Result 192, Combinatorics

Boolean functions violate the square-root degree bound by arbitrary factors

Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every C > 0, there is a sign-valued Boolean function f with ∑if^({i})>Cdeg⁡(f)\sum_i\widehat f(\{i\})\gt C\sqrt{\deg(f)}. Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

A Boolean function can have much more combined correlation with its individual input bits than a proposed degree bound allows. The manuscript reports counterexamples that defeat every constant multiple of that bound.

What changes?

For every positive real C, the manuscript claims a nonconstant function on some finite n-dimensional sign cube whose linear Fourier coefficients sum to more than C times the square root of its degree. Each coefficient averages the product of the output and one input over all inputs; both take values minus or plus one. Degree is the largest number of variables in a term of its exact real multilinear polynomial. The function and dimension may depend on C.

What does that help mathematicians do?

The reported counterexamples rule out repairing the Gopalan–Servedio square-root conjecture simply by increasing its constant. For researchers relating polynomial degree to correlation with individual bits, this eliminates an entire proposed family of universal inequalities, not just one numerical bound. Any replacement must change the dependence on degree, impose additional assumptions, or otherwise alter the statement. The quantity bounded here is the signed sum of correlations.

Are there practical applications?

Its immediate value is foundational: it clarifies a limit on how polynomial representations can constrain the behavior of Boolean functions. Specifically, it shows why degree alone cannot justify this square-root estimate for aggregate single-bit correlation. The supplied abstract describes a counterexample result, not an algorithm or a demonstrated practical application.

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

Unbounded Violations of the Square-Root Degree Bound

September 26, 2026 23 pages

We disprove the Gopalan–Servedio square-root conjecture, even up to an arbitrary constant factor. For every real C > 0, there is a nonconstant Boolean function f:{−1,1}n→{−1,1}f:\{-1,1\}^n\to\{-1,1\} on a finite sign cube such that

∑i=1nf^({i})>Cdeg⁡(f).\displaystyle \sum_{i=1}^n \widehat f(\{i\})\gt C\sqrt{\deg(f)}.

Here f^({i})\widehat f(\{i\}) is the linear Fourier coefficient associated with the ith input, and deg⁡(f)\deg(f) is the degree of the real multilinear polynomial representing f.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026,
  author = {{OpenAI}},
  title = {{Unbounded Violations of the Square-Root Degree Bound}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026.pdf}{OAI:Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/192.md.

Boolean functions violate the square-root degree bound by arbitrary factors

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

Scope

The formalization disproves the Gopalan–Servedio square-root degree conjecture by an unbounded factor. For every C>0C>0, it gives a nonconstant Boolean function on a finite sign cube for which the sum of its linear Fourier coefficients exceeds Cdeg⁡fC\sqrt{\deg f}, where deg⁡f\deg f is its real multilinear degree.

It also proves that the ratio of the sum of the absolute values of those coefficients to deg⁡f\sqrt{\deg f} is unbounded among positive-degree Boolean functions.

Comparator links

Result Comparator statement
Unbounded violations of the square-root Fourier-degree bound SquareRootDegree.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.