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→{−1,+1} with coordinate influences
Infℓ(f), BBT assigns exponent
pℓ=1+Infℓ(f) to butterfly layer
ℓ, yielding the contraction invariant
μ(f)=∏ℓ2−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−Θ(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.