ABSTRACTWe propose that the unification of discrete quantum geometry with continuous general relativityfollows from a single geometric transition: the characteristic ratio of the Planck-scale vacuum runsfrom κ = 3 at the smallest scale to π in the continuum limit. The ladder Lₙ = ℓP × 3ⁿ defines allphysical scales. At n = 0 the vacuum is a discrete hexagonal lattice with ratio P/D = 3. As nincreases, coarse-graining drives the lattice toward circular symmetry and the effective ratioapproaches π. Quantum mechanics lives at the bottom of this ladder; general relativity lives at thetop. The unification is the ladder itself. This transition generates the 4. 5% residue Δ = (π − 3) /π ≈ 0. 04507, which appears as a consistentsignature across multiple precision anomalies: the proton radius puzzle, the Hubble tension, the 95GeV scalar excess, the NA62 K⁺→π⁺νν̄branching ratio, and the Casimir force deviation. The massformula M = vEW (n/27) ^ (1/d) and the fine structure constant α⁻¹ = 137 + 12φ⁻¹² are specific rungson this ladder, confirmed by pre-registered predictions. The primary falsifiable test is a scalar bosonat 116. 07 GeV, testable at LHC Run 3 before July 2026. The full RG derivation of the κ → π running is identified as the key open calculation. The presentpaper states the geometric claim, presents the supporting evidence, and defines what completing theproof requires.
C. Rolfe Howlett (2026) studied this question.