Spontaneous FCC Lattice Formation from Non-Local Scalar Field Theory: A Numerical Demonstration of the Non-Circular Derivation of Newton's G

This paper presents numerical simulations confirming the non-circular derivation of Newton's gravitational constant G in the Field of Resonance (FoR) framework. The companion analytical paper (DOI: 10.5281/zenodo.19141698) derives G = c³ℓ₀²/ħ from FCC lattice mechanics without G as input. This paper establishes numerically that ℓ₀ is determined by a substrate field parameter k₀ containing no G. Key results: A non-local scalar field theory with Swift-Hohenberg kernel spontaneously produces a hexagonal ground state in 2D (6-spot power spectrum confirmed) and a full FCC ground state in 3D (12/12 FCC peaks confirmed, ratio = 1.000 exactly). A scan of five k₀ values confirms ℓ₀ ∝ 1/k₀ with slope agreeing with the geometric prediction 4π/√3 to 1.7%. The analytic formula k₀ = k_peak × (U₀α_k/σ²) / (2c₀² + U₀α_k/σ²) is derived and verified to 0.2% accuracy. The geometric conventions resolving the angular wavenumber and FCC nearest-neighbour distance give the corrected non-circular expression G = 2π²c³/(ħk₀²). The Swift-Hohenberg functional, used here as a simulation tool, is derived from first principles as the propagator of a mediator field χ in the substrate two-field theory, as established in companion Paper 3 of the FoR series.