A Field- and Construction-Independent Blocking-Set Law, and Its Exhaustion Across 22 Rank-23 Parent Schemes for 3×3 Matrix Multiplication

We extend the pair-drop Blocking-Set Law (Beuchert, preprint, "A ProvedBlocking-Set Law and Exhaustion of the Drop-and-Repair Ladder for Rank-22 3x3Matrix Multiplication") -- a proved, field-independent combinatorial factgoverning which atom-pair drops from an exact rank-23 scheme can be repairedacross a substantially widened parent corpus: from the original fourinteger-coefficient schemes to a total of 22 parent schemes spanning sevendistinct construction paradigms (hand-derived integer algorithms, GF(3)flip-graph search lifted to the rationals, and independently sourcedliterature schemes from public catalogs). The law is verified without exception on every tested (row, parent)combination 360 checks across two independent computation passes, zeromismatches. We report a second, methodologically important finding: parentconstruction diversity and susceptibility to drop-and-repair are independentaxes. Some newly incorporated parents exhibit blocking-set profilesstructurally unlike any in the original corpus, including one parent withfull blocking on every row and others with markedly softer local structure yet every parent tested at the single-drop level, and the softest cases alsoat the pair-drop-plus-atom level, yields zero valid rank-22 assemblies.Structural novelty in blocking geometry does not translate into closerproximity to a rank-22 witness. We report this corpus as the current practical exhaustion of thedrop-and-repair paradigm under the tested move classes and move depths, andoutline three concretely different directions for future work none ofwhich is a further parent search: symmetry-reduced or invariant-based search,algebraic-geometric methods capable of demonstrating non-existence directly,and a reversal of research direction toward improved lower-bound proofs. No claim is made about the rank of the 3x3 matrix-multiplication tensor,which remains open (19 <= R <= 23).sb@sb-forschung.com