OpenAI · 192-01 · 2026-09-26 · Combinatorics
Unbounded Violations of the Square-Root Degree Bound
preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026
Family 192. A counterexample to the Gopalan–Servedio 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
Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every , there is a sign-valued Boolean function with . Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor.
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 on a finite sign cube such that Here is the linear Fourier coefficient associated with the ith input, and is the degree of the real multilinear polynomial representing f.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 192: Boolean functions violate the square-root degree bound by arbitrary factors. 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.