Proposed
Provider proposes this result; assessment pending ↗
Observed Oct 10, 2026
Bound · version 2
Proposed depth-three bound beyond the square-root exponent
Statement
A fixed language in P requires more than 2^{A√n} depth-three OR–AND–OR gates for every fixed A > 0, at every sufficiently large input length.
Assumptions
- Unbounded finite fan-in and sharing; free input negations and constants; total gate count; exact computation.
Scope
Restricted depth only; not a fixed n^{1/2+ε} exponent and not unrestricted circuits.
This source presents a theorem with a proposed proof. Its wording does not establish acceptance or independent mathematical review.
Evidence
Source reports
Connections
In this problem · publication v9 · Selected context
P versus NP →
Restricted-model result · Proposed
This statement contributes context in its stated scope; it does not settle the canonical P versus NP question.
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v14
P versus NP →
Restricted-model result · Proposed
This statement contributes context in its stated scope; it does not settle the canonical P versus NP question.
Relationship not independently reviewed. This concerns its use in this problem.
Sources
OpenAI · Mathematics release: complexity proposals →
- Family 112; lean/docs/112.md and DepthThree comparator → · Original passage ↗
Recorded wording
A fixed language in P requires more than 2^{A√n} depth-three OR–AND–OR gates for every fixed A > 0, at every sufficiently large input length.
paraphrase · Only metadata and original OpenMath paraphrases are published; proof PDFs remain at their source.
OpenAI internal model; author attribution per manuscript · Project documentation · Publication date Oct 6, 2026
Source-reported preprint/project; independent mathematical assessment pending.
Depth-three circuit lower bound — formalization scope ↗Source-reported evidence
Revision & attribution
Source revision: fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. Observed Oct 10, 2026.
OpenAI reports internal-model production and accompanying formalizations at mixed stages of assessment. These Claims remain proposed in this curation; external formal evidence does not become an OpenMath check.
Depth-three circuit lower bound — formalization scope: fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. Pinned provider scope document and comparator links. OpenMath has not executed the code or independently reviewed the mathematical correspondence.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proposed
Provider proposes this result; assessment pending ↗
Observed Oct 10, 2026 · Source passage →
Record details
- Claim ID
- claim-pnp-depth-three-proposal
- Statement version
- 2
- SHA-256
- a0ab5bcc8bfbfee020415cd1b6175962f8ef1c523eb3644569a9cad715d049da