Average sensitivity of polynomial threshold functions is at most 3.762 d sqrt(n)

For all positive integers n,d, let p be any real polynomial in n variables with total degree at most d. On the uniform cube {-1,1}^n, define f(x)=1 when p(x)>=0 and f(x)=-1 otherwise. Let average sensitivity be the sum over coordinates of the probability that flipping that coordinate changes f. Then

The theorem includes zeros of p under the positive-at-zero convention and arbitrary polynomials, with no multilinearity assumption. There is no restriction d<=n. The weighted-grading construction is due to OpenAI; the refined coupling estimate is due to Lagdive, Aglave and Damre. This result formalizes their integration and extracts the sufficient coefficient 3.762; optimality is not asserted.