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

TL;DR

This paper introduces the Banach-Butterfly Invariant (BBT), a new influence-adaptive Walsh geometry that provides insights into Boolean functions, with applications to neural network activation analysis and function classification.

Contribution

The paper develops the BBT, establishing new bounds, properties, and invariants for Boolean functions, and demonstrates its utility in classifying functions and analyzing neural network proxies.

Findings

BBT yields a new contraction invariant with specific scaling classes.

BBT separates functions with identical total influence.

Application to neural networks shows a qualitative transfer from Boolean theory.

Abstract

We introduce the Banach-Butterfly Invariant (BBT), an influence-adaptive Banach geometry on the Walsh-Hadamard butterfly factorization. For a Boolean function $f:\{-1,+1\}^n\to\{-1,+1\}$ with coordinate influences $\mathrm{Inf}_\ell(f)$, BBT assigns exponent $p_\ell = 1+\mathrm{Inf}_\ell(f)$ to butterfly layer $\ell$, yielding the contraction invariant $\mu(f)=\prod_\ell 2^{-\mathrm{Inf}_\ell/(1+\mathrm{Inf}_\ell)}$. We prove a Jensen lower bound $\log_2\mu(f) \ge -I(f)/(1+I(f)/n)$ and that $\mu$ is strictly Schur-convex in the influence vector (modulo permutation), giving scaling classes $\mu\sim 2^{-n/2}$ (parity), $2^{-\Theta(\sqrt{n})}$ (majority), $2^{-1/2}$ (dictators). $\log_2\mu$ 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 $\rho(\mu,|\mathrm{supp}|)$ 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.