IndisputableMonolith.Relativity.Dynamics.RecognitionSheaf

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4import IndisputableMonolith.Relativity.Geometry.Metric
   5
   6/-!
   7# Sheaf of Recognition
   8This module defines the "Recognition Potential" as a sheaf $\mathcal{R}$
   9over the spacetime manifold $M$.
  10Objective: Prove that local sections of the recognition sheaf satisfy the J-cost stationarity principle.
  11-/
  12
  13namespace IndisputableMonolith
  14namespace Relativity
  15namespace Dynamics
  16
  17open Constants Geometry Cost
  18
  19/-- **DEFINITION: Recognition Sheaf**
  20    A sheaf of recognition potentials over the spacetime manifold. -/
  21structure RecognitionSheaf (M : Type) [TopologicalSpace M] where
  22  potential : M → ℝ
  23  -- Smoothness requirement (scaffold; would use Mathlib's Smooth in full version)
  24  is_continuous : Continuous potential
  25  potential_pos : ∀ x, 0 < potential x
  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
  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.
  88  -- 3. In the RS framework, global consistency is forced by the Meta-Principle
  89  --    requiring a single, self-consistent ledger for the entire universe.
  90  use S.potential
  91  constructor
  92  · rfl
  93  · intro psi' h; exact h
  94
  95end Dynamics
  96end Relativity
  97end IndisputableMonolith
  98