Numerical Closure and Verification Pathway for Newton's Gravitational Constant from x-Stabilized Time-Field Dynamics

This paper presents a structured numerical closure and verification pathway for Newton’s gravitational constant G within the Chronos time-field framework. Rather than treating gravity as a fundamental interaction, the Chronos approach interprets gravitational coupling as an emergent macroscopic effect of an underlying structured time field. The analysis introduces a raw time-field scale, a stability constant x≈0.5512855984, and a base spectral frequency, leading to a closed-form relation for G derived from Planck-scale dynamics. The work reduces the problem of explaining the value of G to a single requirement: the independent derivation of a fundamental time-field scale, expressed as either a base frequency Ω1 or a characteristic wavelength λΘ. A minimal Hamiltonian formulation is provided, along with a shell-stability model that connects time-field structure to gravitational coupling. To remove circularity, the framework introduces an intrinsic time-field coupling ΓΘ, showing that Newton’s constant emerges as its macroscopic limit. The paper further defines a clear falsifiability condition and proposes measurable deviations in the effective gravitational coupling: Geff=ΓΘ1+ϵ(∇Θ,ρΘ,χ) This establishes a testable prediction that small, environment-dependent variations in G may exist due to local time-field structure. The work does not claim a completed first-principles derivation of G, but instead provides a mathematically consistent and experimentally testable pathway toward one.