LocalSection

formal source

  26
  27/-- **DEFINITION: Local Section**
  28    A local section of the recognition sheaf on an open set U. -/
  29def LocalSection {M : Type} [TopologicalSpace M] (S : RecognitionSheaf M) (U : Set M) : Type :=
  30  { f : U → ℝ // ∀ x : U, f x = S.potential x }
  31
  32/-- The J-cost derivative at the unit ratio.
  33    Since J(1) = 0 is the global minimum, J'(1) = 0. -/
  34noncomputable abbrev J := Cost.Jcost
  35
  36/-- **CORE LEMMA**: The J-cost has a stationary point at r = 1.
  37    This is the key physical property: the recognition ratio Ψ/Ψ₀ = 1 is the
  38    stable equilibrium point of the ledger dynamics. -/
  39theorem J_stationary_at_one : deriv J 1 = 0 := deriv_Jcost_one
  40
  41/-- **THEOREM (PROVED): J-Cost Stationarity Principle**
  42    The local sections of the recognition sheaf satisfy the J-cost stationarity
  43    principle at the unit recognition ratio.
  44
  45    Proof: By definition, a local section f satisfies f(x) = potential(x),
  46    so the ratio is always 1, and J(1) = 0 is the stationary minimum. -/
  47theorem section_stationarity_thm {M : Type} [TopologicalSpace M]
  48    (S : RecognitionSheaf M) (U : Set M) (x : U) (f : LocalSection S U) :
  49    J (f.val x / S.potential x) = J 1 := by
  50  have h_eq : f.val x = S.potential x := f.property x
  51  rw [h_eq]
  52  have h_pos : S.potential x ≠ 0 := (S.potential_pos x).ne'
  53  rw [div_self h_pos]
  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