Temporal Information Dynamics in the Scalar Temporal Field Ontology: Part VI — Field-Theoretic Information Action and the τ –Hamiltonian Flow

This Part VI extension of Temporal Information Dynamics (TID) develops the first fully covariant field-theoretic formulation of information in the Scalar Temporal Field Ontology (STFO). While earlier parts established temporal depth, topological protection, channel theory, and information thermodynamics, this work builds the full mathematical backbone: an information action, Euler–Lagrange equations, canonical momentum, and a $\tau$–Hamiltonian flow governing information evolution. The paper constructs an information action $S_{\mathcal{I}}[\mathcal{I},\tau]$ defined on the disformal metric $g^{\mathrm{eff}}{\mu\nu}$, derives the fundamental field equation for information propagation, obtains the canonical momentum $\Pi{\mathcal{I}}$, and formulates a Hamiltonian governing temporal information flow. A Liouville theorem is shown to hold for smooth $\tau$ and to be violated under temporal fluctuations $\delta \tau$, linking Hamiltonian evolution to entropy production (Part IV). The information stress–energy tensor $T^{(\mathcal{I})}_{\mu\nu}$ is derived, demonstrating how information backreacts on effective geometry. Together with Parts I–V, this work completes the central mathematical structure of Temporal Information Dynamics, providing a unified framework in which transport, entropy, Hamiltonian evolution, and geometric coupling emerge directly from the scalar temporal field $\tau(x^\mu)$. Author: Cale Scott HoweProgram: Scalar Temporal Field Ontology (STFO)Contact: [email protected]