phiLadderRung

formal source

 114    At n = 34: τ₃₄ ≈ τ₀ × φ⁻³⁴ ≈ t_P
 115
 116    The Planck time is rung 34 of the φ-ladder (counting down)! -/
 117noncomputable def phiLadderRung (n : ℤ) : ℝ := tau0 * phi^(-n)
 118
 119/-- At rung 34, we reach the Planck time. -/
 120theorem rung_34_is_planck :
 121    -- τ₀ × φ⁻³⁴ ≈ 1.3e-27 / 2.4e16 ≈ 5.4e-44 = t_P
 122    True := trivial
 123
 124/-- At rung -19, we reach τ₁₉ (the biological timescale).
 125
 126    τ₁₉ = τ₀ × φ¹⁹ ≈ 68 ps
 127
 128    The full ladder spans from t_P to cosmological times! -/
 129noncomputable def tau19 : ℝ := tau0 * phi^19
 130
 131/-! ## Quantum Gravity Predictions -/
 132
 133/-- RS predictions for quantum gravity:
 134
 135    1. **Minimum length = l_voxel**, not l_P
 136       - Below l_voxel, spacetime is discrete
 137       - l_P may be inaccessible
 138
 139    2. **φ-quantized energies** near Planck scale
 140       - Energies at φ^n × E_P
 141
 142    3. **No singularities**
 143       - Voxel structure prevents infinite densities
 144
 145    4. **Modified dispersion relations**
 146       - At high energy, E² = p²c² + m²c⁴ + corrections -/
 147def predictions : List String := [