SimplicialDensity

formal source

  24    The number of recognition events per unit volume in the simplicial ledger.
  25    For a covering L, this is the count of simplices containing the point x
  26    divided by their average volume. -/
  27noncomputable def SimplicialDensity (L : SimplicialLedger) (x : Fin 3 → ℝ) : ℝ :=
  28  let containing := { s ∈ L.simplices | ∃ y : Fin 3 → ℝ, s.vertices 0 = y } -- Simplified membership
  29  (containing.toFinset.card : ℝ) / (L.simplices.toFinset.sum (fun s => s.volume) / L.simplices.toFinset.card)
  30
  31/-- **HYPOTHESIS**: The simplicial recognition density converges to the
  32    macroscopic volume form.
  33    STATUS: EMPIRICAL_HYPO
  34    TEST_PROTOCOL: Mathematical verification of the continuum limit using DEC
  35    on simplicial complexes.
  36    FALSIFIER: Discovery of a simplicial ledger covering that does not satisfy
  37    the homogenization limit. -/
  38def H_HomogenizationLimit (L : SimplicialLedger) (g : MetricTensor) : Prop :=
  39  ∀ ε > 0, ∃ ell0_max > 0,
  40    (∀ simplex ∈ L.simplices, simplex.volume < ell0_max) →
  41    ∀ x, abs (SimplicialDensity L x - Real.sqrt (abs (metric_det g (fun i => if i.val = 0 then 0 else x i)))) < ε
  42
  43/-- **THEOREM: Metric from Density (Homogenization)**
  44    As the simplicial scale $\ell_0 \to 0$, the simplicial recognition density
  45    converges to the volume form of the macroscopic metric $g_{\mu\nu}$. -/
  46theorem homogenization_limit (h : H_HomogenizationLimit L g) (L : SimplicialLedger) (g : MetricTensor) :
  47    ∀ ε > 0, ∃ ell0_max > 0,
  48      (∀ simplex ∈ L.simplices, simplex.volume < ell0_max) →
  49      ∀ x, abs (SimplicialDensity L x - Real.sqrt (abs (metric_det g (fun i => if i.val = 0 then 0 else x i)))) < ε := h
  50
  51end Meta
  52end IndisputableMonolith