Discrete Proper Time and a Tempo Field: Core Axioms and Consequences

This note recasts familiar quantum and QFT structures in an operator-algebraic framework with discrete proper time t = nτ, a unitary one-step evolution U, and a positive tempo field Θ. It is not a new logical system or a new physical theory; it is a compact restatement intended to streamline calculation and interpretation. A conditional expectation E_B onto the principal band B gives an operational reading of apparent probabilities as band-limited outputs of deterministic, stroboscopic evolution. Results are organized into kinematic statements (from the axioms alone) and dynamical statements (under stated sufficient conditions). Einstein-type causality for the discretized picture follows from quasi-local (Lieb–Robinson) bounds, agreeing with metric light-cones in appropriate limits. In the continuum limit τ → 0, standard generators appear via the Cayley–exponential correspondence. Throughout the kinematic part, Θ is treated as a background scale. A minimal dynamics for Θ is described separately; it is not assumed for the kinematic results.