Gravity as Emergent Topology in a Planck-Scale FCC Lattice: Newtonian Sector and Yang-Mills Teleparallel Completion

We derive gravitational interaction from first principles within the Field of Resonance (FoR) framework, in which spacetime at the Planck scale is a face-centred cubic (FCC) lattice occupied by an SU(2)-valued preon field. The paper addresses two questions in sequence. In the Newtonian sector, we show that a massive body embedded in the FCC lattice acts as an Eshelby elastic inclusion, producing an exterior scalar potential Ψ ∝ 1/r satisfying Laplace's equation. A test wave packet couples to this potential through a non-derivative energy interaction uniquely selected by the Speed-Invariance Theorem, yielding the Newtonian force law F = −GMm/r². Newton's constant G = c³ℓ_Pl²/ħ is derived from FCC lattice parameters without G as input. The weak equivalence principle follows as a structural consequence of the wave-packet coupling theorem rather than an independent axiom. The anharmonic coupling constant α = 9/2 is derived from the collective wave-packet mechanism with full FCC elastic tensor contraction. In the relativistic sector (Option A), we extend the substrate Lagrangian by the natural Yang-Mills term ℒ_grav = −(1/4G) F^a_μν F^a_μν, where F^a_μν is the curvature of the SU(2) Maurer-Cartan connection. In the Weitzenböck gauge this term equals the torsion-squared action of the Teleparallel Equivalent of General Relativity (TEGR), placing FoR within a framework observationally identical to Einstein's GR for all local physics. The scalar field Ψ is reinterpreted as the trace/gauge mode of the metric perturbation h_μν rather than an independent mediator, resolving the PPN parameter γ = 1 automatically without fine-tuning. The extended framework passes all nine classical tests of general relativity: Newtonian gravity, weak equivalence principle, gravitational redshift, light deflection (4GM/c²b), PPN γ = 1, Mercury perihelion precession (43.0 arcsec/century), geodetic precession (6642 mas/yr), Lense-Thirring precession (41 mas/yr), and gravitational wave polarisations (2 modes). Papers 1–5 (scalar sector only) pass three of these nine tests; the Yang-Mills extension passes all nine.