The logical boundary of physical theories: a categorical criterion for continuous parameters

The logical boundary of physical theories: a categorical criterion for continuous parameters We ask which quantities a testable physical theory can determine from finitely many geometric data, and we answer this question with a categorical criterion. We construct a base category Cwhose objects are the discrete geometric structures of a concrete unified framework (the internal space N3,0 = S7/Z3, the gauge group, the fermion bundle, the spectral data, the black-hole operator), generated by a finite symbol table (9 objects, 9 morphisms). We prove three theorems. (T1) The set of reals definable from C by finite construction diagrams is countable, whereas the valueset S1 of a continuous parameter such as the CP-type angle θ is uncountable; by Cantor’s diagonal argument (in ZF alone, without Choice), θ /∈ DefinableC. (T2) No consistent categorical extension of C (Yoneda embedding, sheaf topos, derived category, classifying space, covering discretizations,torsion truncations) admits an injection S1 → A into a countable/finite set; a six-candidate table is screened by a pigeonhole argument, and the specific candidate π6(G2) = Z3 is doubly rejected(cardinality bound and layer mismatch). (T3) The theory TC admits conservative parameter extensions TC ∪{θ = θ1} and TC ∪{θ = θ2} with θ1= θ2 as equally valid models; θ is therefore not a categorical invariant of TC, and the divergence of models is an essential property of the structure.We then prove a limiting criterion: if a testable theory T has models diverging on a contin uous parameter θ, then no consistent extension T′ ⊇ T can make θ both definable and testable— any defining extension has language cardinality ≥ 2ℵ0 (non-finite) or is untestable. Locking a continuous parameter and testability are mutually exclusive. All proof steps are machine-checked (23 steps, 5 proofs, valid). We conclude that continuous parameters are categorically irreducible boundary conditions: the discrete skeleton (gauge group, generations, ultraviolet finiteness) is law,the continuous parameters are initial conditions.PACS numbers: 02.10.Ab, 03.65.-w, 04.60.-m, 11.30.E