SAFIRE: Mathematical Analysis of a Differentiable Fuzzy-Inspired Rule-Scoring Surrogate for Medical Tabular Classification

We develop SAFIRE (Self-Attention Fuzzy-Inspired Rule Estimator), a differentiable fuzzy-inspired rule-scoring surrogate for binary medical tabular classification coupling multi-head self-attention, Gaussian membership functions, and Hard Concrete gates for continuous rule scoring. We position SAFIRE as a smooth surrogate of the discrete $L_0$-regularised rule-selection problem and establish five mathematical results and one complexity remark: (1) the relaxed objective is differentiable almost everywhere under positive Gaussian widths (enforced by a Softplus reparameterisation) and fixed batch-normalisation statistics; (2) the deterministic-inference active threshold is strictly stricter than the expected-nonzero training threshold, identifying Hard Concrete gates as continuous rule-scoring devices rather than automatic pruning mechanisms; (3) per-sample forward complexity identifies attention and rule layers as the dominant terms; (4) the Softplus–BatchNorm–linear rule operator violates all four triangular-norm axioms—with necessary and sufficient conditions per axiom and a no-finite-parameterisation impossibility result—while a Softplus reparameterisation restores coordinate-wise monotonicity; (5) a margin-based upper bound characterises disagreement between the full classifier and a top-k rule-only surrogate; and (6) the Softplus-reparameterised constrained variant is provably coordinate-wise monotone with explicit asymptotic regimes. Evaluated on four University of California, Irvine (UCI), medical binary tabular benchmarks under repeated stratified cross-validation, SAFIRE-Prog is statistically competitive with strong interpretable, modern, and gradient-boosting baselines, with one Bonferroni-significant gain over RuleFit on the Diabetic Retinopathy Debrecen corpus. The 48-configuration Hard Concrete sweep, constrained-variant comparison, and a top-k fidelity analysis (per-fold range 0.73–0.95) provide quantitative companion measurements for the mathematical framework. A supplementary large-scale hospital electronic health record (EHR) benchmark (Diabetes 130-US Hospitals, $n=101,766$) shows the rule-scoring mechanism scales to ∼${10}^5$ records and, under severe class imbalance, statistically matches gradient boosting on accuracy while significantly exceeding it on macro-F1. The results offer a mathematically auditable pathway towards interpretable, auditable rule scoring for medical tabular classification, with rule signatures defined in a projected latent space rather than over raw clinical variables.