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

We introduce the Banach-Butterfly Invariant (BBT), an influence-adaptive Banach geometry on the Walsh-Hadamard butterfly factorization. For a Boolean function f:-1,+1n→-1,+1 with coordinate influences Infℓ(f), BBT assigns exponent pℓ = 1+Infℓ(f) to butterfly layer ℓ, yielding the contraction invariant μ(f)=prodℓ 2-Infℓ/(1+Infℓ). We prove a Jensen lower bound log2μ(f) ≥ -I(f)/(1+I(f)/n) and that μ is strictly Schur-convex in the influence vector (modulo permutation), giving scaling classes μ~ 2-n/2 (parity), 2^-Theta(√n) (majority), 2-1/2 (dictators). log2μ is rational but not polynomial in the Fourier coefficients while μ is algebraic, and μ separates functions with identical total influence (122 pairs at n=3). Using the certified n ≤ 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 ρ(μ,|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: μ 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.