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

Abstract

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.