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.

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.