Temporal Hydrodynamics and Entropy Production in a Scalar Time Field Framework (Stand-Alone)

This document develops the hydrodynamic limit of a scalar time field τ(xμ) whose gradient defines causal direction and energy flow. In the infrared regime, the dynamics admit a fluid-like description with density ρτ, pressure pτ, and four-velocity uμ = ∇μτ / √X. Bulk-viscous corrections generate entropy in expanding backgrounds, with phenomenologically relevant signatures in cosmology and gravitational-wave propagation. Explicit entropy production in FRW spacetime is shown, along with axisymmetric temporal vorticity (“eddies in time”). Stability under degenerate higher-order scalar–tensor constraints is analyzed. The resulting fluid can modestly suppress late-time structure growth, potentially alleviating the S8 tension, while avoiding Ostrogradski ghosts and preserving luminal tensor speed. Observable consequences are forecast for Euclid, DESI, Rubin, Roman, and LISA. This manuscript functions as an official companion to the Scalar Temporal Field Ontology (Version 26.1).