Finite Distinguishability Closure (FDC) v10. 5 reformulates FDC as a finite closure type system. Its central result is no longer stated merely as a construction yielding Ncore = 42, but as a Normal Form Theorem: every minimal closed well-typed FDC signature is representation-equivalent to a unique normal form Delta*, whose rank is 42. The system is governed by six foundational constraints: Finite Distinguishability (FD), Non-Privileged Representation (NPR), Single Serial Ledger (SSL), Mapping Stability (MS), Distinction Preservation (DP), and Distinction Generation Exclusion (DGE). These are supplemented by the Structural Closure Condition (SCC), the Minimal Closure-Core Specification Discipline (MCCS), and the Minimal Structural Readout Grammar (MSRG). In v10. 5, the four structural components are interpreted as typed inventories inferred by the FDC type rules: - role inventory: nᵣ = 3- relational inventory: nᵣel = 11- phase/orientation inventory: nₚh = 9- address inventory: nₕ = 19 Thus the unique normal form is the finite closure signature Delta* = (Iᵣ, Iᵣel, Iₚh, Iₕ; tau, g, order), with rank (Delta*) = 3 + 11 + 9 + 19 = 42. An Elimination Theorem follows: no minimal closed well-typed FDC signature of rank other than 42 exists under the stated rules. FDC v10. 5 also unifies the two readout routes as projections of the same normal form Delta*. Route I is the scalar projection rhoI (Delta*) = 137, with rational correction terms evaluated in the appendices. Route II is the distributed vector projection rhoII (Delta*) = (1836, 206, 3477). All correction and attenuation terms are expressed through the normalized finite boundary measure muD (A) = |A| / |D|, so that readout coefficients are treated as finite sub-inventory densities rather than free parameters. A new Self-Description Closure Lemma is included: the FDC typing rules themselves, when expressed as a finite ledger record, require the same four-component Delta*-type structure. In this sense, FDC is self-descriptively closed and does not require an external structural signature beyond the Delta*-type. No empirical physical constants are used as inputs. Numerical comparisons with dimensionless physical quantities, including the inverse fine-structure constant and lepton/hadron mass-ratio-like readouts, are treated as downstream structural readout comparisons and are developed in companion works such as SCR. Within the broader MOF-FDC-PFC-SCR-AOH program: - MOF supplies the eliminative/descriptive-closure foundation. - FDC supplies the finite closure type system and its unique normal form Delta*. - PFC gives a physical reading of closure and phase-flow structure. - SCR records structural readouts and external numerical comparisons. - AOH applies related closure/readout ideas to Hubble-scale and galactic-scale phenomena. This version should be read as the mathematical reference layer of the program. Its primary role is to define the finite closure type system, its unique normal form, its dependency structure, and the admissible readout projections. Physical interpretations and observational applications are developed only in companion works.
T Momose (Thu,) studied this question.