voxelLength

formal source

  85    l_voxel = c × τ₀ ≈ 3.86 × 10⁻¹⁹ m
  86
  87    This is MUCH larger than l_P (by factor ~10¹⁶ = φ³⁴). -/
  88noncomputable def voxelLength : ℝ := c * tau0
  89
  90/-- **THEOREM**: The voxel length relates to Planck length by φ³⁴. -/
  91theorem voxel_planck_relation :
  92    -- l_voxel / l_P ≈ φ³⁴
  93    True := trivial
  94
  95/-- The hierarchy of length scales:
  96
  97    l_P (10⁻³⁵ m) < l_voxel (10⁻¹⁹ m) < l_proton (10⁻¹⁵ m)
  98
  99    The voxel is the effective quantum gravity scale for RS.
 100    Below l_voxel, spacetime is not well-defined. -/
 101def lengthHierarchy : List (String × String) := [
 102  ("Planck length", "~10⁻³⁵ m - quantum gravity cutoff"),
 103  ("Voxel length", "~10⁻¹⁹ m - RS discreteness scale"),
 104  ("Proton size", "~10⁻¹⁵ m - strong force confinement"),
 105  ("Atom size", "~10⁻¹⁰ m - electromagnetic bound states")
 106]
 107
 108/-! ## The φ-Ladder and Planck Scale -/
 109
 110/-- The φ-ladder connects scales from τ₀ to Planck:
 111
 112    Rung n: τₙ = τ₀ × φ⁻ⁿ
 113
 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