A Renormalizable Chronos Lagrangian: Field Dynamics, Linear Stability, and the Emergence of the Chronos Constant χ

This paper introduces the first fully renormalizable field-theoretic formulation of the Chronos framework. It constructs a minimal scalar-field Lagrangian for the time field Θ(t, x) and the density field ρ(t, x), using only dimension-4 operators to ensure renormalizability and physical consistency. By deriving the Euler–Lagrange equations and taking the overdamped/low-energy limit, the model collapses exactly into the reaction–diffusion Chronos PDEs validated in prior work. Linearization around equilibrium reproduces the same 2×2 stability matrix and dispersion relations found in the machine-checked December 2025 Chronos stability papers. The Chronos constant χ emerges directly from Lagrangian parameters as a dimensionless ratioχ = C / √(κσ)and reduces to χ = C / κ in the symmetric regime. The paper demonstrates that the same χ controlling homogeneous stability also determines global k-mode stability in the full dispersion analysis. A verification protocol is included to allow automated reasoning systems to confirm every derivation step without relying on consensus or external references. This paper forms the theoretical bridge between microscopic time-field dynamics and the macroscopic Chronos/CHaSSE behavior.