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

arXiv: https://arxiv.org/abs/2605.01637 · HTML: https://mlsanity.com/paper/2605.01637/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 29 references that are in the map.
- First posted on arXiv 2 May 2026.

## Builds on (most cited first, 5 of 5)

- [Trained Ternary Quantization](https://mlsanity.com/paper/1612.01064/) (2016)
- [Deep Differentiable Logic Gate Networks](https://mlsanity.com/paper/2210.08277/) (2022)
- [Optimize Weight Rounding via Signed Gradient Descent for the Quantization of LLMs](https://mlsanity.com/paper/2309.05516/) (2023)
- [A Scalable Walsh-Hadamard Regularizer to Overcome the Low-degree Spectral Bias of Neural Networks](https://mlsanity.com/paper/2305.09779/) (2023)
- [Differentiable Logic Synthesis: Spectral Coefficient Selection via Sinkhorn-Constrained Composition](https://mlsanity.com/paper/2601.13953/) (2026)

## Built on by (most cited first, 1 of 1)

- [Influence-Inspired Spectral Rotations for Extreme Low-Bit LLM Quantization](https://mlsanity.com/paper/2605.25203/) (2026)

## Abstract (arXiv)

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.