The Problem of Time as a Relational Continuation Theorem: Why a persister forces a relational continuation order, and when two clocks are the same time

In canonical quantum gravity the Hamiltonian is a constraint that annihilates physical states, and the Wheeler–DeWitt equation carries no external time — the “problem of time.” One standard response is relational: change is correlation between dynamical quantities along a constraint orbit, and where a good internal clock exists the timeless constraint deparametrizes into a Schrödinger equation. This paper imports that machinery unchanged. Its own contribution is a derivation. La Profilée fixes, from a pre-physical root, that any decidable identity verdict requires a carrier reidentifiable across the transformation; applied to a first-class constraint, this derives an oriented continuation preorder — a partial order on acyclic branches or on the mutual-reachability quotient — of which a clock is the local geometric representation at regular points of the flow. It follows that a determinate persister necessarily admits that continuation structure wherever it is determinable at all — the relational structure is not posited but forced by the persistence verdict being decidable. On that basis the paper proves an exact continuation theorem: an internal clock whose level surfaces cut the constraint orbits transversally, uniquely, orientedly, and completely induces a unique groupoid of relational evolution maps, so a vanishing canonical Hamiltonian does not mean the absence of change, and physical time is the oriented continuation order, not the value of any particular clock. In the linearly deparametrizable quantum sector a self-adjoint reduced Hamiltonian yields unitary relational evolution. And the multiple-choice problem — which of many clocks is right — is reduced to a decidable test: two clocks are the same time exactly when an isomorphism intertwines their dynamics, observables, probabilities, and identity () classes, a condition strictly stronger than unitary equivalence of reference frames because it carries the identity structure. What remains open is stated throughout: the existence of a globally admissible clock in full general relativity, the Hilbert space and dynamics of the full quadratic constraint, the universal equivalence of all quantum clocks, and any derivation of a thermodynamic arrow from the law.