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

The Hamiltonian constraint of canonical quantum gravity supplies no external time parameter. Relational dynamics describes change through correlations along a constraint history, and a suitable internal clock yields evolution between its sections. This paper derives the necessary continuation structure from LP and proves its classical and quantum representations. A constituted bearer with decidable identity under admitted transformation requires reidentifiable stages and a composition-compatible continuation relation. With a physically specified orientation this is a preorder; an acyclic branch or its mutual-reachability quotient carries a partial order. At a regular point the flow-box theorem supplies a local smooth clock. Global transversality, orientation, unique orbit intersection and interval completeness yield a unique groupoid of relational evolution maps. The clock value is not the continuation structure. Regular increasing recalibrations preserve the same sections and propagator, while orbit-dependent clock changes require event and arrow maps rather than a single synchronization of clock readings. Clock equivalence compares complete constitution, admissible futures, observables, probabilities and Q1/Q2a/Q2b at the same carrier and level. A clock-equivalence quotient removes redundant representations; it does not choose a physical generator. In the linearly deparametrizable sector a specified self-adjoint reduced Hamiltonian gives unitary evolution. We prove why the clock and formal differential expression do not determine that operator: two Robin domains give different ground states and measurable boundary responses for the same clock. We also prove a positive closure theorem: exact intertwining of finite Hermitian generators on coherent isometric embeddings gives an essentially self-adjoint global cylinder-core operator. A further transport example shows that classically equivalent positive constraint rescalings can have different quantum deficiency indices. These results identify the operator-domain and physical-measure data required for determinate quantum continuation.