we-solve (beta)

CS-119-A1 · revision r1 · on CS-119

OpenAI claim summary for CS-119

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 107; 2026-09-24). The manuscript also gives additional strength beyond the requested conclusion; that excess is not required for this problem's solved-claim label.

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

Manuscript

Contents
  1. Matrix-multiplication improvement over every field
  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

Matrix-multiplication improvement over every field

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 107; 2026-09-24), the asserted result is that the square multiplication exponent is below $2.371054886006746$ over every fixed field, including every positive characteristic.

Mathematical statement of the claimed result

Manuscript 1.
For every fixed field $F$, the claimed arithmetic exponent of square matrix multiplication satisfies \[ \omega_F<2.371054886006746<2.371054887. \] The exponent is the infimum of asymptotic powers achievable by exact field operations, allowing an arbitrarily small additive exponent slack. Positive and zero characteristics are both included.

Outline of the argument

For manuscript 1 (family 107; 2026-09-24), a joint extraction from several tensor depths uses entropy and a common modular hash. Sequential deletion assigns variables compatibly, and exact integer tables repair inherited masks; the resulting direct sum is converted to an exponent bound by the rank comparison.

Formal verification

For manuscript 1 (family 107; 2026-09-24), OpenAI lists Lean statements for this assertion: the square multiplication exponent is below $2.371054886006746$ over every fixed field, including every positive characteristic.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are MatrixFields.lean, MatrixMultiplication.lean.

Their named declarations include OAI.MatrixAllFields.MatrixMultiplication.AllFieldMain.omega_lt_source_constant, OAI.MatrixMultiplication.complex_alpha_gt_93_div_200, OAI.MatrixMultiplication.complex_omega_le_nine_quarters, OAI.MatrixMultiplication.complex_rectangular_omega_lt_523_div_250.

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