A Qualitative Theory of Large Games with Strategic Complementarities (TA-2026-003)

2 claims investigated in this paper

1	Theorem 1(i)Theorem · main result	False as stated	AIPrivate humanpartial source check
2	Lemma A1 proof: function-space stepLemma · auxiliary result	Missing assumption	AIPrivate humanpartial source check

Claim 1 of 2 · theorem · main result · p. 503–504

Theorem 1(i)

False as stated

The literal one-way order condition admits a game satisfying Assumption 1 with no distributional equilibrium.

Theorem 1(i). Under Assumption 1, there exists a greatest and a least distributional equilibrium of the game.

Assumptions: 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
Scope of the claim: Every game admitted by the standing framework and Assumption 1.

What was found

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.

What was 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.

Does not establish

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. 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.

Verification

Separate AI recheckOct 2026 · informed. A separate informed hostile-validation session reconstructed the assumptions, tested rescue attempts and confirmed the exact counterexample.
Reproduced by the researcherNot done.
External human verificationNot done.
Written author agreementNot given.
Qualifying human confirmation (private)Oct 2026. A qualifying human confirmation is recorded in the private research file. Its source and correspondence are not disclosed.

Confirmed: 2 kinds of check, 1 by a human. The standard requires two kinds, at least one by a human.

Source

Partial check, pp. 503–504. 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.

Claim 2 of 2 · lemma · auxiliary result · p. 519

Lemma A1 proof: function-space step

Missing assumption

Measurability and isotonicity do not imply continuity, so the asserted inclusion in C(X) does not follow.

Lemma A1 proof: function-space step. In the proof of Lemma A1, M(X)-M(X) is asserted to be a Riesz subspace of the continuous functions C(X).

Assumptions: X is a compact Hausdorff partially ordered space; M(X) is defined as the bounded measurable isotone real-valued functions on X
Scope of the claim: The appendix argument is presented for the stated general class of compact ordered spaces and regular measures.

What was found

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.

What was tested

The appendix definitions and the claimed Riesz-subspace step were traced in the historical audit and independently attacked in the hostile validation.

Does not establish

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. 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.

Verification

Separate AI recheckOct 2026 · informed. A separate informed hostile-validation session confirmed the mismatch between the displayed definition of M(X) and the continuity required by the proof.
Reproduced by the researcherNot done.
External human verificationNot done.
Written author agreementNot given.
Qualifying human confirmation (private)Oct 2026. A qualifying human confirmation is recorded in the private research file. Its source and correspondence are not disclosed.

Confirmed: 2 kinds of check, 1 by a human. The standard requires two kinds, at least one by a human.

Source

Partial check, pp. 519. 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.

Authors and corrections

Authors: Not disclosed.
Correction: None published.

History

October 2026Theorem-level counterexample and appendix defect identified and separately rechecked
October 2026Published-source text checked; canonical VOR level remains partial
7 November 2026Record version 1.0 released

Versions

Version 1.07 November 2026First release.

Released versions are never edited; each keeps its own address.

Use of AI in this case

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.