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CS-137-A1 · revision r1 · on CS-137

OpenAI claim summary for CS-137

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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).

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Depth-five homogeneous lower bounds for iterated matrix multiplication
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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