A Guarded Trainable Exponent Operator: Structure-Matched Extrapolation, Loss Targets, and Numerical Boundaries

Positive power products have established roots in product-unit networks, neural arithmetic modules, and log-linear regression, where log transformation yields a quadratic objective with a unique ordinary least squares (OLS) solution and a compact, interpretable elasticity. Reliable use remains limited by three gaps: how exponent constraints interact with target structure to affect out-of-distribution prediction, how loss targets change the estimand and the interpretation of magnitude-weighted WAPE versus mean absolute relative error, and how numerical guards—input floor ε and cap L—and precision alter bias and finiteness without guaranteeing correctness. We conduct a reproducible audit of a guarded operator, ŷ = exp(clip(Σ q_j log max(x_j, ε) + b, ±L)), against analytic log-OLS using identical observations: 10,368 primary fits across four power families and 24 seeds; 2520 matched-budget fits with noisy-validation checkpointing including a finite Müntz–Szász network and raw/log-input multilayer perceptrons; and 2016 guard and mixture diagnostics with factorial ε, L, and precision paths. Independent exponents improve OOD prediction when the shared constraint is violated (WAPE ~0.09 versus ~0.79–0.96 for shared), whereas the correctly matched shared constraint reduces WAPE from 0.0920 to 0.0605, demonstrating constraint matching rather than universal superiority. Analytic log-OLS attains WAPE 0.0666 versus 0.3443 for same-family Adam, revealing optimization effects. Flooring informative near-zero inputs induces severe bias (WAPE 0.044 at ε = 1 × 10−12 versus 9.71 at 1 × 10−8) despite finite outputs, and a 0.1 anchor-weight non-power mixture exceeds an illustrative 0.10 tolerance. Results support guarded single-product use in which a positive power relation is scientifically plausible and range/precision are validated while delineating boundaries where multi-term bases are preferable.