Contact Order ≥ 4 at the Rank-22 Obstruction Locus of 3×3 Matrix Multiplication

Null-space FD-proxy measurements at the rank-22 obstruction locusof M₃ (3×3 matrix multiplication) — schema-independently confirmed. Version 8 — new: complementarity with Wang (2026).Five experiments (W1–W5) establish the relationship between Wang'sF₂-certificate (arXiv:2603.07280, independently verified atDOI 10.5281/zenodo.20691958) and our FD-proxy measurements:both methods are globally orthogonal (Spearman = −0.068),diagnostically complementary, not operatively bridgeable. Notation: a k=3 direction is the sum of three null-space basisvectors basis_i+basis_j+basis_k for (i,j,k) ∈ C(n,3).c₃ and c₄ denote the 3rd and 4th FD-order ALONG such a direction(c₄ via Method A′, 5 nodes) — not a C(n,4) combination scan. Main results:- L1-obstruction triangle: three rank-22 near-miss points (l0=1, fp=0) on three independent rank-23 schemas- c₃=0 on 228,196 k=3 directions (vertices + edges + face + 18D triple intersection + 3b1f290e schema)- c₄=0 on 306,458 k=3 directions (two parent schemas: 47806bba + laderman76)- All scans scan_complete=true, no sampling- ε-stable over {0.50, 0.25, 0.10} Algebraic support: rank(J)=496 (column count), syzygy_corank=233identical over ℤ and mod 32003 (SYZYGY_BETTI_ZZ_P_MATCH).Betti−nullity = 162 = 2×9² at all three vertices. Wang complementarity (W1–W5):- W1: FD-depth ≥4 in all 48 Wang projection subspaces- W2: 18D triple intersection has 4D F₂-structure (GF(2)-rank 4/18)- W3: lb-growth vs FD-magnitude globally orthogonal (Spearman −0.068)- W4+W5: Wang proof-paths yield no operative ℤ-lift on near-miss baseline Additional searches (all negative): ILP, annealing, GF(2)+GF(3)campaigns, CRT synthesis, CP-ALS over ℂ, substitution scan. R(M₃), R̄(M₃), and border cactus rank remain open.Results do not imply R(M₃)=23.Consistent with cactus barrier theory (Buczyński 2026).German version included. Version history:v1 (2026-06-07): c₃=0 Vertex C (57,155 triples)v2 (2026-06-07): c₃=0 all 3 vertices (153,229 triples)v3 (2026-06-07): c₄=0 complete, FD-depth ≥5v4 (2026-06-08): ε-stability, syzygy over ℤ, terminology fixv5 (2026-06-09): schema-independent, GF campaigns, global orthogonalityv6 (2026-06-10): full definitions, corrected citations, precise claimsv7 (2026-06-14): triple intersection 18D, additivity, L1-uniqueness, Wang v8 (2026-06-14): Wang complementarity W1–W5; precise FD-ledger (k=3 directions, c₄ = 4th order along k=3, no C(n,4) scan) Stefan Beuchert ORCID: 0009-0002-3883-5540 Mail: [email protected]