A Weak-Field Sourcing Law for Temporal Flow in the Scalar Temporal Field Ontology

This paper presents a conservative, weak-field construction within the Scalar Temporal Field Ontology (STFO) addressing how temporal flow responds to matter in the classical regime. Time is treated as a real, monotonic scalar field τ(x), with the local temporal flow rate defined as N = dτ/dt. Building on prior STFO results showing that spatial gradients of τ reproduce gravitational time dilation and free-fall motion, this work constructs a minimal effective energy functional for N. Variation of this functional yields a sourcing law that reproduces the Newtonian Poisson equation under appropriate assumptions, including weak gravitational fields, static non-relativistic sources, and asymptotic flatness. The construction is explicitly non-fundamental and non-unique, serving as a proof of consistency rather than a replacement for General Relativity. All empirical predictions of classical gravity are preserved. The result demonstrates that STFO admits a coherent weak-field dynamics compatible with Newtonian gravity while maintaining an ontology-first framework.