Completing Einstein's Unified-Field Programme at the Planck Scale: General Relativity, Quantum Structure and the Standard Model from the κ = 3 Lattice

Abstract absence of a microscopic principle capable of fixing its non-Levi-Civita sector. That principle is supplied here by an A₂ hexagonal Planck-scale substrate with stability constant κ = 3. Its discrete-to-continuum mismatch, Δ = (π − 3)/π = 0.04507034…, and scale bridge F_n = 9⁻ⁿ determine how microscopic distortion becomes invisible under coarse- graining. The unified connection Γ = Γ_LC + F(K + L) + Γ_EM is inserted into a single action before variation. Gravity, electromagnetic structure, microscopic distortion and quantum matter therefore enter as sectors of one geometry rather than as independently coupled theories. General Relativity is recovered in the macroscopic limit, while the full connection necessarily generates linear, derivative and quadratic distortion terms. The same geometric construction is applied across particle, gravitational and cosmological scales. Quantum probability is selected by the lattice Hilbert-space structure; Born-weighted matter provides the gravitational source; particle masses are organised by A₂ norms; black-hole entropy is recovered from the same cell geometry; and the cosmological sector is governed by the same discrete-to-continuum deformation. The framework is already empirically prospective rather than merely retrospective. Timestamped predictions include observations subsequently reported after registration, while further registered tests remain open with explicit falsification criteria. Most sharply, the framework predicts an observationally Einsteinian tensor ringdown together with a scalar breathing component whose characteristic amplitude is fixed by the holonomy deficit, Δ ≈ 4.5%. A separate structural audit of 600 equations drawn from Einstein’s major programmes finds 53 nonlinear curvature identities displaying the structure generated by the scaled connection, 140 explicit openings for the framework, 272 neutral or compatible equations, 62 requiring re-derivation after action-level insertion of F, 73 outside the geometric scaling sector, and no direct algebraic contradiction. The central claim is therefore not that Einstein’s geometry must be replaced. It is that the microscopic scale, source and constraint structure required to complete it were unavailable to his century.