Sharp decay for the sign-ensemble Fourier coefficients

Prove the conjectured low-degree decay bound for the Fourier coefficients \(\Gamma(X,Y)=\widehat{g_Y}(X)\) of products of threshold functions of Hadamard transforms: for every constant \(\alpha>0\), show that there is a constant \(C(\alpha)\) such that, whenever \(Y\neq\varnothing\) and \(|X|+|Y|\le n^{\alpha}\), one has \(|\Gamma(X,Y)|\le (|X|+|Y|)^{C(|X|+|Y|)}N^{-(\max\{|X|,|Y|\}+|X|)/4}\).

Background

The appendix proposes an explicit sign ensemble intended to replace the paper's non-explicit Gaussian-rounded hard distribution for separating FH_2 from FH_3. The lower-bound analysis reduces to bounding a single Fourier coefficient of a product of threshold functions applied to pairwise orthogonal Hadamard forms. The paper proves the predicted exponent in part of the parameter range, but the complementary range—especially correlations with |X|<|Y|—remains unresolved. Establishing the conjecture would yield an explicitly defined hard distribution for the oracle separation.