{"id":"1c1332e1-78cf-4264-830d-3335c3562710","arxiv_id":"2502.07891","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For causal structures with latent variables and a fixed node order, the paper fully determines the observational dominance order at three visible nodes and bounds it at four nodes, showing that most equivalence classes require non-CI constraints.","lead":"This paper maps which causal structures with hidden variables can be told apart from observational data alone, giving a complete classification for three visible variables and partial results for four. It also shows that inequality constraints, analogous to Bell inequalities, become the norm as the number of variables grows, so algorithms that only use conditional independence tests miss most distinctions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"3-node closure and 4-node lower bounds rest on Fraser's support-realizability algorithm; a flawed direct proof for support S' in Sec. 6.2 makes this external dependence concrete and uncorrected.","rationale":"The paper's main contribution is the complete 3-node order and the partial 4-node characterization with precise counts. The 3-node order is 'complete' only if every pair of classes not joined by a proven dominance is known to be non-dominating; the final split between Instrumental BAC and Instrumental CAB is one such pair. The direct proof of that split is demonstrably wrong, and no other direct argument is given in the text, so the correctness of the 3-node closure currently inherits from an unverified external algorithm. The 4-node lower bound is even more directly a product of that algorithm. This is not an objection to using external computational tools; the code is available and the rules are sound. But the paper treats the algorithm as a black box, and the one place where it attempts to replace the black box with a hand proof contains a concrete error. An independent reimplementation is a cheap check that would either validate the counts or expose a need for corrected proofs. I therefore do not change the reader's conditional verdict: accept only after the algorithm is independently verified and the S' construction is corrected.","tokens_in":52236,"tokens_out":15908,"duration_ms":141538,"concrete_test":"Reimplement support realizability for 4 binary visible variables (e.g., by exhaustive search over deterministic response functions with latent cardinality up to 8, or via SAT/CP) and recompute the block count for 'DEF + DC + e-sep + Supps up to 8 events' in Table 4; verify it equals 1253. Separately, for the 3-node case, determine whether the stated support S' = {(1,0,0),(0,0,1),(0,1,1),(0,0,0)} is realizable by Instrumental CAB under any parameterization; if it is, supply a correct explicit construction, and if it is not, replace S' with a support that genuinely separates Instrumental BAC and Instrumental CAB and prove it directly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof that Instrumental BAC and Instrumental CAB are observationally inequivalent (Sec. 6.2) is needed to reach the 15-block proven-inequivalence partition that closes the 3-node case. The paper offers a direct proof by constructing a distribution with support S' = {(1,0,0),(0,0,1),(0,1,1),(0,0,0)} realizable by Instrumental CAB. But the stated functions Xa = Xλ·Xγ, Xb = Xλ·(Xa⊕1), Xc = Xγ generate the different support {(0,0,0),(0,0,1),(0,1,0),(1,0,1)}. Thus the written construction does not realize S', and the separation falls back on Fraser's algorithm (Sec. 5.2, Appendix B). For the 4-node results, the lower bound 1253 and the 1156 identified classes (Tables 4-5) are obtained by the same algorithm for binary supports up to 8 events, and the bound k≥s from Appendix B is cited from Ref. [39] without proof. If the algorithm misclassifies a support as unrealizable, the proven-inequivalence partition over-splits, so the claimed interval [1253,1444] and the ≥85.2% nonalgebraic fraction could be wrong. This is load-bearing because the central claim includes those exact counts.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the observational dominance partial order on mDAGs (marginalized DAGs) with latent variables, under a fixed ordering of the visible variables. It assembles known dominance- and nondominance-proving rules, adds two new facet-merging rules (Moderate and Strong Facet-Merging), and augments the graphical nondominance rules with a support-based criterion derived from Fraser's algorithm. The central claims are: for three visible nodes, the 72 mDAGs consistent with a fixed nodal ordering form exactly 15 observational equivalence classes whose dominance relations are completely determined; for four visible nodes, the number of classes lies between 1253 and 1444, with 1156 classes fully identified. From these results the paper derives lower bounds on the prevalence of nonalgebraic classes (at least 85.2% for four nodes) and argues that conditional-independence constraints alone identify fewer than 10% of four-node classes, so that richer constraints are needed for causal discovery.","tokens_in":52503,"tokens_out":18459,"duration_ms":168530,"significance":"If the results are correct, the complete 3-node observational partial order and the partially characterized 4-node order are valuable reference points for causal discovery with latent variables, and the evidence that nonalgebraic classes are generic strengthens the case for going beyond conditional-independence constraints. The paper's care in distinguishing proven-equivalence from proven-inequivalence partitions, its systematic application of existing rules, and its public code repository are explicit strengths. The two new facet-merging rules are proved in the appendices and are likely to be useful beyond the specific classification. However, the classification leans heavily on an external computational algorithm, and one of the paper's own written proofs of a load-bearing support separation is incorrect; these issues must be fixed before the central claims can be accepted as established.","major_comments":[{"comment":"The direct proof that Instrumental CAB realizes the support S' is incorrect. With Xa = Xlambda*Xgamma, Xb = Xlambda*(Xa XOR 1), Xc = Xgamma, the four binary assignments of (Xlambda, Xgamma) give the support {(0,0,0),(0,0,1),(0,1,0),(1,0,1)}, not S' = {(1,0,0),(0,0,1),(0,1,1),(0,0,0)}. Since the inequivalence between Instrumental BAC and Instrumental CAB is required to close the 3-node classification, the written proof fails at a load-bearing point. The separation may still be true (indeed, S' is realizable by Instrumental CAB under a different parametrization), but the text must either supply a correct explicit construction or state plainly that this inequivalence rests on Fraser's algorithm and the associated computation.","section":"Sec. 6.2"},{"comment":"The 3-node closure, the 4-node lower bound of 1253, the 1156 identified classes, and the derived 85.2% nonalgebraic fraction all depend on Fraser's algorithm for deciding which supports are realizable, including the bound k >= s on latent cardinalities. This algorithm and bound are cited from Ref. [39] and are not proved in the manuscript. Because a misclassification of a single support would over-split the proven-inequivalence partition and invalidate the interval [1253,1444], the paper should either include a self-contained proof of the algorithm's correctness and of the latent-cardinality bound, or state them as explicit theorems with precise references and make the computational verification (e.g., the repository scripts used to generate Tables 4 and 5) fully reproducible.","section":"Sec. 5.2 and Appendix B"},{"comment":"For three binary visible variables, a support can contain up to 8 events, yet Table 2 and the text report comparisons only up to 4-event supports. The paper does not explain why checking supports of size at most 4 is sufficient to certify the absence of any further inequivalences. If supports of size 5-8 were not checked, the claimed completeness of the 3-node classification is not established. Please either report that all 2^8 possible supports were checked, or provide a reduction argument showing that any separating support of size greater than 4 would imply a separating support of size at most 4.","section":"Sec. 6.2 and Table 2"}],"minor_comments":[{"comment":"The conclusion inverts the conditional-independence statistics: the text says 'for 4-node mDAGs ... 2/3 ... for 3-node mDAGs ... less than 10%', but Section 7.2.2 correctly reports 10/15 = 2/3 for three nodes and fewer than 10% for four nodes. This should be corrected.","section":"Sec. 8"},{"comment":"The text states that the best lower bound on the number of 4-node classes is 1256, while Table 4 and the subsequent calculation use 1253. These numbers must be reconciled.","section":"Sec. 7"},{"comment":"There is a typo in the paragraph after Lemma 7: 'it is possible to do find such a support' should read 'it is possible to find such a support'.","section":"Sec. 5.2.1"},{"comment":"The sentence 'We infern that Evans is an observational equivalence class' contains a typo ('inf ernn' should be 'infer').","section":"Sec. 6.2"},{"comment":"The caption says 'Classification of the different nondominance-proving rules' for what appears to be the dominance-proving rules diagram; the two captions in Fig. 4.1 and Fig. 5.1 appear to be interchanged or duplicated.","section":"Fig. 4.1 caption"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a question of genuine interest and the overall framework is well organized. The direct proof error in Sec. 6.2 is localized and correctable, and the reliance on Fraser's algorithm is a standard computational tool, but the manuscript needs to make the computational provenance of the key counts explicit and verifiable before publication. I do not see grounds for rejection if these issues are resolved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline: this is a real step forward for the 3-node case, and the two new facet-merging rules are original and worth knowing. The 4-node counts are plausible but not fully load-bearing; they depend on an external algorithm, and one of the direct proofs in Sec. 6.2 is wrong as written.\n\nWhat is new: the complete observational partial order for 3-node mDAGs (15 classes) is genuinely new. Ref. [24] had a partial equivalence partition but not the dominance order. The Moderate and Strong Facet-Merging rules are new, and their proofs go through existing lemmas rather than being assumed. Counting at the level of observational equivalence classes, rather than individual mDAGs, is the right way to quantify how generic inequality constraints become. The paper is also honest about what is proven versus merely identified, and the code is available.\n\nSoft spots. First, the direct construction for support S' in Sec. 6.2 does not generate the support claimed: the stated functions produce {(0,0,0),(0,0,1),(0,1,0),(1,0,1)}, not S'. That is a concrete error in the written proof, and it means the separation of Instrumental BAC from Instrumental CAB falls back on Fraser's algorithm as presented. Second, the conclusion swaps two numbers: it says 2/3 of classes are identified by CI alone for 4-node and less than 10% for 3-node; the correct order is the reverse. Minor, but it needs fixing.\n\nThe larger issue is the reliance on Fraser's algorithm. The 3-node closure and the 4-node lower bounds (1253 proven-inequivalence blocks, 1156 identified classes) are obtained by this external algorithm for binary supports up to 8 events. The paper assumes its correctness, and the k ≥ s bound is cited from Ref. [39] without proof. If the algorithm misclassifies a support as realizable, the proven-inequivalence partition over-splits and the interval [1253,1444] and the ≥ 85.2% nonalgebraic fraction shift. This is not a refutation—the algorithm is published—but it makes the exact 4-node counts conditional on external code until someone checks them independently.\n\nBottom line: the 3-node classification is likely correct, and the new rules are solid. The Sec. 6.2 construction and the swapped statistics are easy repairs. The 4-node counts should be read as conditional on Fraser's algorithm. The paper deserves a serious referee; I would ask the referee to verify the Sec. 6.2 correction and to sample-check the support computations rather than trust the counts wholesale.","headline":"A genuine advance on the 3-node observational partial order and two new facet-merging rules, but the 4-node counts rest on an external algorithm and one direct proof in Sec. 6.2 is wrong as written.","tokens_in":53047,"tokens_out":2197,"would_cite":true,"duration_ms":23688,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62D20","05C20","68T37"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper gives a complete observational dominance order for causal structures with latent variables on three visible variables and a partial order for four.","keywords":["observational dominance","causal discovery","latent variables","mDAGs","observational equivalence","inequality constraints","partial order","support realizability"],"falsifier":"Take the support set with three binary events that the paper uses to separate two of the three-node classes and independently search for a latent-variable model realizing it, allowing latent cardinalities beyond the enumeration's bound; a successful construction would invalidate the claimed three-node nondominance and with it the completeness of the fifteen-class order.","tokens_in":52038,"feed_emoji":"🧩","tokens_out":10625,"duration_ms":94807,"temperature":0.7,"pith_summary":"The paper asks which causal structures with hidden variables can be told apart from observational data alone, and in what direction their explanatory power runs. It defines observational dominance as containment of the sets of realizable distributions over the visible variables, so a structure that can explain more distributions dominates one that can explain fewer. The paper's main result is a complete map of this dominance order for all causal structures on three visible variables (fifteen equivalence classes) and a partial map for four visible variables (between 1,253 and 1,444 classes, with 1,156 fully identified). Along the way it proves two new rules for showing when adding a common-cause facet does not change the observational profile, and it shows that a brute-force support-comparison method subsumes all previously known graphical tests for non-dominance. This matters for causal discovery because it tells practitioners when conditional-independence tests are too weak: for four variables, fewer than 10 percent of classes can be singled out by conditional independence alone, and at least 85 percent of classes obey nontrivial inequality constraints.","feed_headline":"Causal structure order solved for three visible variables","feed_subtitle":"Four-variable map is bracketed between 1,253 and 1,444 classes; inequality constraints dominate.","key_machinery":"The workhorse is the mDAG (marginalized directed acyclic graph): a DAG whose visible nodes carry directed edges for direct causation together with a simplicial complex of faces recording which visible subsets share an unobserved common cause. The observational profile of an mDAG is the tuple, over all cardinalities of the visible variables, of the sets of distributions it can realize; dominance is elementwise set inclusion of these profiles. Dominance is established by structural edge and facet additions plus two new facet-merging rules (Moderate and Strong), which show when merging common-cause facets preserves or extends the realizable-distribution set. Nondominance is decided hierarchically: skeleton comparison, d-separation, e-separation, densely connected node pairs, the directed-edge-free rule, and finally comparison of unrealizable supports, where an enumeration of response functions decides which support sets are realizable. A key structural result is that the support-comparison step subsumes all the cheaper graphical nondominance rules, so the remaining gaps in the four-node case are purely about support sets at higher cardinalities.","core_discovery":"The central claim is that the observational partial order of latent-permitting causal structures is accessible in full for three visible variables and in large part for four, when one fixes the temporal ordering of the visible nodes. For three variables, the 72 candidate graphs collapse into 15 observational equivalence classes, and every dominance relation among them is identified, yielding the first complete such order. For four variables, the paper brackets the number of equivalence classes between 1,253 and 1,444 and completely identifies 1,156 of them, leaving the rest as well-defined open cases. The paper further claims that nonalgebraic classes, meaning those whose realizable-distribution sets are cut out by inequalities rather than equalities alone, are the majority: at least 85.2 percent of four-variable classes, up from one third at three variables. It also claims that conditional-independence relations alone identify fewer than 10 percent of four-variable classes, establishing that richer constraints are needed for causal discovery.","pith_inferences":["Beyond the paper, the monotone trend in the fraction of nonalgebraic classes suggests that for five or more visible variables nearly every observational equivalence class will carry inequality constraints; this is an extrapolation, not a result in the paper.","The seven mDAGs that realize every binary support but may not be saturated form a natural test bed for whether support-level equality implies distribution-level equality; a single counterexample would separate possibilistic from probabilistic causal compatibility.","The facet-merging rules point toward a combinatorial closure operation on facets that might decide observational equivalence for arbitrary node counts; the paper does not claim such an operation exists.","For quantum causation, the ubiquity of nonalgebraic classes implies that new quantum-classical gaps should be sought in generic four-variable networks rather than specially designed ones; this is an implication the paper only gestures at."],"forward_implications":["For three visible variables, the complete fifteen-class order means any new causal-compatibility constraint found for one class automatically transfers to its class, and realizability of a distribution can be propagated up or down the order.","For four variables, conditional-independence-based algorithms can single out fewer than 10 percent of classes, so nested constraints and inequality constraints are not optional extras for causal discovery.","With at least 85.2 percent of four-variable classes nonalgebraic, most causal structures can in principle show quantum-classical gaps, since such gaps require inequality constraints.","The two new facet-merging rules tighten the upper bound on four-node classes from 1,481 to 1,444, showing that previously known equivalence rules missed real equivalences.","Because support comparison subsumes all graphical nondominance rules, the unresolved four-node cases are isolated to finite computational searches over supports, not to missing graphical criteria."],"supporting_citations":[{"why":"Defines mDAGs, establishes that structural dominance implies observational dominance, and supplies the exogenization and redundant-latent-node reductions used to form equivalence classes.","marker":"[24]"},{"why":"Supplies the edge-adding rule used to prove many observational equivalences among three-node and four-node mDAGs.","marker":"[6]"},{"why":"Gives the algorithm that enumerates realizable supports; it underlies all support-comparison nondominance results and the four-node lower bounds.","marker":"[39]"},{"why":"Classifies causal structures with inequality constraints and supplies the trinary support examples reused for the non-saturated four-node classes.","marker":"[8]"},{"why":"Proves an mDAG is algebraic if and only if it is observationally equivalent to a confounder-free mDAG, used to bound the number of algebraic classes.","marker":"[33]"},{"why":"Introduces e-separation and the graphical inequality constraints that support comparison later subsumes.","marker":"[34]"},{"why":"Establishes the perfect-correlation criterion for densely connected nodes used as a nondominance rule.","marker":"[36]"},{"why":"Provides the instrumental inequalities used to certify the specific three-event supports separating instrumental classes from the other three-node class.","marker":"[20]"},{"why":"Shows perfect correlation requires a common ancestor, supporting the directed-edge-free nondominance rule.","marker":"[29]"}],"fun_headline_variants":["Causal dominance order fully mapped for three visible variables","Complete observational order for three-variable causal structures","Four-variable causal order: 1,253–1,444 classes, inequality constraints dominate","Conditional independence alone underperforms in causal discovery","Inequality constraints now crucial for causal structure inference"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything the paper concludes about non-dominance presupposes that the combinatorial enumeration of realizable supports, including its bound on how large the hidden variables need to be, is correct and complete; if that enumeration misses a support, the claimed three-node classification and the four-node lower bounds would not be established.","fun_headline_variants_meta":{"raw":{"variants":["Causal dominance order fully mapped for three visible variables","Complete observational order for three-variable causal structures","Four-variable causal order: 1,253–1,444 classes, inequality constraints dominate","Conditional independence alone underperforms in causal discovery","Inequality constraints now crucial for causal structure inference"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000354,"raw_usage":{"total_tokens":1941,"prompt_tokens":978,"completion_tokens":963,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":594,"completion_tokens_details":{"reasoning_tokens":879}},"tokens_in":594,"tokens_out":963,"duration_ms":9208,"temperature":1.0,"reasoning_tokens":879,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T11:31:25.640027+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the support set with three binary events that the paper uses to separate two of the three-node classes and independently search for a latent-variable model realizing it, allowing latent cardinalities beyond the enumeration's bound; a successful construction would invalidate the claimed three-node nondominance and with it the completeness of the fifteen-class order.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines mDAGs, establishes that structural dominance implies observational dominance, and supplies the exogenization and redundant-latent-node reductions used to form equivalence classes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the edge-adding rule used to prove many observational equivalences among three-node and four-node mDAGs."},{"cited_title":"A Combinatorial Solution to Causal Compatibility","cited_arxiv_id":null,"evidence_quote":"Gives the algorithm that enumerates realizable supports; it underlies all support-comparison nondominance results and the four-node lower bounds."},{"cited_title":"Pusey, and Elie Wolfe","cited_arxiv_id":null,"evidence_quote":"Classifies causal structures with inequality constraints and supplies the trinary support examples reused for the non-saturated four-node classes."},{"cited_title":"Latent-free equivalent mDAGs","cited_arxiv_id":"2209.06534","evidence_quote":"Proves an mDAG is algebraic if and only if it is observationally equivalent to a confounder-free mDAG, used to bound the number of algebraic classes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces e-separation and the graphical inequality constraints that support comparison later subsumes."},{"cited_title":"Dependency in DAG models with Hidden Variables","cited_arxiv_id":"2106.07523","evidence_quote":"Establishes the perfect-correlation criterion for densely connected nodes used as a nondominance rule."},{"cited_title":"On the Testability of Causal Mod- els with Latent and Instrumental Variables","cited_arxiv_id":null,"evidence_quote":"Provides the instrumental inequalities used to certify the specific three-event supports separating instrumental classes from the other three-node class."},{"cited_title":"Information- theoretic Inference of Common Ancestors.En- tropy, 17(4):2304–2327, 2015","cited_arxiv_id":null,"evidence_quote":"Shows perfect correlation requires a common ancestor, supporting the directed-edge-free nondominance rule."}],"review_version":1}