Two Inequivalent Ternary Rank-176 Decompositions of ⟨5,7,7⟩ over ℤ with Fewer Atom Types than the Published Solution

We report two inequivalent exact algorithms for the product of a 5×7 by a 7×7 matrix, each using 176 multiplications in which every coefficient is −1, 0 or +1. Rank 176 is the best rank published for ⟨5,7,7⟩ in any coefficient domain; the rank-176 scheme in the public catalogue carries coefficients up to absolute value 11, the widest range we have encountered at a record rank. To the best of our knowledge these are the first ternary decompositions at the record rank for this format over ℤ. Beyond the coefficient alphabet the schemes are also structurally narrower: each is assembled from six types of rank-one atom where the catalogued solution needs eight, and neither carries an atom of mode-rank type (1,1,4) or (1,1,5). Their de Groote profiles are distinct from one another and from the catalogued solution, so the two schemes lie in two new orbits of the sandwiching action of GL5(ℤ)×GL7(ℤ)×GL7(ℤ) rather than being rescalings of a known one. Each scheme reconstructs all 60 025 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.