The Born Rule as the Entropy-Uniform Limit of Record Formation: A Falsifiable Measurement Framework

The Born rule sits at the heart of quantum mechanics as an unexamined assumption: probabilities are fundamental. For a statistician, that is uncomfortable. Probabilities are tools for describing ignorance — so what, precisely, are we ignorant of? That question leads to a deeper one: what is measurement, really? If measurement is a physical event, something in the world must change when it happens — something records it. WF-S formalises this with two physical inputs and one Lagrangian. Postulate 1 (WF-S kernel): a complex amplitude field φ(x) couples to a real substrate field s(x) via L = Lφ + Ls − g|φ|²s. Postulate 2 (record condition): macroscopic outcomes are weighted by cumulative entropy export σi = ΔSenv,i/kB during record formation. The substrate carries the irreversible imprint of that record. The Schrödinger equation and the Born rule are not independent postulates: they are the same Lagrangian in two regimes, separated by the physical threshold of record formation. The Born rule emerges as the entropy-uniform limit of a more general selection rule — derivable, not assumed. The entropy increments σi are thermodynamic quantities, defined through physical irreversibility independently of any probability assignment. The arrow runs: physical event → thermodynamic σi → outcome weighting → Born rule. The framework makes falsifiable predictions. For two outcome channels, the log-odds shift is ln(P1/P2) = ln(|a1|²/|a2|²) + Δσeff, where Δσeff ≈ εη Δσeng, Δσeng is an independently calibrated entropy contrast and εη is an experimentally bounded suppression factor. Swap-symmetry of the tilt (F1), linear response in engineered entropy contrast (F2), universality of the coarse-graining parameter across platforms (F3), and exact Born recovery in the entropy-uniform limit (F4-F5) are testable on existing platforms independently of the numerical value of εη.