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Auditing an AI-Generated Mathematical Proof: A Correction to a Greedy Conditioning Lemma in Quantum Parallel Repetition
Mikołaj Sienicki, Krzysztof Sienicki
TL;DR
The note identifies a polarity error in a greedy conditioning proof that confuses average success with individual failure. It gives a counterexample, states the correct continuation condition, and proves the lemma after a local repair, without independently verifying the broader theorem.
Problem
The printed proof incorrectly infers a coordinate with conditional failure probability above δ from an average-success condition.
Method
The note supplies an explicit counterexample, identifies the intended continuation criterion, and gives a complete corrected greedy-conditioning proof.
Results
The lemma statement is valid, and after correcting the continuation condition, the greedy proof establishes the required average conditional success probability.
Takeaways & Limitations
The conditioning argument is repaired locally, but the remaining parallel-repetition proof requires separate specialist verification.
Takeaways & Limitations
The note does not independently verify the deeper parallel-repetition theorem or the later arguments in Chapter 6.
Abstract
from arXiv · showhide
Chapter 6 of OpenAI's *Ten Advances in Mathematics and Theoretical Computer Science* claims an exponential parallel-repetition theorem for all finite two-player, one-round entangled games. Early in the proof, the chapter uses a quantitative greedy conditioning lemma. The lemma is meant to select a small set of coordinates (D) such that, after conditioning on winning every coordinate in (D), a randomly chosen remaining coordinate is won with average probability at least (1-δ). The statement is correct, but the proof as printed contains a polarity error. Its continuation test is written in terms of average success, while the next step requires a coordinate with a large conditional failure probability. That implication is false, and even simple examples can leave the printed procedure without a valid next move. This note gives an explicit counterexample, identifies the intended continuation condition, and supplies a complete corrected proof. The repair is local: it leaves the statement of the lemma and the parameters used later in the chapter unchanged. It should not, however, be read as an independent verification of the main parallel-repetition theorem. More broadly, the example shows how a mathematically plausible AI-generated argument can hide a small but decisive reversal between complementary events.
1 Introduction
The note examines Chapter 6’s claimed exponential parallel-repetition theorem for finite two-player entangled games and focuses on a polarity error in its quantitative greedy conditioning lemma. It states that the lemma remains valid, provides a corrected proof, and limits its conclusions to this local repair rather than independently verifying the chapter’s full argument.
- Chapter 6 context: Chapter 6 claims exponential decay for repeated finite two-player, one-round entangled games whenever ω∗(G) = 1 − ε < 1.In G⊗n, the referee accepts only if the players win every one of n independent copies.
- Proof strategy: The proof uses conditioning and rounding alongside a postselection-stable quantum sampleability argument.Its preliminary Lemma 3.1 is the quantitative greedy conditioning lemma, which starts from an all-coordinate win probability ϑ > 0 and seeks a small conditioning set D.
- Central correction: The lemma’s statement is valid, but Appendix A.2 reverses success and failure in the continuation condition for the greedy procedure.The note isolates this inference, explains why it fails, states the correct condition, and proves the lemma in full.
- Scope and source: The repair is presented narrowly and does not independently verify the rest of Chapter 6’s later argument.The examined version is the official OpenAI PDF dated 1 August 2026, with the discussed passage in Chapter 6, Appendix A.2, on printed page 175.
2 The conditioning problem
This section formulates a quantitative conditioning problem for coordinate-winning events. Its lemma asserts that conditioning on a limited number of wins can make the average success probability of an unconditioned coordinate at least 1−δ.
- Setup: The framework represents coordinate wins by events W_1, . . . , W_n in a probability space and considers conditioning on subsets D of coordinates.The events W_i denote winning coordinate i.
- Setup: P(W_D) ≥ ϑ because W_[n] ⊆ W_D, so every conditional probability below is well defined when ϑ > 0.The positivity condition ϑ > 0 ensures valid conditioning.
- Lemma 1: The guarantee is that conditioning on winning only a limited number of coordinates leaves a uniformly chosen unconditioned coordinate won with average probability at least 1−δ.This is the lemma’s plain-language interpretation.
3 The defective continuation criterion
The printed proof contains a polarity error: its average-success continuation test does not imply the large conditional-failure coordinate required next. Replacing it with an average-failure test restores the averaging argument, likely correcting a dropped complement sign.
- The defective continuation criterion: The procedure starts with D = ∅ and continues using a condition expressed in terms of average success.The next instruction, however, requires choosing an index i /∈ D with conditional failure probability greater than δ.
- The defective continuation criterion: The required coordinate satisfies P(W_i | W_D) < 1 − δ, equivalently its conditional failure probability exceeds δ.This is the selection condition used by the subsequent step of the printed proof.
- The defective continuation criterion: Average success > δ does not imply that some coordinate has failure probability > δ.The printed proof therefore makes an invalid inference between complementary events.
- The defective continuation criterion: The continuation test should instead require average conditional failure probability > δ.Then averaging guarantees a remaining coordinate with conditional failure probability greater than δ, as required.
- The defective continuation criterion: The most natural correction is that a complement sign was dropped in the printed condition.The corrected condition should use the complement event W_i^c rather than W_i.
4 A counterexample to the printed procedure
A concrete probabilistic example exposes a fatal reversal in the printed greedy procedure: its continuation test is satisfied even though no valid next coordinate exists. The procedure should instead stop because its desired conclusion has already been reached, and the issue does not depend on independence.
- 4 A counterexample to the printed procedure: n = 2 and δ = 0.1 with P(E) = 0.95 gives an explicit counterexample to the printed procedure.The example uses an event E with probability 0.95.
- 4 A counterexample to the printed procedure: At D = ∅, the average success probability is 0.95, so the printed continuation condition is satisfied.The procedure therefore continues from the empty selected set.
- 4 A counterexample to the printed procedure: No coordinate satisfies the procedure’s next instruction, so the printed algorithm has no valid next move.This failure occurs for both coordinates.
- 4 A counterexample to the printed procedure: The procedure should already have stopped because its desired conclusion has already been reached.The printed test forces continuation at exactly the point where stopping is appropriate.
- 4 A counterexample to the printed procedure: The counterexample does not rely on independence because its purpose is only to refute the probabilistic implication used in the printed greedy step.The lemma permits win events generated by an arbitrary repeated-game strategy, so the events need not be independent.
5 Corrected statement and proof
The corrected proof constructs a proper subset D of coordinates satisfying all three conclusions in Equation (1). It ensures the conditioning quantity remains well defined, decreases sufficiently when coordinates are added, and therefore forces termination before all n coordinates are selected.
- Corrected statement and proof: The lemma guarantees a subset D ⊊ [n] satisfying all three conclusions in Equation (1).
- Corrected statement and proof: The construction maintains q_D as well defined at every stage.
- Corrected statement and proof: Each added coordinate reduces the conditioning-event probability by a factor strictly smaller than 1 − δ.
- Corrected statement and proof: Because the probability decreases multiplicatively, the procedure cannot add all n coordinates, so D remains a proper subset of [n].
6 Comparison with the printed proof
The comparison finds that nearly all of Appendix A.2 remains valid, with one localized defect in the initial continuation test. Replacing Equation (2) with Equation (5) repairs the proof without changing the lemma’s statement.
- Most of Appendix A.2 survives unchanged.
- The multiplicative decrease in Equation (4) and the logarithmic estimate in Equation (8) are correct.
- The only defect is the initial continuation test, which is repaired by replacing Equation (2) with Equation (5).The lemma’s statement requires no change.
7 Effect on the parallel-repetition argument
The corrected conditioning proof preserves the lemma’s conclusions and the downstream definitions and estimates involving p, q, and η. The repair is local and does not by itself refute the main theorem, while later proof components still require separate verification.
- Downstream use: The conditioning lemma is used to control |D| and retain a lower bound needed by the later argument.The chapter also introduces η as the information cost per remaining coordinate.
- Repair: The corrected proof delivers exactly the lemma’s conclusions, preserves the bound on |D|, and leaves later definitions and estimates involving p, q, and η intact.It need not select the same D as the printed procedure, which can fail to specify a valid next step.
- Limitations: The local error does not by itself refute the main theorem, but the conditioning argument must be repaired before the rest of the proof can be assessed.Sampleability, correlated sampling, state alignment, and rounding remain separate questions requiring specialist verification.
8 Relevance to the auditing of AI-generated mathematics
The example shows that a fluent, plausible proof can fail because a single condition reverses complementary events. Careful auditing must verify that each hypothesis guarantees the object or inequality used next, including in elementary probabilistic lemmas.
- Core auditing lesson: The proof’s elementary error is a reversal between the complementary events W_i and W_i^c.The issue is not a difficult operator inequality or subtle quantum-mechanical argument.
- Core auditing lesson: A displayed continuation condition conflicts with the intended procedure of selecting substantial failure, conditioning on a win, and stopping at high average success.The surrounding argument sounds convincing, but its stated test does not support the next step.
- Core auditing lesson: A careful audit must check whether each stated hypothesis guarantees the selected object or subsequent inequality, especially when average success, average failure, and individual failure differ.Placing these quantities side by side exposes the mismatch.
- Core auditing lesson: The two-coordinate counterexample shows that a plausible proof template can hide a local reversal that invalidates the procedure as written, without supporting broad claims about AI-generated proofs.Testing an elementary probabilistic lemma outside its advanced setting reveals the limited scope of the conclusion.
9 Conclusion
The greedy conditioning lemma is correctly stated but incorrectly proved because its continuation test uses the wrong polarity. Replacing that test with the appropriate failure-probability criterion repairs the proof locally without independently verifying the deeper theorem.
- Proof correction: The lemma’s statement is correct, but the published proof incorrectly infers a conditional failure probability above δ from Equation (2).Equation (5) supplies the appropriate continuation criterion.
- Proof correction: With Equation (5), each added coordinate reduces the conditioning-event probability by a factor strictly below 1 −δ while that probability remains at least ϑ.This bounds the number of additions by Equation (8) and yields the required average conditional success probability.
- Scope of repair: The repair is local: it preserves the lemma’s statement and the quantitative parameters used later in Chapter 6.It removes the identified obstruction from the chapter’s argument.
- Scope of repair: The correction should not be mistaken for an independent verification of the deeper parallel-repetition theorem.It addresses only the particular obstruction in the Chapter 6 argument.
- Broader lesson: The example illustrates that sophisticated AI-generated mathematics can contain small, local mistakes resembling familiar human errors.The broader gap between advanced AI-generated mathematics and ordinary human mathematical practice is becoming harder to locate.