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) — three proved theorems. Version 10 — new since v9:(1) Syzygy regularity (Yang 2026, arXiv:2606.14637): reg(coker J) = 0 globally — partially linear 2-term structure (0 → R^496 → R^567 → coker(J) → 0, d₀=1, no violations). Global CM-regularity ≠ local contact order (different charts). Theorem 1 (multiplicity) is unaffected. (2) Sheshadri 2026b (ECCC TR26-097, 9 June 2026): sdc(Σxᵢⁿ) ≥ (1/2e − o(1))n² over ℂ via polar degree for symmetric determinantal complexity. Strengthens CONDITIONAL_GO from v9 §8: Sheshadri actively extends the polar-degree method that motivates our contact order formalization. (3) Wang verification package complete: verify_standalone.py (no matrix_lab required), two modes: --check-log (instant) and --cert rmms_n333.pb.txt (~3.3 hours). Zenodo record fully updated. Cumulative results (Theorems 1–3):- Theorem 1: contact order ≥4 at A,B,C (multiplicity, Macaulay2)- Theorem 2: κ(J)−ν(J) = m−n = 729−567 = 162 (elementary)- Theorem 3: {6,13,18} ⊂ Sedoglavic-R = {6,13,18,19} (3/4 overlap)- Proposition 6.1: A=full_333, B/C=bridge_mixed (Tichavsky taxonomy) FD ledger: c₃=0 on 228,196 k=3 directions;c₄=0 on 306,458 k=3 directions (ε-stable, scan_complete=true).(c₄ = 4th FD-order along a k=3 direction, not a C(n,4) scan.) Wang complementarity W1–W5: globally orthogonal (Spearman −0.068).Wang F₂-certificate independently verified:DOI 10.5281/zenodo.20691958. Additional searches (all negative): ILP, annealing, GF(2)+GF(3),CRT, CP-ALS, substitution scan, Sedoglavic 19+3, ℤ[i]/Eisenstein. R(M₃), R̄(M₃), border cactus rank remain open.Results do not imply R(M₃)=23.Consistent with Buczyński (2026, arXiv:2602.11309).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 W1–W5; precise FD-ledgerv9 (2026-06-15): Theorem 1–3, Proposition 6.1, Sheshadri §8v10 (2026-06-16): Yang reg=0, Sheshadri 2026b, Wang package complete Stefan Beuchert ORCID: 0009-0002-3883-5540 Mail: [email protected]