Improving a Matrix Multiplication Bound with a MacBook and a Proof Assistant

We give the first machine-checked bound on the matrix multiplication exponent: a Lean 4 theorem, quantified over 1.5 MB of published canonical certificate bytes, establishing $\omega \le 10935605172023554189/2^{62} = 2.371281376\ldots$, below the best published level-three bound by $5.76 \times 10^{-5}$. The trusted base is two enumerated axioms: the published feasibility-to-$\omega$ theorem, and one native evaluation (profile CN) corroborated by an independent Rust checker. The numerical search is untrusted and unpublished by design; the certificate is the entire argument. A stronger level-four bound ($2.371177$) is reported without formal verification; we do not claim the state of the art.