Stratified Coexistence of Time Exact Linear–Gaussian Update Identities and Descriptive EKF Regimes

We present an inferential framework in which effective temporal direction is characterized through posterior precision dynamics. In the linear–Gaussian setting, we derive an exact additive identity for the precision update and obtain a closed-form expression for its expected increment, thereby establishing a theorem-level result under explicit assumptions. We then examine two controlled extensions beyond this regime: an EKF-local update statement that remains exact only conditional on the stepwise Jacobian, and full-step nonlinear behavior that is treated strictly as descriptive and supported only by reproducible numerical evidence. The central contribution of the paper is therefore twofold: an exact linear–Gaussian precision-update result, and a disciplined separation between exact and descriptive layers in nonlinear inference. This yields a reproducible inferential structure for temporal ordering that is mathematically exact in the linear–Gaussian update regime, empirically structured beyond it, and explicitly bounded in scope. No universal irreversibility law, no microphysical derivation, and no application-layer claims are asserted here.