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def definition def or abbrev

H_HomogenizationLimit

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formal statement (Lean)

  38def H_HomogenizationLimit (L : SimplicialLedger) (g : MetricTensor) : Prop :=

proof body

Definition body.

  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 $g_{\mu\nu}$. -/

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