Temporal Rate and Quantum Probability: Consistency Theorems and a Falsifiability Framework for Temporal Rate Ontology

Abstract

Temporal Rate Ontology (TRO) proposes that the locally realized rate of temporal progression is more fundamental than spacetime geometry. In its simplest representation, dτ = ρ(x,t) dt, where ρ is the local temporal-rate field and spacetime geometry is interpreted as an effective representation of relations among locally accumulated temporal rates. This paper develops a phenomenological, falsifiable test of TRO at the interface between temporal-rate structure and quantum measurement, and formulates the proposal around three explicit mathematical consistency conditions rather than a single postulated weighting law. The primitive object is a physically instantiated history γ, admissible only as a member of a decoherent partition C = {γ_1,…,γ_n} in the sense of consistent- or decoherent-histories quantum mechanics. Within such a partition, ordinary quantum-measure theory already supplies well-defined probabilities p_Q(γ_i|C), satisfying Σ_i p_Q(γ_i|C) = 1; this paper does not modify Sorkin's quantum measure itself, which is not in general additive outside a decoherent partition, but reweights the ordinary probabilities that quantum measure theory already assigns within one. The proposed TRO probability is P_TRO(γ_i|C) = p_Q(γ_i|C) exp(ηΘ[γ_i]) / Σ_{γ_j∈C} p_Q(γ_j|C) exp(ηΘ[γ_j]), where Θ[γ] = ∫_γ dτ = ∫_γ ρ dt is the accumulated temporal measure along γ, well-defined only for a timelike physical history. We require three consistency conditions. First, shift invariance: normalized probabilities must be unchanged under a common additive shift of every history's Θ value. Under positivity and mild regularity, this fixes the weighting function uniquely up to normalization as F(Θ)=C exp(ηΘ). Second, history-partition invariance: two admissible decoherent partitions describing the same physically coarse-grained outcome must assign it the same TRO probability. At a single coarse-graining step this forces the effective temporal composition law Θ_eff[Γ]=(1/η) ln[(Σ_i p_i e^{ηΘ_i})/(Σ_i p_i)], which reduces to the ordinary probability-weighted mean proper time as η→0 and is associative under arbitrary finite regrouping. Third, no-signalling: temporal weighting must not allow a spacelike-separated measurement choice to alter a remote marginal. A local-reweighting plus constrained information-projection construction guarantees no-signalling marginals by definition but encounters a support obstruction for maximally correlated states. For binary full-support quantum tables, the constrained reconstruction can nevertheless be reduced exactly to an odds-ratio equation for the reconstructed correlation E_xy; the resulting correlation differs from the quantum value only at second order in the local temporal biases, so CHSH deviations begin at O(η²) for fixed finite temporal contrasts. This does not establish quantum realizability or the Tsirelson bound, which remain additional consistency tests. The antecedent physical premise, that an accumulated temporal history may enter outcome probabilities at all, is not derived here from the deeper dynamics of ρ. It is the phenomenological TRO hypothesis tested by this paper; the propositions below determine what mathematical form and consistency constraints such a dependence must satisfy if it exists. We also state explicitly, that in the minimal model developed here temporal weighting does not modify coherent unitary evolution; it enters only in the probability assigned to members of an already-decohered history set, after ordinary quantum dynamics and decoherence have run their course. We separate this proposal from the decoherence-based proper-time witness of Zych, Costa, Pikovski, and Brukner (2011), note that a photon cannot carry Θ since dτ = 0 identically on a null worldline, and record the dimensional content [η] = T⁻¹, with η = κ/τ* for dimensionless κ and characteristic temporal scale τ*.