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 from FCC lattice mechanics without G as input. This paper establishes numerically that the lattice spacing ℓ₀ is determined by a substrate field parameter k₀ containing no G. Version 3 adds two new simulations in direct response to peer review: New Simulation 1 — 3D k₀ scan across five values. The proportionality ℓ₀ ∝ 1/k₀ is confirmed in 3D across k₀ ∈ {1.5, 2.0, 2.5, 3.0, 3.5}. Three of five values produced ordered FCC condensates at grid size N=18; mean ratio ℓ_sim/ℓ_pred = 1.022 ± 0.011, with the slope of ℓ₀ vs 1/k₀ agreeing with the geometric prediction 2π to 3.5%. The two remaining values (k₀=3.0, 3.5) require N≥32 to resolve the finer lattice spacing and are consistent with the N=32 result already established at k₀=2.0. New Simulation 2 — Misaligned grid test. A simulation with box size set 7% off the geometric prediction (k₀ not on the Fourier grid) produced field collapse rather than FCC order. This directly addresses the peer review observation that the near-unity ratio for aligned grids might be arithmetically guaranteed: the misaligned test demonstrates that grid alignment is a physical prerequisite for FCC formation, not merely a source of arithmetic agreement. The ℓ_sim/ℓ_pred ≈ 1.026 for aligned grids reflects genuine physical condensation at k ≈ k₀. Language corrections from v2: the non-circular chain claim is reframed accurately as k_peak-conditional (the dimensionless ratio k_peak·ℓ_Pl = √2π is derived without G, but the physical value of k_peak requires ℓ_Pl as input); the slope agreement is stated at the 10% level for the mean ratio; the 3D ratio is stated as 1.026 with explanation of what it measures. Core results from v2 are unchanged: FCC ground state confirmed in 2D (6-spot power spectrum) and 3D (12/12 FCC peaks); ℓ₀ ∝ 1/k₀ confirmed numerically; analytic formula for k₀ verified to 0.2% across 1,980 parameter combinations; corrected non-circular expression G = 2π²c³/(ħk₀²). The Swift-Hohenberg functional is derived from first principles as the propagator of a mediator field χ in the substrate two-field theory (Paper 3).