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CS-154 · computer science · mathematics · Computational complexity

Average sensitivity of polynomial threshold functions

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Must every Boolean threshold of a real multilinear degree-at-most-d polynomial on the uniform n-cube, with $sign(0)=1$ and $1\leq d\leq n$, have total influence at most $8d\surd n$?

Short English statement

Must every Boolean threshold of a real multilinear degree-at-most-d polynomial on the uniform n-cube, with $sign(0)=1$ and $1\leq d\leq n$, have total influence at most $8d\surd n$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-154-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-154

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 127; 2026-09-25).

Average sensitivity of polynomial threshold functions

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 127; 2026-09-25), the asserted result is that degree-$d$ Boolean polynomial threshold functions have total influence at most $8d\sqrt n$, uniformly for $1\le d\le n$, assigning positive sign at zero.

Mathematical statement of the claimed result

Manuscript 1.
For integers $n\ge1$ and $1\le d\le n$, take a multilinear polynomial $p$ with real coefficients and $\deg p\le d$. Set $f(x)=\operatorname{sgn}(p(x))$ on $\{-1,1\}^n$, taking $\operatorname{sgn}(0)=1$. With uniform input and $x^{\oplus i}$ obtained by reversing coordinate $i$, the asserted average-sensitivity estimate is \[ I(f)=\sum_{i=1}^n\Pr[f(x)\ne f(x^{\oplus i})]\le8d\sqrt n. \]

Outline of the argument

For manuscript 1 (family 127; 2026-09-25), a one-sided coupling estimates changed cube edges using weighted Fourier grading. An anticommutator recovers the edge contribution, and parity together with a binomial coupling controls the final degree and dimension dependence.

Formal verification

For manuscript 1 (family 127; 2026-09-25), OpenAI lists Lean statements for this assertion: degree-$d$ Boolean polynomial threshold functions have total influence at most $8d\sqrt n$, uniformly for $1\le d\le n$, assigning positive sign at zero.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are GotsmanLinial.lean.

Their named declarations include OAI.LeanBlast.GotsmanLinial.gotsmanLinialStatement.

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

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