sphereVolume

formal source

  80noncomputable def sphereArea (radius : ℝ) : ℝ := 4 * π * radius^2
  81
  82/-- Volume of a sphere. -/
  83noncomputable def sphereVolume (radius : ℝ) : ℝ := (4/3) * π * radius^3
  84
  85/-- **THEOREM**: Information scales as R², not R³.
  86    This is surprising because you'd expect interior degrees of freedom ~ R³. -/
  87theorem information_scales_as_area (R : ℝ) (hR : R > 0) :
  88    maxInformation (sphereArea R) (by unfold sphereArea; positivity) =
  89    4 * π * R^2 / (4 * planckArea) := by
  90  unfold maxInformation sphereArea
  91  ring
  92
  93/-- The "holographic" ratio: Area/Volume ~ 1/R.
  94    As regions get larger, the surface-to-volume ratio shrinks. -/
  95noncomputable def holographicRatio (R : ℝ) (hR : R > 0) : ℝ :=
  96  sphereArea R / sphereVolume R
  97
  98theorem holographic_ratio_scales (R : ℝ) (hR : R > 0) :
  99    holographicRatio R hR = 3 / R := by
 100  unfold holographicRatio sphereArea sphereVolume
 101  -- (4πR²) / ((4/3)πR³) = 3/R
 102  have hR_ne : R ≠ 0 := ne_of_gt hR
 103  have hπ_ne : (π : ℝ) ≠ 0 := Real.pi_ne_zero
 104  field_simp
 105
 106/-! ## The Ledger Explanation -/
 107
 108/-- In RS, the holographic bound comes from **ledger projection**:
 109
 110    1. Ledger entries are fundamentally 2-dimensional
 111    2. They live on causal horizons (surfaces)
 112    3. The "bulk" (interior) is encoded on the "boundary"
 113    4. This is automatic, not a choice