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 $μ(f)=\prod_\ell 2^{-\mathrm{Inf}_\ell/(1+\mathrm{Inf}_\ell)}$. 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.

翻译：摘要：我们引入Banach-Butterfly不变量（BBT），这是一种在Walsh-Hadamard蝶形分解上的影响自适应Banach几何。对于布尔函数$f:\{-1,+1\}^n\to\{-1,+1\}$，其坐标影响为$\mathrm{Inf}_\ell(f)$，BBT为蝶形层$\ell$赋予指数$p_\ell = 1+\mathrm{Inf}_\ell(f)$，从而得到压缩不变量$μ(f)=\prod_\ell 2^{-\mathrm{Inf}_\ell/(1+\mathrm{Inf}_\ell)}$。我们证明了Jensen下界$\log_2μ(f) \ge -I(f)/(1+I(f)/n)$，并且$μ$在影响向量（模置换意义下）上是严格Schur-凸的，从而得到缩放类别：$μ\sim 2^{-n/2}$（奇偶函数）、$2^{-Θ(\sqrt{n})}$（多数函数）、$2^{-1/2}$（独裁函数）。$\log_2μ$在傅里叶系数上是有理函数而非多项式，而$μ$是代数函数；$μ$能区分具有相同总影响的函数（$n=3$时有122对）。利用来自配套综合手稿的认证$n \le 4$三项Walsh-阈值全域作为有限测试平台，我们计算了所有65,536个$n=4$布尔函数的精确MILP最小支持度证书（均值6.42，最大值9，通过奇偶性论证全为奇数），以及在$n=5$时从616,126个NPN典型代表中枚举的10,000个函数（匹配OEIS A000370）。在固定总影响条件下，$n=4$最大分层中的条件Spearman相关系数$ρ(μ,|\mathrm{supp}|)$为$+0.571$，但在$n=5$时，无论是在函数均匀采样还是NPN典型采样下，都反转为$-0.38$：$μ$是一个有效的Schur-凸集中不变量，而非跨$n$的通用最小支持度单调预测器。一篇配套应用论文基于此理论，在五个预训练大语言模型上以W2A16量化验证了一个实数值WHT活化能代理，相较于标准auto-round，将wikitext-2困惑度降低15-58%；从布尔理论到实数值代理的迁移是定性的，而非形式化的。