Result 135, Theoretical computer science

Homogeneous depth-five lower bounds for iterated matrix multiplication

Over every characteristic-zero field, the $(1,1)$ entry of a product of n independent $n\times n$ variable matrices requires $n^{\Theta(\sqrt n)}$ gates in homogeneous depth-five sum–product circuits. This sharp bound allows shared gates and bottom linear forms involving all variables.

Lean formalization New or sharp bound

The bigger picture

Why it matters

Multiplying n independent n-by-n matrices of variables produces polynomials that are easy to specify but can be costly to compute with shallow circuits. This manuscript reports a sharp cost for computing the product's top-left entry.

What changes?

The model has five alternating addition and multiplication layers, starting and ending with addition. Syntactic homogeneity requires matching total degrees at additions. Over any characteristic-zero field, the claimed lower bound is n to the power (square root of n divided by 400), for sufficiently large n with a field-independent threshold. A construction over every field uses at most n to the power (square root of n plus 4) gates for n at least 2.

What does that help mathematicians do?

The bound permits shared gates, any finite number of inputs and outgoing connections per gate, and bottom linear forms, meaning weighted sums that may involve all variables. Researchers can therefore rule out polynomial-size circuits in this model without assuming sparse inputs or forbidding reused calculations. Together with the construction, it pins down the exponent's scale to the square root of n, rather than merely showing that some large circuits are necessary.

Are there practical applications?

The immediate value is foundational: the result quantifies the cost of restricting both circuit depth and polynomial degree structure for iterated matrix multiplication. It identifies a precise limitation of this arithmetic computing model, not a slowdown for ordinary matrix multiplication algorithms or a lower bound for unrestricted circuits.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Homogeneous depth-five lower bounds for iterated matrix multiplication

September 25, 2026 22 pages

Let $\mathop{\mathrm{IMM}}\nolimits _{n,n}$ 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 $\Sigma\Pi\Sigma\Pi\Sigma$ circuit computing $\mathop{\mathrm{IMM}}\nolimits _{n,n}$ has at least $n^{\sqrt n/400}$ 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 $n^{\sqrt n+4}$ gates for n ≥ 2. Thus the gate complexity over characteristic-zero fields is $n^{\Theta(\sqrt n)}$ .

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:Homogeneous-depth-five-lower-bounds-for-iterated-matrix-multiplication-September-25-2026,
  author = {{OpenAI}},
  title = {{Homogeneous depth-five lower bounds for iterated matrix multiplication}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Homogeneous-depth-five-lower-bounds-for-iterated-matrix-multiplication-September-25-2026/Homogeneous-depth-five-lower-bounds-for-iterated-matrix-multiplication-September-25-2026.pdf}{OAI:Homogeneous-depth-five-lower-bounds-for-iterated-matrix-multiplication-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/135.md.

Homogeneous depth-five lower bounds for iterated matrix multiplication

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Homogeneous depth-five lower bounds for iterated matrix multiplication

Scope

Let $\mathrm{IMM}_{n,n}$ be the $(1,1)$ entry of the product of $n$ independent $n\times n$ variable matrices. The formalization proves that, over every characteristic-zero field, syntactically homogeneous depth-five $\Sigma\Pi\Sigma\Pi\Sigma$ circuits computing this polynomial require at least $n^{\sqrt n/400}$ gates for all sufficiently large $n$ , with a threshold independent of the field. Arbitrary bottom support, finite fan-in and fan-out, and sharing are allowed.

It also gives, over every field and for $n\ge2$ , circuits with at most $n^{\sqrt n+4}$ gates. These are the lower and upper bounds selected from the paper.

Comparator links

Result	Comparator statement
Homogeneous depth-five bounds for iterated matrix multiplication	DepthFive.lean

At a glance

Subject: Theoretical computer science
Manuscripts: 1
Pages: 22
Dated: Sep 25, 2026
Lean formalization: Yes
Reasoning summary: No

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