Three Inequivalent Ternary Rank-110 Decompositions of <5,6,5> over Z from the N-Base Engine

We report three mutually inequivalent exact algorithms for the product of a 5×6 by a 6×5 matrix, each using 110 multiplications in which every coefficient is −1, 0 or +1. Rank 110 is the best rank published for ⟨5,6,5⟩ in any coefficient domain; the published rank-110 scheme carries coefficients up to absolute value 9, obtained by Hensel-lifting a flip-graph solution found over 𝔽₂. To the best of our knowledge, these are the first ternary schemes at the record rank for this format over ℤ. Each scheme reconstructs all 22,500 entries of the matrix-multiplication tensor with zero residual and multiplies integer matrices correctly under direct evaluation. Their de Groote profiles are pairwise distinct, and none of them matches any catalogued rank-110 solution under any admissible reading of its indexing convention: the curated catalogue of fast matrix multiplication schemes (six rank-110 entries) and the Perminov FastMatrixMultiplication corpus (169 schemes falling into 165 distinct orbits, not one of them ternary) were both checked, and AlphaTensor does not cover this format. The three schemes are therefore inequivalent to one another and to every rank-110 solution we could locate, under the sandwiching action of GL₅(ℤ) × GL₆(ℤ) × GL₅(ℤ): three new orbits, not rescalings of an existing solution. The coefficient archives are not included. They are committed by SHA-256 in the note, and a live, input-free verification endpoint at https://verify.n-base.io/verify5 re-derives every claim inside an isolated container and signs the transcript with Ed25519, including the check that the three de Groote profiles are pairwise distinct. The schemes were produced by the N-Base Engine; method internals are withheld, and every claim in the note is independently checkable without them.