OpenAI · 127-01 · 2026-09-25 · Theoretical computer science
Average sensitivity of polynomial threshold functions
preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026
Family 127. The asymptotic Gotsman–Linial conjecture.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves that a degree-at-most- polynomial threshold function on the uniform -dimensional Boolean cube has average sensitivity at most , uniformly for . Average sensitivity counts expected output changes under single-bit flips. This establishes the asymptotic Gotsman–Linial conjecture, allowing polynomial zeros with .
For every n ≥ 1 and , we prove that a polynomial threshold function of degree at most d on the uniform Boolean cube has average sensitivity at most . This proves the asymptotic form of the Gotsman–Linial conjecture. The bound is uniform in both parameters and uses the convention .
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 127: Average sensitivity of polynomial threshold functions. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.