Result 127, Theoretical computer science

Average sensitivity of polynomial threshold functions

Proves that a degree-at-most-d polynomial threshold function on the uniform n-dimensional Boolean cube has average sensitivity at most 8dn8d\sqrt n, uniformly for 1≤d≤n1\le d\le n. Average sensitivity counts expected output changes under single-bit flips. This establishes the asymptotic Gotsman–Linial conjecture, allowing polynomial zeros with sign(0)=1\mathop{\mathrm{sign}}\nolimits (0)=1.

Lean formalization Proof

The bigger picture

Why it matters

How often can a decision based on a low-degree polynomial change when just one input bit flips? The manuscript reports a bound linking this sensitivity to polynomial degree and the number of input bits.

What changes?

A polynomial threshold function assigns a binary output according to whether a polynomial in n input bits is nonnegative or negative, assigning output 1 at zero. Its average sensitivity is the expected number of single-bit flips that change the output, averaged uniformly over all inputs. The manuscript claims this quantity is at most 8 times d times the square root of n for degree at most d, uniformly for every n at least 1 and every integer d from 1 through n.

What does that help mathematicians do?

The claimed bound establishes the asymptotic Gotsman-Linial conjecture and gives a precise average-case stability guarantee. Dividing by n shows that flipping one uniformly chosen bit of a uniformly chosen input changes the output with probability at most 8 times d divided by the square root of n. Thus, when d grows more slowly than the square root of n, this probability tends to zero. This does not guarantee stability at every input.

Are there practical applications?

Its immediate value is foundational for studying binary decision rules represented by polynomial thresholds. The bound limits how much their outputs can fluctuate across neighboring inputs, providing a quantitative constraint for theoretical analyses of sensitivity. The supplied material establishes no practical deployment or algorithmic improvement, and the guarantee specifically concerns uniformly sampled inputs.

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

Average sensitivity of polynomial threshold functions

September 25, 2026 13 pages Main result formalized in Lean

For every n ≥ 1 and 1≤d≤n1\le d\le n, we prove that a polynomial threshold function of degree at most d on the uniform Boolean cube has average sensitivity at most 8dn8d\sqrt n. This proves the asymptotic form of the Gotsman–Linial conjecture. The bound is uniform in both parameters and uses the convention sgn(0)=1\mathop{\mathrm{sgn}}\nolimits (0)=1.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026,
  author = {{OpenAI}},
  title = {{Average sensitivity of polynomial threshold functions}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026/main.pdf}{OAI:Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/127.md.

Average sensitivity of polynomial threshold functions

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

Scope

The formalized result proves the average-sensitivity bound 8dn8d\sqrt n for every Boolean threshold function defined by a real multilinear polynomial of degree at most dd on the nn-dimensional cube. The sign convention assigns value 11 at zero. This establishes the stated asymptotic bound, rather than the separate conjecture identifying exact extremizers.

Comparator links

Result Comparator statement
Average sensitivity of polynomial threshold functions GotsmanLinial.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 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.