{ "public_id": "TA-2026-003", "version": "1.0", "released": "2026-11-07", "status": "published", "sample": false, "kind": "defect", "track": "game_theory", "title": "Balbus et al. (2019)", "headline": "Theorem 1(i) is false under the order assumptions as printed, and the appendix contains an independent function-space defect.", "work": { "authors": [ "Łukasz Balbus", "Paweł Dziewulski", "Kevin Reffett", "Łukasz Woźny" ], "title": "A Qualitative Theory of Large Games with Strategic Complementarities", "venue": "Economic Theory", "year": 2019, "volume": "67", "pages": "497–523", "doi": "10.1007/s00199-017-1075-7" }, "claims": [ { "id": "c1", "label": "Theorem 1(i)", "kind": "theorem", "role": "main", "page": "503–504", "gist": "Under Assumption 1, greatest and least distributional equilibria exist.", "statement": "Under Assumption 1, there exists a greatest and a least distributional equilibrium of the game.", "conditions": [ "compact perfect Hausdorff type space with a regular singleton-null probability measure", "weakly measurable complete-sublattice-valued feasible actions", "the printed partial-order condition that pair comparison implies action comparison", "quasisupermodularity, single-crossing differences and the stated Carathéodory condition" ], "quantifiers": "Every game admitted by the standing framework and Assumption 1.", "result": { "outcome": "defect_found", "candidate_status": "confirmed", "defect_class": "statement_false", "affects": "main_result", "depth": "deep_audit", "summary": "The literal one-way order condition admits a game satisfying Assumption 1 with no distributional equilibrium.", "witness": "Take types [0,1] with Lebesgue probability, actions {0,1}, a constant feasible-action set, and equality as the partial order on type-action pairs. Let p be the mass choosing action 1. Make action 1 uniquely optimal when p<1/2 and action 0 uniquely optimal when p≥1/2. Equality is permitted by the printed one-way order condition and makes the induced stochastic order equality, so the single-crossing condition is vacuous. Any equilibrium with p<1/2 would force p=1; any equilibrium with p≥1/2 would force p=0. Hence no distributional equilibrium exists.", "tested": "The exact assumption matrix, the induced stochastic order, strict best replies and the equilibrium contradiction were checked in a separate informed hostile validation. A strengthened four-action witness also defeats the possible rescue that greatest/least feasible distributions or chain completeness were intended as additional premises.", "not_claimed": [ "The record does not claim that the authors' intended stronger order condition is false.", "Applications that independently impose a product or same-type action order are not claimed to fail.", "No downstream paper is claimed to be incorrect without checking its own assumptions." ], "repair": "local_repair_proposed", "repair_note": "A same-type fiber condition requiring action order to imply pair order excludes the counterexample and supplies a comparison used in the proof. The appendix chain-completeness argument still needs a separate valid justification." }, "vor_check": { "level": "partial", "pages": "503–504", "detail": "The canonical source lock is partial because the historical decisive check used extracted final publisher text rather than a complete rendered-page verification. The theorem wording and order assumptions were rechecked on the publisher-hosted article during release preparation." }, "checks": [ { "kind": "ai_recheck", "date": "2026-10", "result": "confirms", "detail": "A separate informed hostile-validation session reconstructed the assumptions, tested rescue attempts and confirmed the exact counterexample.", "session": "informed" }, { "kind": "private_human_confirmation", "date": "2026-10", "result": "confirms", "detail": "A qualifying human confirmation is recorded in the private research file. Its source and correspondence are not disclosed." } ], "novelty": null }, { "id": "c2", "label": "Lemma A1 proof: function-space step", "kind": "lemma", "role": "auxiliary", "page": "519", "gist": "The proof treats M(X)-M(X), where M(X) contains bounded measurable isotone functions, as a Riesz subspace of C(X).", "statement": "In the proof of Lemma A1, M(X)-M(X) is asserted to be a Riesz subspace of the continuous functions C(X).", "conditions": [ "X is a compact Hausdorff partially ordered space", "M(X) is defined as the bounded measurable isotone real-valued functions on X" ], "quantifiers": "The appendix argument is presented for the stated general class of compact ordered spaces and regular measures.", "result": { "outcome": "defect_found", "candidate_status": "confirmed", "defect_class": "missing_or_hidden_assumption", "affects": "lemma_or_proof_step", "depth": "deep_audit", "summary": "Measurability and isotonicity do not imply continuity, so the asserted inclusion in C(X) does not follow.", "witness": "Under the equality order, every bounded measurable function is isotone. Discontinuous indicator functions therefore belong to M(X), so M(X) and M(X)-M(X) need not be contained in C(X). The Stone-Weierstrass step cannot be justified from the displayed definition as written.", "tested": "The appendix definitions and the claimed Riesz-subspace step were traced in the historical audit and independently attacked in the hostile validation.", "not_claimed": [ "This defect alone is not claimed to falsify the conclusion of Lemma A1.", "The record does not claim that no alternative proof of the relevant measure-order result exists.", "The record does not publish or rely on any private proposed correction." ], "repair": "local_repair_proposed", "repair_note": "The function class or the proof must be strengthened so that the functions used in the Stone-Weierstrass argument are continuous; a complete repair of the constrained-measure chain-completeness argument is not claimed here." }, "vor_check": { "level": "partial", "pages": "519", "detail": "The canonical source lock is partial because the historical check used extracted final publisher text. The publisher-hosted article was rechecked during release preparation and still defines M(X) as measurable and bounded before asserting the subspace inclusion." }, "checks": [ { "kind": "ai_recheck", "date": "2026-10", "result": "confirms", "detail": "A separate informed hostile-validation session confirmed the mismatch between the displayed definition of M(X) and the continuity required by the proof.", "session": "informed" }, { "kind": "private_human_confirmation", "date": "2026-10", "result": "confirms", "detail": "A qualifying human confirmation is recorded in the private research file. Its source and correspondence are not disclosed." } ], "novelty": null } ], "author_position": { "position": "not_disclosed", "summary": null }, "correction": null, "evidence": { "package_doi": null, "package_sha256": null, "licence": "No public evidence package released.", "files": [] }, "history": [ { "date": "2026-10", "type": "audit", "label": "Theorem-level counterexample and appendix defect identified and separately rechecked" }, { "date": "2026-10", "type": "source", "label": "Published-source text checked; canonical VOR level remains partial" }, { "date": "2026-11-07", "type": "release", "label": "Record version 1.0 released" } ], "challenges": [], "changelog": [ { "version": "1.0", "date": "2026-11-07", "change": "First release." } ], "statements": { "ai_use": "AI systems were used for discovery and separate hostile rechecking. Confirmation also includes a qualifying human check recorded privately. Private correspondence is not disclosed, and AI agreement is not treated as human verification." } }