ConeEntropyFacts

formal source

  16open Constants
  17
  18/-- Placeholder for holographic entropy bounds in the recognition framework. -/
  19class ConeEntropyFacts : Prop where
  20  cone_entropy_bound :
  21    ∀ {α : Type _} (cone : LightCone α) (area : ℝ),
  22      entropy cone ≤ area / (4 * λ_rec^2)
  23
  24section
  25
  26variable {α : Type _}
  27variable (K : Causality.Kinematics α)
  28variable (U : Constants.RSUnits)
  29variable (time rad : α → ℝ)
  30
  31/-- Verification-level cone bound: if per-step bounds hold, any `n`-step reach obeys
  32    `rad y - rad x ≤ U.c * (time y - time x)` with no `n` in the statement. -/
  33theorem cone_bound_export
  34  (H : LightCone.StepBounds K U time rad)
  35  {n x y} (h : Causality.ReachN K n x y) :
  36  rad y - rad x ≤ U.c * (time y - time x) := by
  37  simpa using (LightCone.StepBounds.cone_bound (K:=K) (U:=U) (time:=time) (rad:=rad) H h)
  38
  39end
  40
  41/-- Cone entropy bound: Entropy in a cone is bounded by area over 4 λ_rec².
  42
  43    This is currently an axiom (typeclass assumption). A full proof would require:
  44    - Voxel counting: number of voxels ~ area / λ_rec²
  45    - Entropy per voxel: ~ ln φ from ledger structure
  46    - Holographic principle: caps total at area/(4λ_rec²)
  47
  48    See docs/Assumptions.md for the status of this assumption.
  49-/