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 FD-obstruction depth sequence. Main result (Version 4): FD-obstruction depth ≥5 at all three vertices of the L1-triangle, confirmed ε-stable across ε ∈ {0.10, 0.25, 0.50}. Definition: The FD-obstruction depth at a near-miss vertex is the largest order k for which all finite-difference approximations of k-th order directional derivatives in the null-space vanish. This is a precisely defined numerical obstruction measure (not algebraic-geometry contact order). Complete FD-obstruction sequence:• c₁ = 0 — formal (Macaulay2 over ℤ, Singular mod 32003)• c₂ = 0 — empirical (Hessian, depth 2)• c₃ = 0 — ε-stable: 381,425 triples × 3 ε-values, all vertices + edges + face• c₄ = 0 — ε-stable: 306,458 quadruples × 3 ε-values, all vertices, both schemas Total: 534,654 units × 3 ε-values, c_k_nonzero = 0 everywhere. Algebraic support (characteristic-independent):Syzygy over ℤ: rank(J) = 496, syzygy_corank = 233.Syzygy mod 32003: rank(J) = 496, syzygy_corank = 233.Verdict: SYZYGY_BETTI_ZZ_P_MATCH — Betti structure is characteristic-independent. Results exemplify the cactus barrier theory of Buczyński (2026, arXiv:2602.11309). 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, FD-depth ≥5v4 (2026-06-08): ε-stability validated, syzygy over ℤ confirmed, terminology corrected ORCID: 0009-0002-3883-5540 Mail: [email protected]