OpenAI · 135-01 · 2026-09-25 · Theoretical computer science

Homogeneous depth-five lower bounds for iterated matrix multiplication

preprints/Homogeneous-depth-five-lower-bounds-for-iterated-matrix-multiplication-September-25-2026

Family 135. Sharp homogeneous depth-five complexity of matrix products.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Over every characteristic-zero field, the $(1, 1)$ entry of a product of $n$ independent $n \times n$ 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.

Let $IMM_{n, n}$ be the $(1, 1)$ entry of a product of n independent $n \times n$ 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 $n^{Θ (n)}$ .

Preprint source and Lean scope

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This preprint is named in the Lean scope note for 135: Homogeneous depth-five lower bounds for iterated matrix multiplication. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.

DepthFive.lean · DepthFive.lean

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