Relational Time from Physical Records

Time acquires meaning through changes in the physical relationships between all the objects of a system. No background time is required within this framework. A single structureless point in an otherwise empty universe supplies no changing intrinsic spatial record. A minimum of two points is required for a spatial relationship to form. A duration for a change in spatial relationships requires a dynamical law, physical calibration, and information about the correct branch of movement. For finite classical systems, the Gram matrix of positions and momenta contains the centered phase configuration of the particles, up to a common orthogonal transformation. The closed equations of motion yield a duration functional that remains unchanged by multiplying the constraint by any positive factor. One example of this type of clock is the counterrotor, with its backreaction retained within the system. Special relativity can be incorporated through positive-energy internal clocks that provide proper intervals. Together with measured proper intervals, invariant momentum records determine event durations in the system's total-momentum rest frame. The four-spin model bounds deviations of conditional evolution from its Schrödinger comparator. The quadratic-rotor family is an exactly soluble model yielding a trace-distance discrepancy and the tradeoff between increasing clock resources and sensitivity to finite angular resolution. The record-sufficiency criteria identify which records determine duration, and the numerical noise tests quantify uncertainty in duration reconstructed from finite classical canonical measurements. The quantum models separately bound conditional-state errors. These questions about clock choice, calibration, recurrence, and quantum conditioning are all part of the gravitational problem of time.“Preprint submitted to Classical and Quantum Gravity on 14 September 2026. Not yet peer reviewed.”