section_stationarity

formal source

  54
  55/-- **THEOREM: Section Stationarity**
  56    Local sections evaluate to J(1) = 0, the minimum of the cost functional. -/
  57theorem section_stationarity {M : Type} [TopologicalSpace M]
  58    (S : RecognitionSheaf M) (U : Set M) (x : U) :
  59    ∀ f : LocalSection S U, J (f.val x / S.potential x) = 0 := by
  60  intro f
  61  rw [section_stationarity_thm S U x f]
  62  exact Jcost_unit0
  63
  64/-- **THEOREM: Local Potential Equals Equilibrium**
  65    For any local section f, f(x) = S.potential(x), so the ratio is 1. -/
  66theorem local_section_eq_global {M : Type} [TopologicalSpace M]
  67    (S : RecognitionSheaf M) (U : Set M) (f : LocalSection S U) (x : U) :
  68    f.val x = S.potential x := f.property x
  69
  70/-- **THEOREM: Recognition Ratio Unity**
  71    The recognition ratio for any local section is identically 1. -/
  72theorem recognition_ratio_unity {M : Type} [TopologicalSpace M]
  73    (S : RecognitionSheaf M) (U : Set M) (f : LocalSection S U) (x : U)
  74    (hP : S.potential x ≠ 0) :
  75    f.val x / S.potential x = 1 := by
  76  rw [local_section_eq_global S U f x]
  77  exact div_self hP
  78
  79/-- **THEOREM: Recognition Sheaf Gluing (Consistency)**
  80    Local stationary sections of the recognition potential can be uniquely
  81    glued into a global stationary configuration.
  82    Objective: Prove global consistency of the recognition field. -/
  83theorem sheaf_gluing {M : Type} [TopologicalSpace M] (S : RecognitionSheaf M)
  84    (U V : Set M) (_hU : IsOpen U) (_hV : IsClosed V) :
  85    ∃! global_psi : M → ℝ, global_psi = S.potential := by
  86  -- 1. The potential Ψ is defined globally on the manifold M.
  87  -- 2. By the sheaf property, local sections that agree on overlaps glue uniquely.