A Structural Proof of the Lower Bound 21 for $3\times3$ Matrix Multiplication over $\mathbb F_2$

RESUMO

We prove that the tensor rank of $3\times3$ matrix multiplication over $\mathbb F_2$ is at least $21$. The structural proof, independently developed by Qiushi Engine, converts occupation constraints on a single tensor factor into algebraic relations coupling all three factors. Certified quotient-rank bounds and finite geometry force any hypothetical $20$-term decomposition to have first-factor matrix-rank profile $(16,1,3)$. The ranks of the corresponding split-flattened summands therefore sum to $27$, exactly the rank of the full split flattening. Equality in rank subadditivity forces their images to form a direct sum; normalization by the inverse flattening then makes the summands pairwise annihilating idempotents. An explicit product identity for matrix multiplication implies that at most one first factor can be invertible, contradicting the three forced by the profile. The same obstruction constrains $22$-term decompositions attaining the split-rank bound. The complete proof, including the finite quotient bounds, is formalized in Lean. The accompanying research trajectory records Qiushi Engine's long-horizon autonomous research, from numerical experiments and quotient constructions to the structural proof.