The Banach-Butterfly Invariant: Influence-Adaptive Walsh Geometry for Ternary Polynomial Threshold Functions

arXiv:2605.01637v1 Announce Type: new Abstract: We introduce the Banach-Butterfly Invariant (BBT), an influence-adaptive Banach geometry on the Walsh-Hadamard butterfly factorization. For a Boolean function f:{-1,+1}^no{-1,+1} with coordinate influences Inf_ell(f), BBT assigns exponent p_ell = 1+Inf_ell(f) to butterfly layer ell, yielding the contraction invariant mu(f)=prod_ell 2^{-Inf_ell/(1+Inf_ell)}. We prove a Jensen lower bound log_2mu(f) ge -I(f)/(1+I(f)/n) and that mu is strictly Schur-convex in the influence vector (modulo permutation), giving scaling classes musim 2^{-n/2} (parity), 2^{-Theta(sqrt{n})} (majority), 2^{-1/2} (dictators). log_2mu is rational but not polynomial in the Fourier coefficients while mu is algebraic, and mu separates functions with identical total influence (122 pairs at n=3). Using the certified n le 4 ternary Walsh-threshold universe from a companion synthesis manuscript as a finite testbed, we compute exact MILP minimum-support certificates for all 65,536 Boolean functions at n=4 (mean 6.42, max 9, all-odd by a parity argument) and on 10,000 of the 616,126 NPN-canonical representatives we enumerate at n=5 (matching OEIS A000370). Conditional Spearman rho(mu,|supp|) at fixed total influence is +0.571 in the largest stratum at n=4 but reverses to -0.38 at n=5 under both function-uniform and NPN-canonical sampling: mu is a valid Schur-convex concentration invariant, not a universal monotone predictor of minimum support across n. A companion application paper validates a real-valued WHT activation-energy proxy inspired by this theory on five pretrained LLMs at W2A16, cutting wikitext-2 perplexity by 15-58% versus vanilla auto-round; the transfer from Boolean theory to the real-valued proxy is qualitative, not formal.

Source: arXiv cs.LG | 2026-05-05