Time-Scalar Field Theory: Comprehensive Field Lemmas A Scalar-Time Derivation of the Physical Universe

We present a compendium of tightly stated lemmas showing how a single generative law, ∇μ(α T^μν)= 0, α ≡ ∂tΘ > 0, produces the effective equations of physics across domains. Each lemma gives a minimal derivation sketch from the scalar-time continuity principle and a concrete empirical note indicating how the statement is, or can be, tested. The results synthesize quantum dynamics (Schrodinger/Dirac), gravitation (Einstein limit and weak-field exterior tail), electromagnetism (Maxwell structure from Θ-phase), thermodynamics/information (Clausius, Boltzmann, Landauer, decoherence), and cosmology (Hubble relation and solar-system predictions). Together, they exhibit a single, falsifiable pathway from scalar-time geometry to the observed laws of nature.