Flow-Time Theory: A Fundamental Scalar Field Theory of Entropy

This paper presents a self-contained field-theoretic formulation of Flow-Time Theory. The flow-time density ρ_t is not introduced as a phenomenological parameter but is derived from a fundamental scalar field Φ(x) via ρ_t = ∂_t Φ. A complete Lagrangian density is constructed, from which all core axioms follow as equations of motion. The theory includes a higher-derivative spatial term D (∇²Φ)² that stabilizes the vacuum, generates diffusion dynamics, and improves ultraviolet behavior. Entropy production is driven solely by the spatial gradient of ρ_t, without dependence on matter currents for its sign. The theory naturally reproduces the 10^122 ratio between Planck-scale and cosmic flow-time densities and provides a closed mathematical framework suitable for quantization. A key feature is the restoration of full symmetry between entropy increase and decrease, which appear as exact mirror processes determined only by the sign of ∇ρ_t and the internal sign of ρ_t.