CoarseGrain

formal source

   5namespace ClassicalBridge
   6
   7/-- Coarse graining with an explicit embedding of ticks to cells and a cell volume weight. -/
   8structure CoarseGrain (α : Type) where
   9  embed : Nat → α
  10  vol   : α → ℝ
  11  nonneg_vol : ∀ i, 0 ≤ vol (embed i)
  12
  13/-- Riemann sum over the first `n` embedded cells for an observable `f`. -/
  14def RiemannSum (CG : CoarseGrain α) (f : α → ℝ) (n : Nat) : ℝ :=
  15  (Finset.range n).sum (fun i => f (CG.embed i) * CG.vol (CG.embed i))
  16
  17/-- Statement schema for the continuum continuity equation (divergence form in the limit). -/
  18structure ContinuityEquation (α : Type) where
  19  divergence_form : Prop
  20
  21/-- **HYPOTHESIS**: Coarse-grained Riemann sums converge to a finite limit. -/
  22def H_Convergence (CG : CoarseGrain α) (f : α → ℝ) (I : ℝ) : Prop :=
  23  ∀ ε > 0, ∃ N, ∀ n ≥ N, |RiemannSum CG f n - I| < ε
  24
  25/-- Discrete→continuum continuity: if the coarse-grained Riemann sums of a divergence observable
  26    converge to a finite limit `I`, the divergence-form statement holds.
  27
  28    STATUS: SCAFFOLD — The existence of the limit I is a hypothesis.
  29    TODO: Prove convergence for specific LNAL distributions. -/
  30def discrete_to_continuum_continuity {α : Type}
  31  (CG : CoarseGrain α) (div : α → ℝ) : Prop :=
  32  ∃ I : ℝ, H_Convergence CG div I
  33
  34/-- **THEOREM**: Trivial convergence for zero field.
  35    Replaces the vacuous `∃ I, True` with a constructive witness. -/
  36theorem zero_field_converges {α : Type} (CG : CoarseGrain α) :
  37    discrete_to_continuum_continuity CG (fun _ => 0) := by
  38  use 0