Homogeneous depth-five lower bounds for iterated matrix multiplication

What OpenAI claims

Over every characteristic-zero field, the (1,1) entry of a product of n independent n× n variable matrices requires n^Θ(√n) gates in homogeneous depth-five sum–product circuits. This sharp bound allows shared gates and bottom linear forms involving all variables. (Lean)

Paper

1. Homogeneous depth-five lower bounds for iterated matrix multiplication

   25 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract Let IMM _n,n be the (1,1) entry of a product of n independent n× n matrices of variables. Over every field of characteristic zero, every syntactically homogeneous ΣΠΣΠΣ circuit computing IMM _n,n has at least n^√n/400 gates for all sufficiently large n, with an absolute threshold independent of the field. Bottom linear forms may have arbitrary support, and arbitrary finite fan-in, fan-out, and gate sharing are allowed. Over every field, a block expansion gives such circuits with at most n^√n+4 gates for n ≥ 2. Thus the gate complexity over characteristic-zero fields is n^Θ(√n).

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.