The Entropic Ladder: Geometric Unification from κ = 3

Abstract We present a geometric unification of quantum mechanics, general relativity, and the Standard Model from the A₂ hexagonal lattice at the Planck scale, governed by κ = 3. The constant κ = 3 is derived from the honeycomb conjecture, the E₈ Dynkin index ratio 60/20 = 3, and the super-attractive fixed point f (κ) =½ (κ+9/κ). The scale ladder Lₙ = ℓₚ × 3ⁿ spans all physical scales. The electroweak scale is predicted at n=35: vEW = Mₚ × 3⁻³⁵ = 244. 03 GeV, confirmed by experiment at 246. 22 GeV (0. 9% deviation consistent with geometric residue). Particle masses follow from the A₂ lattice norm n = a²+ab+b² and the exceptional Jordan algebra dimension 27: M = vEW (n/27) ^1/d, with d=1 for vector bosons, d=2 for scalars/fermions. The n=6 scalar at 116. 07 GeV is identified as a quasicrystal diffraction peak arising from the cut-and-project method of E₈ onto the 3D icosahedral group H₃. The fine structure constant is derived as the projection residue of E₈ onto H₃ via the McKay correspondence: α⁻¹ = 137 + 12/φ¹². Here 12 is the vertex count of H₃, and φ¹² is the known scaling invariant of the icosian ring (Elser & Sloane, 1987). Gravity emerges from the information stress-energy tensor Tⁱnfo_μν = -2κ (J_μ J_ν - ½ g_μνJ²). The geometric residue Δ = (π-3) /π explains the proton radius puzzle and the 95 GeV scalar excess. Light fermion masses are explained by ladder depth: the electron corresponds to the n=27 pure vertex at a deeper rung (k=12), with no exponential screening needed. The primary falsifiable test is a scalar resonance at 116. 07 ± 0. 05 GeV (n=6, d=2). Exclusion of a resonance in the 114. 0–118. 0 GeV window at 95% CL by the end of LHC Run 3 (July 2026) would falsify the n=6 assignment. The framework contains zero free parameters and zero external inputs.