Two Inequivalent Ternary Rank-104 Decompositions of ⟨4,5,7⟩ over ℤ

Abstract

We report two inequivalent exact algorithms for the product of a 4×5 by a 5×7 matrix, each using 104 multiplications in which every coefficient is −1, 0 or +1. Rank 104 is the best rank published for ⟨4,5,7⟩ in any coefficient domain; the rank-104 scheme in the public catalogue carries coefficients up to absolute value 4. 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 distinct from one another and from the catalogued solution, so the schemes lie in two new orbits of the sandwiching action of GL4(ℤ)×GL5(ℤ)×GL7(ℤ) rather than being rescalings of a known one. Both are also structurally simpler than the catalogued entry: each is built from only three types of rank-one atom where the catalogued solution needs four, and neither carries an atom of type (1,2,2). Each scheme reconstructs all 19 600 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.