The square-root degree bound

OpenAI's theorem (2026) that the linear Fourier coefficients of a Boolean function can sum to more than any constant multiple of the square root of its degree; through observations of independent signs, a tagged report on Gaussian intervals, and amplification by independent copies.

Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Linear Fourier coefficients and degree
2. Asymptotically normal scores
3. A tagged report
4. Average degree costs

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Probability on finite spaces, conditional expectation, the normal distribution and the central limit theorem (core course Measure and Integration).