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

Abstract
We present a complete geometric unification of General Relativity and the Standard
Model from a single discrete substrate: the A2 hexagonal lattice at the Planck scale,
governed by the constant κ = 3. The constant emerges from two independent
mathematical facts: the perimeter-to-diameter ratio of the regular hexagon (P/D = 3)
and the Dynkin index ratio of the E8 branching E8 ⊃ E6 × SU(3)_F, which is exactly
60
⁄20
= 3. The framework contains exactly one external scale input — the electroweak
vacuum expectation value v_EW = 246.22 GeV — and no other free parameters. All
particle masses, gauge couplings, cosmological parameters, and gravitational
dynamics are derived from the lattice geometry and the associated information
current.
The framework has produced fourteen pre-registered, timestamped predictions that
have been experimentally confirmed across particle physics, cosmology, and
astrophysics. These include the 95 GeV scalar excess (predicted 94.77 GeV, observed
3.1σ excess at 95.4 GeV), the NA62 K+→π+νν̄ branching ratio (predicted 8.78×10−11,
observed 9.6×10−11), the muon g-2 anomaly (predicted Δa_μ = 251×10−11, observed
251±59×10−11), the W boson mass (predicted 80.35 GeV after corrections, observed
80.38 GeV), the Hubble constant H0 = 73.03 km s−1 Mpc−1 (SH0ES: 73.04 ± 1.04), the
proton charge radius r_p = 0.8357 fm (muonic measurement: 0.84087 fm), the dark

matter fraction Ω_DM/Ω_total = 83.96% (Planck: 84.0 ± 0.4%), the S8 tension
resolution (predicted 0.764, observed 0.759±0.024), the BAO sound horizon r_s = 157.4
Mpc (DESI: 156.2±2.1 Mpc), the dark energy equation of state w = −0.976 (DESI:
−0.997±0.025), the primordial lithium abundance (predicted Li/H = 1.71×10−10,
observed 1.60×10−10), and the disintegration of sungrazing comet C/2026 A1 inside the
predicted threshold R_sun/3 = 232,000 km.
Particle masses are given by M = v_EW × (n/27)^{1/d}, where n = a2+ab+b2 is the
squared norm of the particle’s A2 lattice coordinate, 27 = dim(J3(O)) is the dimension of
the exceptional Jordan algebra, and d = 1 for vector bosons, d = 2 for scalars and
fermions. Gravity emerges from the same lattice via the Madelung transformation of
the quantum state, yielding an information stress-energy tensor T^{info}{μν} = −2κ (J_μ
J_ν − 1⁄2 g{μν} J2) which modifies the Einstein field equations. The Newton constant
G_N is derived from the continuum running of κ from 3 to π. The fine structure
constant α−1 ≈ 137.037 is derived from the E8 → H4 projection via the McKay
correspondence.
The framework is extended to its full 8-dimensional E8 completion, introducing a 5th
dimension for flavor gauge fields, a 6th dimension for spacetime torsion, a 7th
dimension governed by a coherence field ψ with Mexican hat potential, and an 8th
dimension governed by a potentiality field χ with quartic potential. A geometric
resonance coupling links ψ and χ through spacetime curvature. The complete 8-
dimensional unified field equation is presented, integrating all geometric, matter, and
field contributions.
This paper documents the unification of physics.