Theorem: Clock separation before coordinate identification

Retain the declarations of Sections sec:typed–sec:incidence, including the Cauchy data, positive timelike branch, clock calibration, spatial boundary condition, prescribed parameter table, and fixed ramp.

The four labelled objects ${\mathsf{C}}_{\mathrm{VI}}$, ${\mathsf{C}}_{\mathrm{E}}$, ${\mathsf{C}}_{\mathrm{VIII}}$, and ${\mathsf{R}}_{\mathrm{X}}$ are not interchangeable functional prescriptions. The following five separations hold:

1. The harmonic relational scalar and scalar-clock relational lapse are distinct. A scalar value alone does not determine the lapse. Classical descent does not imply regulator removal.

2. Original and modified rods are distinct, and the compact perturbation (eq:rod-witness) distinguishes their quadratic relational lapses. Their formulas and asymptotic coefficients cannot be transferred without a connecting construction.

3. The density ramp is a prescribed non-clock. Its first harmonic response has a transition-region pure-gauge defect that a second-order addition cannot remove while the first prescription remains fixed.

4. The incidence parameter, its finite code, and the branch-normal speed do not supply the matter-defined clock, original rods, or relational lapse. A foliation-defined scalar can reproduce the normal speed only after that additional scalar is declared.

5. The supplied Family 271 catalogue and reported Lean scope supply no premise or identification for these constructions.

The substantive detecting calculations are on the stated kinematic domains. The theorem makes no universal claim that numerical coincidences or future explicitly constructed comparison maps are impossible.

Proof. Item (i) is Lemma lem:scalar-lapse together with Lemma lem:descent-not-removal. Item (ii) is Lemma lem:rod-first, Theorem thm:rod-detection, Lemma lem:elliptic-hyperbolic, and Lemma lem:asymptotic-map. Item (iii) is Lemma lem:ramp-nonclock, Proposition prop:ramp-defect, and Corollary cor:first-order. Item (iv) is Lemma lem:incidence-lapse and Propositions prop:retiming and prop:foliation-clock. Item (v) is Proposition prop:catalogue.

The objects also have the different domains, targets, equations, and auxiliary data recorded in Definition def:object. No cross-model identification is supplied by the labels. The detecting witnesses prove the stated nonidentities without assuming that every evaluation must differ.