On the Obstruction to Rank-22 Matrix Multiplication via Single-Product Drops of Known Rank-23 Algorithms

We study whether rank-22 algorithms for 3×3 matrix multiplication can be obtained byremoving a single product from a known exact rank-23 algorithm and solving for a newW-factor. For four independently verified exact rank-23 algorithms (Perminov 2025,Smirnov-variant, Sun 2026, Stapleton 2025), we show that no such repair exists at thespine cells (5,5,8), (2,0,2), and (2,1,5) of the matrix multiplication tensor. Two algebraic obstruction mechanisms are identified and proven: (a) Singletonobstruction — the dropped product is the sole contributor to a spine cell, reducingthe repair equation to 0=1 (domain-independent); (b) Overdetermined obstruction — theremaining 22 products cannot span the target output slice, so the system A·w = b isinconsistent over ℝ. Both are verified algebraically (exact rational arithmetic)and numerically (CP-SAT). We additionally show empirically that the overdeterminedobstruction generalises to randomly constructed rank-22 UV matrices (96% of l0=24frontier candidates, n=25). No rank-22 witness was found. No global rank claim is made; R(⟨3,3,3⟩)remains open (19 ≤ R ≤ 23). sb@sb-forschung.com