Average sensitivity of threshold functions

OpenAI's proof (2026) of the asymptotic Gotsman–Linial conjecture: a polynomial threshold function of degree d on the n-dimensional Boolean cube has average sensitivity at most 8d√n. The course proves a one-sided coupling inequality, builds a grading of the functions on the cube weighted by the polynomial, counts sensitive edges through an anticommutator, and turns the degree bound into a binomial coupling; it also derives a bound on noise sensitivity.

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

1. Threshold functions and a one-sided coupling
2. A weighted grading of the cube
3. The Gotsman–Linial bound

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Hilbert spaces and compact operators · Foundations of von Neumann algebras
• Sensitivity, block sensitivity and composition (Sensitivity and block sensitivity)
• Finite-dimensional complex inner product spaces, orthogonal projections and adjoints (core course Linear Algebra, and Theorem 2.2 and Section 3 of the programme lesson Hilbert spaces and compact operators).