Algebraic and Empirical Obstructions Against Rank-22 Completions Near Known Rank-23 Schemes of 3×3 Matrix Multiplication

This document makes no claim about the rank of ⟨3,3,3⟩, which remains open (19 ≤ R ≤ 23). We investigate whether known rank-23 decompositions of the 3×3 matrix multiplication tensor ⟨3,3,3⟩ (Laderman-type, Perminov, Sedoglavic families) can be completed to exact rank-22 decompositions via local algebraic operations. We introduce an exact, rank-preserving completion order c(S,R), sharper than the conventional residual support metric ℓ₀, and calibrate it against a known exact control witness. On eight identified frontier candidates we show: (1) no candidate admits direct one- or two-term absorption of the residual; (2) an algebraically certified non-existence result for joint W-solutions under modular reduction (GF(2), GF(3)) on 20 stratified candidates; (3) this pattern extends consistently to modular computation over {2,3,5,7,11,13}. We additionally document a broadly supported but only empirically evidenced reentry invariant under ε-perturbation. All results are negative with respect to rank 22 within the investigated search space; no global impossibility claim is made.