What question did this study set out to answer?

The aim is to analyze contact order at the rank-22 obstruction locus of 3x3 matrix multiplication.

June 18, 2026Open Access

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

Key Points

The aim is to analyze contact order at the rank-22 obstruction locus of 3x3 matrix multiplication.
Proof of three theorems regarding syzygy regularity and symmetric determinantal complexity.
Use of null-space FD-proxy measurements to assess contact order.
Correction of previous versions to clarify scope and results.
Theorem 1 confirms contact order ≥ 4 at key loci.
Theorem 2 establishes κ(J)−ν(J) = m−n = 162.
Theorems 1 to 3 have retained results indicating a specific overlap in determined properties.

Abstract

Null-space FD-proxy measurements at the rank-22 obstruction locusof M₃ (3×3 matrix multiplication) — three proved theorems. Version 11 — correction of v10 scope: v10 incorrectly included Wang verification package details (verifyₛtandalone. py, --check-log/--cert modes) in this preprint. These belong to the separate Wang verification preprint (DOI: 10. 5281/zenodo. 20691958, 6) and have been removed. Wang appears only as reference 5 and verified reference 6. New results since v9 (retained from v10): (1) Syzygy regularity (Yang 2026, arXiv: 2606. 14637): reg (coker J) = 0 globally — partially linear 2-term structure. Global CM-regularity ≠ local contact order (different charts). Theorem 1 (multiplicity) is unaffected. (§7) (2) Sheshadri 2026b (ECCC TR26-097, 9 June 2026): sdc (Σxᵢⁿ) ≥ (1/2e − o (1) ) n² via polar degree for symmetric determinantal complexity. Strengthens CONDITIONALGO (§8). Cumulative results (Theorems 1–3, unchanged): - 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=full₃33, B/C=bridgeₘixed (Tichavsky taxonomy) FD ledger: c₃=0 on 228, 196 k=3 directions;c₄=0 on 306, 458 k=3 directions (ε-stable, scancomplete=true). Wang complementarity W1–W5: globally orthogonal (Spearman −0. 068). Wang certificate independently verified (separate preprint 6). R (M₃) remains open: 19 ≤ R ≤ 23. Results do not imply R (M₃) =23. 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 scope errorv11 (2026-06-16): Wang scope corrected verification package belongs to separate preprint DOI 10. 5281/zenodo. 20691958 Stefan Beuchert ORCID: 0009-0002-3883-5540 Mail: sb@sb-forschung. com

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