The matrix multiplication exponent: is omega at most 9/4?

Theorem 1.1: for every $\varepsilon>0$ two complex $n\times n$ matrices can be multiplied with $O_\varepsilon(n^{9/4+\varepsilon})$ arithmetic operations, so $\omega\le9/4$ over $\mathbb C$ . The companions prove $\alpha>0.465$ over every field of characteristic zero, $\omega(1,0.709,1)<2.092$ , $\omega<2.258$ over all fields outside one uncomputed finite set of positive characteristics, and $\omega<2.371054886006746$ over every field. Not shown: anything about $\omega=2$ , bit complexity, or a practical crossover size; the 9/4 proof gives no competitive finite matrix size and its transfer to positive characteristic is not claimed.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Oct 2, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The matrix multiplication exponent: is omega at most 9/4? · Matrix multiplication exponent at most 9/4

Confidence: Not scored

Registry verification: lean checked · announcement · partial

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VibeMathed record: The matrix multiplication exponent: is omega at most 9/4?

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No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of 'An Upper Bound of 9/4 for the Matrix Multiplication Exponent', read against the exponent as usually defined (arithmetic operations over C, epsilon slack). The proof (a separation inequality for tensors sharing one leg, polynomial-multiplication inequalities and Strassen's asymptotic spectrum) was not refereed. Lean: formalization.yaml lists OAI.MatrixMultiplication.complex_omega_le_nine_quarters (lean/OAI/LinearAlgebra/MatrixMultiplication/Main.lean, comparator ComparatorChallenges/MatrixMultiplication.lean). Its statement was read: omega(C) <= 9/4, where omega is the infimum of exponents tau such that for every eps > 0 some constant C bounds the cost of correct division-free straight-line programs (add, sub, mul, constants) for n x n products by C n^(tau+eps) at every n. That is the headline claim. The same file states alpha > 93/200 and omega(1, 0.709, 1) < 523/250; OAI.MatrixAllFields...omega_lt_source_constant states omega(F) < 2.371054886006746 for every field F. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The lean/docs page for this family names only the two September 24 companions, while formalization.yaml lists the 9/4 paper as a source.