SimplicialSheaf

formal source

  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
  55/-- Variation of local J-cost w.r.t potential. -/
  56noncomputable def local_variation (_s : Simplex3) (psi : ℝ) : ℝ :=
  57  -- Simple derivative of J(x) at psi
  58  (Jcost (psi + 0.001) - Jcost psi) / 0.001
  59
  60/-- Predicate for J-cost stationarity. -/
  61def J_stationary (psi : ℝ) : Prop := psi = 1
  62
  63/-- **HYPOTHESIS**: Global stationarity implies local simplicial stationarity.
  64    STATUS: EMPIRICAL_HYPO
  65    TEST_PROTOCOL: Verify that global J-cost minimization on a simplicial manifold
  66    forces every local potential Ψ to its unit value.
  67    FALSIFIER: Discovery of a global minimum that contains local non-stationary sections. -/
  68def H_LocalGlobalUnification (L : SimplicialLedger) (S : SimplicialSheaf L) [Fintype L.simplices] : Prop :=
  69  (∀ s : L.simplices, local_variation s (S.potential s) = 0) →
  70  ∀ s : L.simplices, J_stationary (S.potential s)
  71
  72/-- **THEOREM: Local-Global Unification**