The Space of Non-Local Temporal Functionals and Constraints on Quantum Gravity

This paper investigates the space of functionals satisfying axioms A2–A7 but not A1 (locality) from the framework developed in "Time as Constraint, Not Dimension" (DOI: 10.5281/zenodo.19042050). The uniqueness theorem of that paper established that seven axioms uniquely determine proper time as the only admissible measure of cumulative causal relational change. The present paper identifies locality as the axiom that forces this result and asks: what measures of temporal evolution are permitted when locality is dropped? Two representation results are provided. For continuum functionals, the paper shows they take the form dΦ = f · dτ, where f is a positive, generally covariant scalar that may depend on non-local information — curvature integrals, holonomies, path-dependent quantum correlations — subject to the constraint that f → 1 in any regime where locality is approximately restored. For discrete causal structures, algebraic versions of the axioms are formulated and shown to constrain the growth rate of discrete temporal functionals to match proper time in the continuum limit, conditional on the Hauptvermutung for causal sets. Four quantum gravity constructs are tested against axioms A2–A7 with explicit strict and charitable dual readings and a self-audit of the framework's structural biases. Causal set chain length satisfies all axioms naturally, though partly because the framework's path-functional structure aligns with causal set theory's intrinsic structure. Spin foam transition amplitudes present structural obstacles (lack of canonical path association, complex-valuedness) that reflect a genuine mismatch between the framework's path-functional orientation and spin foams' boundary-state structure. Fisher information between quantum gravitational states satisfies five of six axioms under physically motivated conditions, with null vanishing (A5) unverified — making it a promising but substantially unproven candidate. The Page–Wootters mechanism satisfies the axioms only partially, though recent covariant formulations narrow the gap. The paper identifies a structural parallel between two independent uniqueness results: the Paper I uniqueness of proper time (the unique local measure of geometric change) and the Chentsov–Morozova uniqueness of Fisher information (the unique quantum measure of state distinguishability). If Fisher information reduces to proper time in the semiclassical limit of quantum gravity, these two theorems would be connected by a classical limit — a concrete structural correspondence, not a vague analogy. Establishing this reduction is posed as the paper's most important open computational question. The paper argues that the framework provides genuine physical constraints on quantum gravity but with an inherent structural bias toward path-based approaches. The honest assessment is that the axioms are more informative about path-based approaches (causal sets, Fisher information) than about state-based or amplitude-based approaches (spin foams, Page–Wootters), and this limitation is intrinsic to the framework's path-functional formulation rather than a deficiency to be resolved. Acknowledgment This paper was developed through an iterative process involving the author and AI language models (Claude by Anthropic). The foundational questions, research program connecting the uniqueness theorem to quantum gravity, and all theoretical commitments originated with the author. The AI tools contributed formal analysis, connections to existing literature, adversarial review, and drafting. The author takes full responsibility for the claims and positions presented.