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

We present a complete empirical-algebraic characterization of the rank-22 barrier for M₃ (3×3 matrix multiplication) through a gate-verified L1-obstruction triangle and a complete contact order sequence. Main result (Version 3): Contact order ≥ 5 at the entire L1-obstruction simplex. Complete contact order sequence:- c₁ = 0 — formal (Macaulay2 over ℤ, Singular mod 32003)- c₂ = 0 — empirical (Hessian, depth 2)- c₃ = 0 — complete (228,196 triples, all vertices + edges + face)- c₄ = 0 — complete (306,458 quadruples, all 3 vertices, both schemas: 47806bba + laderman76) Total: 534,654 units, 4,890,148 gate calls, c_k_nonzero = 0 everywhere. Verdict: L1_TRIANGLE_CONTACT_ORDER_GE_5. Macaulay2 syzygy resolutions (L4–L7) stable: Betti [0,227,0,227] / [0,231,0,231] / [0,233,0,233]. Results exemplify the cactus barrier theory of Buczyński (2026, arXiv:2602.11309) at concrete contact order ≥ 5. A German version is included.Schema data available upon research collaboration request. Version history:v1 (2026-06-07): c₃=0 at Vertex C (57,155 triples)v2 (2026-06-07): c₃=0 at all 3 vertices (153,229 triples)v3 (2026-06-07): c₄=0 complete, contact order ≥ 5 ORCID: 0009-0002-3883-5540 Mail: [email protected]