Finite-Distinguishability Closure (FDC) v5: A Constructive Derivation of Non-Arbitrary Numerical Residues from Minimal Axioms

This work presents Version 5 (v5) of the Finite-Distinguishability Closure (FDC) framework, a constructive mathematical approach derived from minimal axioms of finite distinguishability. The central objective is not to claim absolute uniqueness or complete formal classification, but to demonstrate that under a small set of structural constraints, admissible configurations become strongly restricted, leading to the emergence of specific numerical values as non-arbitrary residues. The framework proceeds as follows:- A minimal axiomatic structure is established based on distinguishability and non-arbitrariness.- Structural constraints (including exclusion principles such as Distinction Preservation and Distinction Generation Exclusion) progressively eliminate inadmissible configurations.- The remaining admissible structure yields numerical outputs without parameter fitting or empirical adjustment. Importantly, the derivation does not rely on calibration or post hoc fitting. The resulting numerical values arise as consequences of structural exhaustion, where degrees of freedom are fully constrained and residual quantities emerge. While the present work does not attempt exhaustive enumeration of all conceivable alternatives, it shows that under the imposed axioms, any admissible structure must satisfy strong constraints, leading to a minimal non-arbitrary solution. The numerical results obtained exhibit a level of structured consistency that is difficult to attribute to simple coincidence. Rather than asserting definitive correctness, this work argues that the observed agreement is more plausibly interpreted as a consequence of underlying structural constraints. This version focuses on improving mathematical clarity through minimal formalization, including:- explicit definitions using equivalence relations and quotient structures,- formalization of at least one No-Go argument in proposition-proof form,- clarification of numerical outputs as geometric/structural residues. FDC v5 should be understood as a baseline formulation: a structurally constrained framework demonstrating non-arbitrary emergence, open to further formal refinement and extension.