Result 112, Theoretical computer science

Beyond the square-root exponent for depth-three circuits

Constructs a single language in deterministic polynomial time whose n-bit membership function requires 2ω(n)2^{\omega(\sqrt n)} total gates in unbounded-fan-in OR–AND–OR circuits, at every sufficiently large input length. This crosses the square-root-exponent threshold for explicit depth-three Boolean circuit lower bounds.

Lean formalization New or sharp bound

The bigger picture

Why it matters

The manuscript reports a decision problem that a polynomial-time algorithm can solve, but three layers of particular Boolean operations cannot represent compactly. This separates computational power from the restrictions of a very shallow architecture.

What changes?

The construction gives a single language, meaning a set of binary strings, with membership decidable in deterministic polynomial time. For every sufficiently large input length n, any circuit computing membership with three layers ordered OR, AND, OR requires a number of gates whose base-two logarithm grows faster than the square root of n. Each gate may take arbitrarily many inputs, and the count includes every layer, including the bottom one.

What does that help mathematicians do?

The claimed bound rules out every gate budget of the form two raised to a fixed constant times the square root of n, eventually at all input lengths, not just an infinite subsequence. Researchers would thus have a concrete polynomial-time problem witnessing a stronger limitation of this circuit model. The restriction to OR-AND-OR matters: it is not a lower bound for circuits of arbitrary depth.

Are there practical applications?

Its immediate value is foundational: it sharpens the study of how circuit depth and gate count constrain Boolean computation, even when gates can combine arbitrarily many signals. The reported separation concerns this mathematical model, not measured hardware performance. Any implications for practical hardware design would require additional evidence.

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

Beyond the Square-Root Exponent for Depth-Three Boolean Circuits

September 23, 2026 20 pages Main result formalized in Lean

We construct a language in deterministic polynomial time whose n-bit membership function requires 2ω(n)2^{\omega(\sqrt n)} gates in an unbounded-fan-in OR–AND–OR circuit. The bound holds at every sufficiently large input length and counts all gates, including the bottom layer.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Beyond-the-Square-Root-Exponent-for-Depth-Three-Boolean-Circuits-September-23-2026,
  author = {{OpenAI}},
  title = {{Beyond the Square-Root Exponent for Depth-Three Boolean Circuits}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Beyond-the-Square-Root-Exponent-for-Depth-Three-Boolean-Circuits-September-23-2026/main.pdf}{OAI:Beyond-the-Square-Root-Exponent-for-Depth-Three-Boolean-Circuits-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/112.md.

Beyond the square-root exponent for depth-three circuits

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

Scope

The formalized result gives one polynomial-time Boolean language whose exact depth-three OR–AND–OR circuit size eventually exceeds 2An2^{A\sqrt n} for every fixed A>0A>0. The language and its polynomial-time algorithm are fixed before AA is chosen, while circuits may vary with the input length. Gates have unrestricted finite fan-in and sharing, with free input negations and constants. The result does not assert a fixed exponent n1/2+εn^{1/2+\varepsilon} or a lower bound for unrestricted depth.

Comparator links

Result Comparator statement
Depth-three circuit lower bound beyond every square-root constant DepthThree.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.