Matrix Multiplication from Polynomial Packings: A Constructive and Elementary Proof of $\omega \le 9\mathrm{/}4$

We give a self-contained constructive proof that the matrix multiplication exponent satisfies $\omega \le 9\mathrm{/}4$. This bound was recently claimed by OpenAI~\cite{original} through non-constructive arguments in Strassen’s asymptotic spectral framework, counting exact scalar arithmetic operations over the complex numbers. We explicitly construct a Strassen-style matrix multiplication identity and obtain the bound by ordinary block recursion. For every $\gamma >0$, the resulting algorithm multiplies two $n\times n$ integer matrices using $O(n^{9\mathrm{/}4+\gamma })$ integer arithmetic operations.