Finite Distinguishability Closure: Axiomatic Determination of a Unique Structure and Its Readouts (FDC v7)

This paper presents the formal closure framework behind non-fitted structural readouts near several dimensionless quantities, including - α⁻¹ ≈ 137.035999,- proton/electron mass ratio ≈ 1836.152673,- muon/electron mass ratio ≈ 206.768283,- tau/electron mass ratio ≈ 3477.23. The empirical physical constants are not used as inputs. The aim of FDC is to define a finite-distinguishability closure grammar and determine which numerical and structural residues survive its admissibility conditions. External empirical values, such as CODATA reference values, are used only later as comparison targets in companion readout papers such as SCR. The framework is based on finite distinguishability, non-privileged representation, and a single serial ledger, together with closure/admissibility conditions such as distinction preservation, mapping stability, and distinction-generation exclusion. The paper is intentionally written as a physics-free formal layer: its claims should be evaluated as mathematical and structural claims, not as empirical physical assumptions. Within the broader MOF–FDC–PFC–SCR–AOH program: - MOF supplies the eliminative ontological foundation.- FDC supplies the formal closure grammar.- PFC gives a physical reading of the closure structure.- SCR records structural readouts near known dimensionless constants.- AOH applies the same grammar to Hubble-scale and galactic-scale phenomena. Readers interested in the numerical readout tables and external comparisons should consult SCR. Readers interested in physical interpretations should consult PFC and AOH. Readers interested in the ontological basis should consult MOF. This version should be read as the mathematical reference layer of the program. Physical interpretations and observational applications are developed only in the companion works.