Binary Regression: Universal Ising Model, Binary Expansion and Beyond

We introduce a constructive universality principle for structural interpretability in binary regression. Let $A\in \\{-1,1\\}^{p}$ be binary covariates and $B\in\\{-1,1\\}$ be a binary response. We represent the binary-regression target $\mathbb{E}[B|A]$ by Ising Hamiltonians over observed and auxiliary binary variables. This construction encodes covariate configurations as Boolean states and maps input-output relationships through faithful reductions from Boolean satisfiability (SAT) to ground-state-energy Ising problems. This gives a interpretable graph representation of high-order binary dependence. We showcase this principle with the *Two-body, Linear, Universal, Binary Effect* (T-LUBE) framework. Unlike full power-vector expansion, T-LUBE does not enumerate high-order product features. It recasts these effects through interpretable auxiliary variables and pairwise interactions on an augmented binary graph. Proof-of-concept experiments corroborate our theory.