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 ρₜ is not introduced as a phenomenological parameter but is derived from a fundamental scalar field Φ (x) via ρₜ = ∂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 ρₜ, without dependence on matter currents for its sign. The theory naturally reproduces the 10¹22 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 ∇ρₜ and the internal sign of ρₜ.