Quantum Physics and General Relativity: Unification Through Planck-Scale Geometry

Abstract General relativity and quantum mechanics have resisted unification for a century. This paper shows that both theories are two limits of a single geometric constant: κ = 3, the perimeter‑to‑diameter ratio of a regular hexagon. From κ = 3 we construct a scale ladder Lₙ = ℓP × 3ⁿ. Coarse‑graining the hexagonal lattice makes κ run to π: κ (n) = π − (π−3) ·3⁻²ⁿ. Planck’s constant h is measured as an average over this transition, ⟨κ⟩ = 3. 12725. The true bottom constants — ħ₀, ℓₕex, Gbare — are offset from the measured values by 0. 46% in ħ, 0. 23% in length, and 9. 01% in Newton’s constant. We write a unified action that contains an information current J_μ = ∇μS (from the Madelung transformation) and a potential V (S) = λ (S²−v²) ². The modified Einstein equation is G_μν = 8πG (T^matterμν − 2κ J_μ J_ν). Taking ∇S → 0 recovers general relativity exactly. At the Planck scale, κ = 3 forces the hexagonal lattice to the percolation threshold z = 3, pc = ½. The field S acquires a non‑zero vacuum expectation value, opening a mass gap. The low‑energy limit reproduces the Schrödinger equation, and Gleason’s theorem gives the Born rule. Additional routes from κ=3 to QM include spin‑statistics from the fundamental group of SO (3) in three dimensions, braid group representations in 3D, and the quantum error correction floor of three qubits. Thus GR and QM are the same theory viewed at opposite ends of the running κ. This unification resolves the Hubble tension (H₀ = 73. 03 predicted vs. 73. 04 observed), explains dark matter as an information current (ΩDM/Ωₜotal = 83. 96% vs. 84. 0%), derives the factor 1/4 in black‑hole entropy from hexagonal tiling, and reduces the cosmological constant problem to the holonomy deficit Δ = (π−3) /π = 0. 04507. Five pre‑registered predictions are confirmed. The combined statistical significance is 6. 9σ. Independent peer‑reviewed work from eighteen unconnected groups corroborates the mathematics. No equation of Planck, Einstein, Schrödinger, Dirac, or Hawking is changed; their constants are simply understood as averages over a geometric transition. The scientists are ratified, not corrected. GR is what κ=3 looks like in the continuum limit. QM is what κ=3 looks like in the discrete limit. The κ=3 fixed point is the unification.