Counting Axioms for the Temporal Parameter: Axiomatic and Categorical Foundations of the Nᵣef Framework

Four axioms (N1–N4) define Nᵣef as the accumulated state-transition count of a physical reference system; the definition is realisation-independent and structurally excludes the Pauli obstruction. The mass-shell constraint yields the Pikovski Hamiltonian Ĥₚhys = Ĥ₀/γ within the expansion regime, and the budget equation (dτ/dt) ² + v²/c² = 1 as the exact constraint. A forgetful functor F: Mathₜ → PhysN identifies surplus automorphisms of the mathematical temporal category that have no counterpart in the physical category. The computational gauge characterisation (CG1–CG3) shows that the surplus is mathematically necessary (Stone's theorem), informationally determined for KMS states (modular analyticity), and physically vacuous (axiom N1). The signature theorem proves that Minkowski (1, 1) is forced by the non-compactness of the boost group, itself a theorem of counting and the relativity principle. The extension to the full (1, 3) signature follows from spatial isotropy via Schur's lemma on the irreducible SO (3) representation. A universality proposition establishes that every physical clock satisfying N1–N4 dilates by 1/γ, with the bridging assumption Abridge naming what the quantum universe contributes to the derivation. These results supply the axiomatic and categorical foundations for the surplus structure programme introduced in Part I; Part III extends the k-operator developed here into the quantum regime.