Geometric Unification of General Relativity and the Standard Model from the κ = 3 Lattice

Abstract We derive General Relativity and the quantum structure of matter from a single discretesubstrate: the A2 hexagonal lattice at the Planck scale. The framework is governed bythe stability constant κ = 3. This constant emerges from the perimeter-to-diameter ratioof the regular hexagon (P/D = 3) and the Dynkin index ratio of the E8 ⊃ E6 × SU(3)Fbranching. We introduce no continuously adjustable parameters. The geometry yields theBekenstein-Hawking black hole entropy, the cosmological constant ratio, and the Hubbletension correction via a universal holonomy deficit ∆. General Relativity is recovered exactlyin the exhausted-current limit (Jμ → 0). Quantum Mechanics emerges via the Born rulefrom non-commuting magnetic translations. The framework yields timestamped predictions(viXra:2512.0067) that are experimentally confirmed, including the dark energy equationof state w = −0.976 and the proton charge radius. Several falsifiable predictions remainpending.