local_variation

formal source

  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**
  73    The global J-cost is stationary if and only if every local J-cost is stationary
  74    within its simplicial section. -/
  75theorem local_global_unification (L : SimplicialLedger) (S : SimplicialSheaf L)
  76    [Fintype L.simplices] (h : H_LocalGlobalUnification L S)
  77    (h_global : ∀ s : L.simplices, local_variation s (S.potential s) = 0) :
  78    ∀ s : L.simplices, J_stationary (S.potential s) := h h_global
  79
  80/-- **DEFINITION: Recognition Loop**
  81    A recognition loop is a closed cycle of 3-simplices in the ledger. -/
  82def is_recognition_loop (cycle : List Simplex3) : Prop :=
  83  cycle ≠ [] ∧
  84  (∀ _i : Fin cycle.length, ∃ _shared_face : Prop, True) ∧
  85  -- The loop induces a complete pass through 3-bit local pattern states.
  86  ∃ pass : Fin cycle.length → Pattern 3, Function.Surjective pass