Matrix Multiplication Exponent < 2.371177

Statement. For every universe level and every field $K$ (any characteristic, finite or infinite, no further assumptions):

where $\omega_K$ is the real number matMulExp K, defined below. The inequality is strict.

Definition of $\omega_K$. For $n\in\mathbb N$, let $\langle n,n,n\rangle$ be the matrix-multiplication tensor over $K$: its three mode spaces are $K^{n\times n}$ (functions $\{0,\dots,n-1\}^2\to K$), and

Let $R(T)$ denote the ordinary (textbook) tensor rank: the least $r\in\mathbb N$ such that $T$ is a sum of $r$ pure tensors $u_1\otimes u_2\otimes u_3$ (this is rank, not border rank; $R(0)=0$). Then

with $\log$ the natural logarithm (the ratio does not depend on the base).

Edge cases.

• The infimum is over all $n\in\mathbb N$; the values $n=0,1$ contribute the constant $3$, so $\omega_K\le 3$ always.
• For $n\ge 2$ the tensor $\langle n,n,n\rangle$ is nonzero, so $R\ge 1$ and $\log R\ge 0$, while $\log n>0$; hence every $f(n)\ge 0$, the family is bounded below and nonempty, and the infimum is the genuine real infimum (no junk value from an unbounded-below infimum, no $\log 0$ junk).
• Only square products $\langle n,n,n\rangle$ are used, and only exact rank (not border rank or asymptotic rank) enters the definition.
• $\omega_K$ is defined separately for each field $K$; the claim is asserted for every field simultaneously.