Certified local rank and uniqueness barriers for a 48-term matrix-multiplication decomposition

Abstract

We study replacements in fixed bilinear tensor decompositions, counting changes to complete rank-one summands, including output factors. The shortening frontier records the maximum rank defect of a fixed-size subset and determines the minimum length attainable within a change budget. For the rational 48-term Li--Wang--Hu decomposition \(D(2)\) of \(4\times4\) matrix multiplication over \(\mathbb{C}\), we prove rank radius at least 12, strong radius exactly 11, and border radius at least 8. Every shorter complex decomposition therefore changes at least thirteen original summands. An exact rational twelve-term replacement attains the equal-length barrier. The proofs combine exhaustive support reductions with saturated projected kernels and zero-corner completion arguments controlling arbitrary minimal competitors. A reduced-incidence argument transfers kernel certificates to tensor-space neighborhoods. A Laurent normal form gives strong radius exactly 11 for the sixteen-term core at every nonzero complex parameter. On a nonempty Zariski-open subset of the actual parameter curve, the rank radius is at least 12, the strong radius exactly 11, and the border radius at least 8. We also prove incomparability of the full Kothari--Moitra--Wein sufficient criterion and the Sylvester-equipped kernel criterion. These results describe local decomposition structure rather than a new rank bound for full matrix multiplication.