Correction to Greedy Conditioning Lemma in Quantum Parallel Repetition Proof

An arXiv preprint (2608.14673) has recently been released, addressing a polarity mistake in a proof from OpenAI's *Ten Advances in Mathematics and Theoretical Computer Science*. Chapter 6 of the original proof asserts an exponential parallel-repetition theorem applicable to all finite two-player, one-round entangled games. The identified error lies within a quantitative greedy conditioning lemma that aims to choose a limited set of coordinates, ensuring that winning those coordinates leads to a randomly selected remaining coordinate being won with an average probability of at least (1-δ). Although the lemma is correctly stated, the proof contains a polarity mistake: the continuation test is framed in terms of average success, while the subsequent step necessitates a coordinate with a high conditional failure probability. This leads to a false implication, and straightforward examples can result in the procedure lacking a valid next move. The note details a specific counterexample, clarifies the intended continuation condition, and offers a complete corrected proof. Authored by an anonymous researcher, the preprint was published on arXiv on August 26, 2025. This correction holds considerable importance for quantum information and computational complexity, reinforcing the validity of the exponential parallel-repetition theorem, which is crucial for understanding the capabilities of entangled games and quantum algorithms.