H_LocalGlobalUnification

formal source

  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
  87
  88/-- Every recognition loop carries a surjective pattern pass. -/
  89theorem recognition_loop_has_surjection {cycle : List Simplex3}
  90    (hloop : is_recognition_loop cycle) :
  91    ∃ pass : Fin cycle.length → Pattern 3, Function.Surjective pass := by
  92  exact hloop.2.2
  93
  94/-- **THEOREM: Eight-Tick Cycle Uniqueness**
  95    The 8-tick closure cycle is the unique minimal sequence for a self-consistent
  96    recognition loop on a simplicial manifold. -/
  97theorem eight_tick_uniqueness (_L : SimplicialLedger) :
  98    ∀ cycle : List Simplex3,