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CS-137 · computer science · mathematics · Algorithms and data structures · Computational complexity

Depth-five homogeneous lower bounds for iterated matrix multiplication

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Must every syntactically homogeneous depth-five $\Sigma \Pi \Sigma \Pi \Sigma $ circuit over C computing the (1,1) entry of a product of n independent n×n variable matrices have size at least $n^{c\surd n}$ for some $c>0$ and all sufficiently large n?

Short English statement

Must every syntactically homogeneous depth-five $\Sigma \Pi \Sigma \Pi \Sigma $ circuit over C computing the (1,1) entry of a product of n independent n×n variable matrices have size at least $n^{c\surd n}$ for some $c>0$ and all sufficiently large n?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-137-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-137

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 135; 2026-09-25).

Depth-five homogeneous lower bounds for iterated matrix multiplication

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 135; 2026-09-25), the asserted result is that characteristic-zero homogeneous depth-five circuits for iterated $n$-by-$n$ matrix multiplication need at least $n^{\sqrt n/400}$ gates, while block expansion supplies an upper bound of the same exponent scale.

Mathematical statement of the claimed result

Manuscript 1.
Over $\mathbb C$, let $\operatorname{IMM}_{n,n}$ be the $(1,1)$ entry of the product of $n$ generic $n\times n$ matrices on disjoint variable sets. The claimed lower bound gives absolute $n_0$ such that every syntactically homogeneous depth-five sum-product circuit computing this polynomial obeys \[ \operatorname{size}(\Sigma\Pi\Sigma\Pi\Sigma)\ge n^{\sqrt n/400}\quad(n\ge n_0). \] Syntactic homogeneity is part of the model; the assertion is not a lower bound for arbitrary nonhomogeneous circuits.

Outline of the argument

For manuscript 1 (family 135; 2026-09-25), a rank measure built from differentiation followed by multiplication bounds each homogeneous circuit product. First and second trace moments for iterated multiplication give a much larger rank; pairing-type estimates control the second moment and force the gate lower bound.

Formal verification

For manuscript 1 (family 135; 2026-09-25), OpenAI lists Lean statements for this assertion: characteristic-zero homogeneous depth-five circuits for iterated $n$-by-$n$ matrix multiplication need at least $n^{\sqrt n/400}$ gates, while block expansion supplies an upper bound of the same exponent scale.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are DepthFive.lean.

Their named declarations include OAI.Problem335.main.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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