Non-A Priori Construction of Time III: Physical Realization of Clocks, the Quantum Time Axiom, and Atemporal Quantum Mechanics

AbstractAs the third paper on the non-a-priori construction of time, this paper constructs clocks from mathematical temporal-like structures, defines time in quantum theory axiomatically, and then forgets its temporal interpretation to obtain a formulation of quantum evolution in which physical time is not a primitive variable. Numberless time and temporal numbers from Papers I and II are reintroduced as repetition and recurrence through group actions, and as succession and history through the Peano successor operation, respectively. Physical clocks are defined by equipping their combination with readout and calibration, and an ideal equivalence relation between different types of clocks is examined. A “quantum time axiom” is adopted to define clock time through the energy gap and accumulated relative phase of a two-level quantum clock. The mutual reference between quantum theory and clocks is thereby treated as an axiomatic stopping point grounded in concrete physical realization. A forgetting operation then removes the temporal interpretation while retaining successor operations, group actions, and transformations on Hilbert space as pure operations. A dimensionless unitary group is constructed from a self-adjoint energy operator and the energy scale of a reference clock, and Peano updates, continuous extensions, and invariance under changes of reference clock are examined. The standard Schrödinger equation is recovered through the quantum time axiom, and the dimensionless differential equation is presented as one construction of an atemporal Schrödinger equation. A possible adjunction between free construction and forgetting for time in general is also considered. Finally, comparison with the Page–Wootters mechanism and related approaches distinguishes the nonprimitiveness of a time variable from timelessness in a completely closed system. The paper presents a coherent foundational construction of quantum theory: construct time, realize it as a clock, axiomatize time in quantum theory, and then forget its temporal interpretation. Detailed mathematical derivations and explanations of the constructions presented in this paper are provided in a separate supplementary manuscript. AI Assistance DisclosureGenerative AI was used as an auxiliary tool in drafting, adjusting expressions, and organizing the structure of this paper. The author bears final responsibility for the text, claims, citations, and factual verification. Keywords: non-a-priori construction of time, numberless time, temporal numbers, rhythm time, count time, quantum clocks, quantum time axiom, forgetting the temporal interpretation, atemporal quantum mechanics, group actions, Peano arithmetic, reference-clock invariance