Bounds on the real tensor rank of octonion multiplication

Abstract

The tensor rank of a bilinear map is the least number of multiplications any bilinear algorithm needs to compute it; for the multiplication of an algebra it measures how cheaply the algebra can be multiplied at all. For the even-dimensional real normed division algebras it is $3$ for the complex numbers and $8$ for the quaternions, both classical, while for the octonions $\mathbb{O}$ only a range was known: at least $15$ (Fiduccia and Zalcstein, 1977) and at most $30$ (Cariow and Cariowa). We prove $$18 \le \operatorname{R}_{\mathbb{R}}(T_{\mathbb{O}}) \le 25.$$ The lower bound peels the eight slices of $T_{\mathbb{O}}$ down to two and bounds the rank of the surviving pencil through the octonion norm. Nothing in it is special to dimension $8$: the same steps give $\operatorname{R}_{\mathbb{R}}(T_A) \ge \frac{5}{2}n - 2$ for every real normed division algebra $A$ of even dimension $n$, sharp for $\mathbb{C}$ and $\mathbb{H}$ and the best bound we know for $\mathbb{O}$. The upper bound is a separate construction, an explicit rank-$25$ decomposition certified by a Krawczyk argument, in exact rational arithmetic, to sit within $10^{-6}$ of an exact one. The same two arguments pin down the rank of a smaller three-slice quaternion tensor $\tau$, giving $\operatorname{R}_{\mathbb{R}}(\tau) = 7$. The Lean 4 kernel checks the lower bounds and the Krawczyk existence principle; the accompanying scripts check the certificate's finitely many exact-rational inequalities.