Three Inequivalent Ternary Rank-144 Decompositions of ⟨4,7,7⟩ over ℤ

Abstract

We report three mutually inequivalent exact algorithms for the product of a 4×7 by a 7×7 matrix, each using 144 multiplications in which every coefficient is −1, 0 or +1. Rank 144 is the best rank published for ⟨4,7,7⟩ in any coefficient domain; the rank-144 scheme in the public catalogue carries coefficients up to absolute value 7. To the best of our knowledge these are the first ternary decompositions at the record rank for this format over ℤ. Their de Groote profiles are pairwise distinct and none of them matches the catalogued solution, so the three schemes are inequivalent to one another and to the published entry under the sandwiching action of GL4(ℤ)×GL7(ℤ)×GL7(ℤ): three new orbits, not rescalings of an existing solution. The separation is structural: the catalogued entry carries atoms of types (1,2,4) and (1,2,5), and no scheme presented here has either. Each scheme reconstructs all 38 416 entries of the matrix-multiplication tensor with zero residual over the integers and multiplies integer matrices correctly under direct evaluation. The coefficient archives are committed by SHA-256 in the paper, and a live verification endpoint re-derives every claim inside an isolated container.