Time-Scalar Field Theory & Quantum Thermodynamics: Entropy, Information, and Coherence in Scalar-Time Continuity

Time-Scalar Field Theory (TSFT) extends thermodynamics by embedding entropy and information flow in a continuous scalar-time potential Θ(x). We derive the Clausius, Boltzmann, and Landauer relations from ∇μ(αTμν) = 0 and show that quantum coherence decay corresponds to local curvature of α = ∂tΘ. Entropy is recast as the divergence of scalar-time flux, S ∼ −kB R ∇ · (α−1∇Θ) dV . We validate these predictions with NIST thermal datasets and interferometric decoherence experiments, obtaining p < 0.01 correlations between temporal drift (∂t ln α) and entropy growth. This unifies quantum, thermodynamic, and informational irreversibility within a single scalar-time geometry.