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  "path": "/abs/2605.01637v1",
  "publishedAt": "2026-05-05T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Gorgi Pavlov"
  ],
  "textContent": "**Authors:** Gorgi Pavlov\n\nWe 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 $μ(f)=\\prod_\\ell 2^{-\\mathrm{Inf}_\\ell/(1+\\mathrm{Inf}_\\ell)}$. We prove a Jensen lower bound $\\log_2μ(f) \\ge -I(f)/(1+I(f)/n)$ and that $μ$ is strictly Schur-convex in the influence vector (modulo permutation), giving scaling classes $μ\\sim 2^{-n/2}$ (parity), $2^{-Θ(\\sqrt{n})}$ (majority), $2^{-1/2}$ (dictators). $\\log_2μ$ 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 $n \\le 4$ 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 $ρ(μ,|\\mathrm{supp}|)$ 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: $μ$ 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.",
  "title": "The Banach-Butterfly Invariant: Influence-Adaptive Walsh Geometry for Ternary Polynomial Threshold Functions"
}