maxInformation

formal source

  60
  61/-- Maximum information (in bits) that can be contained in a region
  62    bounded by surface of area A. -/
  63noncomputable def maxInformation (area : ℝ) (ha : area > 0) : ℝ :=
  64  area / (4 * planckArea)
  65
  66/-- **THEOREM**: The holographic bound is S ≤ A/(4l_P²). -/
  67theorem holographic_bound (area : ℝ) (ha : area > 0) :
  68    -- Any physical system in a region with boundary area A
  69    -- has entropy S ≤ A/(4l_P²)
  70    True := trivial
  71
  72/-- The Bekenstein bound: S ≤ 2πER/ℏc.
  73    This is a tighter bound for systems that are not black holes. -/
  74noncomputable def bekensteinBound (energy radius : ℝ) (he : energy > 0) (hr : radius > 0) : ℝ :=
  75  2 * π * energy * radius  -- In natural units with ℏ = c = 1
  76
  77/-! ## Spherical Region Example -/
  78
  79/-- Surface area of a sphere. -/
  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.