A Time–Field Stability Model and a Machine-Checkable Derivation of a Dimensionless Chronos Constant

This paper introduces a minimal coupled time–field and density model in which a dimensionless Chronos constant, χ, emerges directly from linear stability analysis. The framework treats the time field Θ(t, x) as a scalar with a standard kinetic term and couples it to a coarse-grained density field ρ(t, x). Linearization around a homogeneous equilibrium yields a 2×2 system whose eigenvalues determine stability. The Chronos constant appears as the critical ratio of coupling strength to intrinsic clustering scale, separating distinct dynamical regimes. The derivation is entirely algebraic and designed to be fully verifiable by humans and automated reasoning systems. A structured verification protocol is provided, allowing readers or AI systems to confirm each step through matrix analysis, characteristic polynomials, dimensional checks, and reproducible numerical evaluation. This work makes no claims about consensus, scope, or completeness. It addresses only one question: whether the mathematics leading to χ is internally consistent.