A lower bound of 21 for 3 × 3 matrix multiplication over every field

The formal proof establishes $R_K\langle3,3,3\rangle\geqslant21$ for every field $K$ , including finite and infinite fields in every characteristic. The input linear forms and output matrices are unrestricted; there are no minimality, nonzero-term, symmetry or coefficient-rank assumptions. The argument passes to an algebraic closure, chooses a decomposition minimizing a coefficient-rank sum, and uses conductor growth and residual reconstruction to rule out at most 20 terms. This is partial progress on exact rank: it does not determine the exact rank over every field and gives no border-rank or asymptotic-exponent bound. Results establishing 22 or 23 over $\mathbb F_2$ or for integer-coefficient algorithms do not imply the same numerical bound over arbitrary fields. The README places this result alongside the classical general lower bound 19 and recent field-specific results.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Oct 12, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-co-developed. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: A lower bound of 21 for 3 × 3 matrix multiplication over every field · 3 × 3 matrix multiplication: bilinear rank at least 21

Confidence: Not scored

Registry verification: lean checked · announcement · partial

Open the source record ↗

Attribution

VibeMathed
registry · event recorded by

GPT-6 Astra (gpt-6-astra)
model · ai model contributor · OpenAI

with GPT-6 Sol (gpt-6-sol) used for part of the work
model · ai model contributor · OpenAI

Andrey Ivanov
human · human collaborator

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: A lower bound of 21 for 3 × 3 matrix multiplication over every field

specialist registry · passed

No independent mathematician has checked this yet. Read by this site on 15 October 2026, not rebuilt. The headline theorem MMRank.Palomar.lower_bound was read in Challenge.lean, which imports only Mathlib: for any field K and finite index set I, if $AB=\sum_i a_i(A)b_i(B)W_i$ for all $3\times3$ matrices $A,B$ , with arbitrary linear forms $a_i,b_i$ and matrices $W_i$ , then $|I|\geqslant21$ . That is exactly bilinear rank (tensor rank) of the 3x3 matrix multiplication tensor: exact equality, not border rank, no restriction on the algorithm, and an arbitrary Field includes finite fields and characteristic 2. Solution.lean proves the same statement and comparator.json pairs the two. A grep of all 198 Lean files (18,829 lines) found only the deliberate sorry in Challenge.lean, which nothing imports; no axiom declarations, native_decide, unsafe, implemented_by or extern. The repository's CI run 38306224074 on commit cb1557b printed propext, Classical.choice and Quot.sound for the headline and audited 1,842 project declarations; Palomar run 38306239835 fetched the same commit and its Comparator step passed. Both logs were read here; neither build was repeated. The proof itself was not read.

Scope and record

Verify, challenge, or extend the result.