A Proved Blocking-Set Law and Exhaustion of the Drop-and-Repair Ladder for Rank-22 3×3 Matrix Multiplication

We study a natural escalation of the single-product-drop-and-repair question for3x3 matrix multiplication (Beuchert 2026, DOI 10.5281/zenodo.21188750): given anexact rank-23 scheme, can removing k products and replacing them with k-1 newatoms ever restore an exact rank-22 scheme? We report three results. First, a proved combinatorial lemma (the Blocking-SetLaw): for pair drops, a pair renders an output row infeasible if and only if atleast one of its two atoms lies in the row's single-drop blocking set. Thisforces an exact count formula, verified without exception across fourstructurally distinct exact rank-23 parent schemes and unchanged undercoefficient-magnitude escalation. Second, an exhaustive three-rung empirical ladder: single-product drop (norepair found, prior work), pair-drop-plus-one-atom, and triple-drop-plus-two-atoms. Across 7,084 fully scanned triples and 759 fully scanned pairs, zerocandidates reach a valid rank-22 assembly. The atom-completion search isverified eligibility-complete on the two most structurally different parentschemes. Third, a guard-discipline episode: an automated search run produced one objectsatisfying a naive success criterion that was in fact a disguised rank-23restoration rather than a rank-22 result. A purpose-built guard caught it, anda retrospective audit of prior search output found no historical instance ofthe same misclassification. We report this as a drop-and-repair ladder exhaustion candidate. The pair-levelmechanism is proved as a lemma; the triple-level result is an exhaustive scanunder a documented search cap. No claim is made about the rank of the 3x3matrix-multiplication tensor, which remains open (19 <= R <= 23), aboutunbounded coefficient magnitude, about drops of more than three atoms, or aboutparent schemes beyond the four studied here.sb@sb-forschung.com