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CS-139-A1 · revision r1 · on CS-139

OpenAI claim summary for CS-139

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 126; 2026-10-05).

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Exponential semidefinite complexity for perfect matching
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Exponential semidefinite complexity for perfect matching

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 126; 2026-10-05), the asserted result is that each fixed positive shift below one in the matching slack matrix has exponential real PSD rank, implying exponential size for exact semidefinite lifts of the matching polytope.

Mathematical statement of the claimed result

Manuscript 1.
For even $n\ge4$, index rows by odd subsets $U\subseteq[n]$ and columns by perfect matchings $M$ of $K_n$, and define \[ A_n(\rho)_{U,M}=|M\cap\delta(U)|-1+\rho. \] Real PSD rank is the smallest $r$ permitting exact factorizations $A_{U,M}=\operatorname{tr}(F_UG_M)$ with real symmetric positive semidefinite $r\times r$ matrices. The asserted bound is \[ \exists c>0\ \forall\rho\in(0,1)\ \exists n_0(\rho)\ \forall\text{ even }n\ge n_0(\rho):\quad \operatorname{rank}_{\rm psd}A_n(\rho)\ge2^{cn}. \] The exponential constant is uniform in $\rho$, but the threshold may depend on it. No symmetry or precision restrictions are imposed on the factors.

Outline of the argument

For manuscript 1 (family 126; 2026-10-05), coordinatewise product averages are expressed as sums of matrix squares. Orthogonal frequency splitting bounds a Gram potential through the averaging process; matching smoothing and an expanding parity system force a rank large enough to support the remaining Fourier mass.

Formal verification

For manuscript 1 (family 126; 2026-10-05), the documented Lean scope is superpolynomial PSD rank for the selected matching slack matrix and superpolynomial size of exact lifts; the claimed exponential bound for every fixed positive shift is outside the selected statement.

This coverage does not establish a formal proof of the complete problem as stated here.

The linked family Comparator checks are MatchingAffineLift.lean, MatchingPSD.lean.

Their named declarations include OAI.PerfectMatchingPSD.affine_lift_lower_bound, OAI.PerfectMatchingPSD.main.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources