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CS-125 · computer science · mathematics · Algorithms and data structures · Computational algebra and number theory

Complex square and rectangular multiplication bounds

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Over C, do the arithmetic matrix-multiplication exponents satisfy $\omega <2.258$, the dual rectangular exponent $\alpha >0.465$, and $w(0.709)<2.092$, using exact arithmetic operations?

Short English statement

Over C, do the arithmetic matrix-multiplication exponents satisfy $\omega <2.258$, the dual rectangular exponent $\alpha >0.465$, and $w(0.709)<2.092$, using exact arithmetic operations?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-125-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-125

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).

Complex square and rectangular multiplication bounds

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 characteristic-zero matrix multiplication has square exponent below $2.258$, dual exponent above $0.465$, and rectangular exponent at aspect $0.709$ below $2.092$, with the stated finite-characteristic exceptions.

Mathematical statement of the claimed result

Manuscript 1.
In exact complex arithmetic, let $\omega$ be the square matrix-multiplication exponent, $w(k)$ the exponent for multiplying an $n\times n^k$ matrix by an $n^k\times n$ matrix, and $\alpha=\sup\{k:w(k)=2\}$. The claimed bounds are \[ \omega<1129/500=2.258,\qquad\alpha>0.465,\qquad w(0.709)<2.092. \] These are arithmetic exponents, rather than bit-complexity or finite-precision bounds.

Outline of the argument

For manuscript 1 (family 107; 2026-09-24), an auxiliary tensor supplies a conditional label to the missing third leg while retaining a dot-product block. Recursive completion trees and finite-family entropy optimization produce certified parameters; a strict comparison argument verifies the advertised boundaries.

Formal verification

For manuscript 1 (family 107; 2026-09-24), the documented Lean scope is the stated matrix-multiplication exponent estimates over $\mathbb C$, including the rectangular bounds; the declarations do not extend these estimates to all characteristic-zero fields or the claimed finite-characteristic range.

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

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