Simplex3

formal source

  21
  22/-- **DEFINITION: Simplicial Voxel**
  23    A 3-simplex (tetrahedron) representing the atom of volume in the ledger. -/
  24structure Simplex3 where
  25  vertices : Fin 4 → (Fin 3 → ℝ)
  26  volume   : ℝ
  27  vol_pos  : volume > 0
  28
  29/-- **DEFINITION: Simplicial Ledger**
  30    A collection of 3-simplices that form a manifold covering. -/
  31structure SimplicialLedger where
  32  simplices : Set Simplex3
  33  /-- The simplices form a non-empty set (non-vacuity). -/
  34  non_empty : simplices.Nonempty
  35  /-- SCAFFOLD: Manifold covering property.
  36    Proof requires simplicial complex axioms and manifold topology.
  37    See: LaTeX Manuscript, Chapter "Gravity as Recognition", Section "Simplicial Ledger". -/
  38  is_covering : Prop
  39
  40/-- **DEFINITION: Simplicial Sheaf**
  41    A sheaf assigning a recognition potential to each simplex in the ledger. -/
  42structure SimplicialSheaf (L : SimplicialLedger) where
  43  potential : Simplex3 → ℝ
  44  /-- The potential is consistent across simplex boundaries (placeholder). -/
  45  is_consistent : Prop
  46
  47/-- Local J-cost on a single simplex. -/
  48noncomputable def local_J_cost (s : Simplex3) (psi : ℝ) : ℝ :=
  49  Jcost psi * s.volume
  50
  51/-- Global J-cost summed over the ledger (for finite ledgers). -/
  52noncomputable def global_J_cost (L : SimplicialLedger) (S : SimplicialSheaf L) [Fintype L.simplices] : ℝ :=
  53  ∑ s : L.simplices, local_J_cost s (S.potential s)
  54