Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 192-01. Unbounded Violations of the Square-Root Degree Bound
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OpenAI@2026-10-07#rem:oai-p-192-01
Authors: OpenAI. Family 192.
Title: Unbounded Violations of the Square-Root Degree Bound.
Abstract: 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 $f:{}{-1,1{}}{}n{}to{}{-1,1{}}$ on a finite sign cube such that ${}displaystyle {}sum_{i=1}{}n {}widehat f({}{i{}}){}gt C{}sqrt{{}deg(f)}.$ Here ${}widehat f({}{i{}})$ is the linear Fourier coefficient associated with the ith input, and ${}deg(f)$ is the degree of the real multilinear polynomial representing f.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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