Finite Distinguishability Closure (FDC): Derivation Record for a Distinguished Scalar and Three Mass Hierarchies

This document records the mathematical construction underlying theFinite Distinguishability Closure (FDC) program. Starting from a finite set of structural axioms concerningdistinguishability, closure, and scalar readout, the constructionproduces • a distinguished scalar quantity• three hierarchical scale outputs whose numerical values coincide with the CODATA values of • the inverse fine-structure constant• the proton–electron mass ratio• the muon–electron mass ratio• the tau–electron mass ratio to high numerical accuracy. The present document serves as a derivation record establishingpriority of the mathematical construction. A full exposition andformal presentation of the theory will appear in subsequentpublications.