completion_tokens_details":{"reasoning_tokens":764}},"tokens_in":691,"tokens_out":835,"duration_ms":8145,"temperature":1.0,"reasoning_tokens":764,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:39:54.377752+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit one satisfiable Schönfinkel-Bernays sentence whose translation in the proof of Theorem 7 is not satisfied by any Markovian SCM with the prescribed DAG, or is satisfied when the original sentence is not. Concretely, translate a small sentence such as $\\exists x\\,\\forall y\\,((R(x)\\wedge R(y))\\to x=y)$: if some assignment of exogenous variables with positive probability makes two copies of the relation $R$ disagree on equal declared arguments, then the graph edges alone have not forced the dependency the proof relies on, and the reduction fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves satisfiability of Schönfinkel-Bernays sentences is NEXP-complete, the source of the lower bound."},{"cited_title":"NEXP -completeness and universal hardness results for justification logic","cited_arxiv_id":null,"evidence_quote":"Shows the NEXP-complete satisfiability remains hard when all variables are binary, justifying the reduction's binary-domain assumption."},{"cited_title":"Is causal reasoning harder than probabilistic reasoning? The Review of Symbolic Logic, pages 1--26, 2022","cited_arxiv_id":null,"evidence_quote":"Establishes the ∃R-completeness of polynomial probabilistic satisfiability and that the PCH layer alone does not change complexity; provides the baseline the graph-constrained results extend."},{"cited_title":"A logic for reasoning about probabilities","cited_arxiv_id":null,"evidence_quote":"Introduced the probabilistic satisfiability framework, proved NP-completeness for basic and linear arithmetic, and supplied the small-model linear-algebra argument used throughout."},{"cited_title":"The hardness of reasoning about probabilities and causality","cited_arxiv_id":null,"evidence_quote":"Introduced compact marginalization and the class succ-∃R, showing that the summation operator increases complexity; the paper's landscape builds directly on this work."},{"cited_title":"On the computational complexity and geometry of the first-order theory of the reals","cited_arxiv_id":null,"evidence_quote":"Gives the PSPACE algorithm for the existential theory of the reals used in the exponential-time upper bound for small-model polynomial arithmetic."},{"cited_title":"a ser, Julian D \\","cited_arxiv_id":null,"evidence_quote":"Defines the class ∃RΣ and settles the small-model probabilistic-layer case, which the paper transfers to SCM-based small models via its Fact 3."}],"review_version":1}