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Phys-54 · physics · mathematics · Physics

An exponential threshold-repetition bound for arbitrary finite entangled games

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every finite two-player one-round Boolean-acceptance game G with finite-dimensional entangled value $v<1$, is there universal $\kappa _0>0$ such that any finite-dimensional joint k-repetition strategy wins at least ceil((v+delta)k) coordinates with probability at most $\exp (-\kappa _0$ $\delta ^5$ k/(1+log d)), for $0<\delta <1-v$ and d the source answer-alphabet parameter? Correlated question distributions are allowed.

Short English statement

For every finite two-player one-round Boolean-acceptance game G with finite-dimensional entangled value $v<1$, is there universal $\kappa _0>0$ such that any finite-dimensional joint k-repetition strategy wins at least ceil((v+delta)k) coordinates with probability at most $\exp (-\kappa _0$ $\delta ^5$ k/(1+log d)), for $0<\delta <1-v$ and d the source answer-alphabet parameter? Correlated question distributions are allowed.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-54-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-54

Entangled-game repetitions are claimed to have exponentially small probability of winning above the single-game value by any fixed gap, for arbitrary question correlations and finite-dimensional joint strategies. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

An exponential threshold-repetition bound for arbitrary finite entangled games

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Entangled-game repetitions are claimed to have exponentially small probability of winning above the single-game value by any fixed gap, for arbitrary question correlations and finite-dimensional joint strategies.

Mathematical statement of the claimed result

Result 1.
Consider a finite game $G$ with two players, one simultaneous answer round and a Boolean acceptance rule. Its entangled value is $v=\omega^*(G)<1$, the supremum over finite-dimensional strategies. Put $d=|\mathcal A||\mathcal B|$, the product of answer-set sizes. For $k$ independently sampled question pairs and any joint finite-dimensional strategy, let $W_k$ count accepted coordinates. The claimed universal $\kappa_0>0$ gives \[ \mathbb P[W_k\ge\lceil(v+\delta)k\rceil] \le\exp\!\left[-\frac{\kappa_0\delta^5}{1+\log d}k\right] \qquad(0<\delta<1-v,\ k\ge1). \] The same estimate holds after taking the supremum over all finite-dimensional joint strategies.

Outline of the argument

Conditioning on a tested subset is represented by one common purification of effects. Information and error amplification bounds convert excessive threshold success into a better single-game strategy; universal estimates give the fifth-power gap rate and fixed question laws permit a cubic refinement.

Formal verification

For family 277, the documented exponential repetition rate has a $\delta^{13}$ factor, rather than the manuscript improved $\delta^5$ rate; the distribution-dependent cubic refinement is also not selected. The listed Comparator files are EntangledGames.lean. Named declarations include OAI.ThresholdParallelRepetition.threshold_parallel_repetition. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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