IndisputableMonolith.Foundation.RecognitionForcing

   1import Mathlib
   2import IndisputableMonolith.Foundation.LawOfExistence
   3import IndisputableMonolith.Foundation.LedgerForcing
   4import IndisputableMonolith.Recognition
   5
   6/-!
   7# Recognition Forcing: From Cost to Recognition Structure
   8
   9This module proves that **recognition is forced** by the cost foundation.
  10-/
  11
  12namespace IndisputableMonolith
  13namespace Foundation
  14namespace RecognitionForcing
  15
  16open Real
  17
  18/-! ## Part 0: Recognition-Cost Bridge -/
  19
  20/-- The J-cost of a recognition event. -/
  21noncomputable def recognition_cost (e : LedgerForcing.RecognitionEvent) : ℝ :=
  22  LedgerForcing.J e.ratio
  23
  24/-- Recognition events with ratio = 1 are cost-free. -/
  25theorem self_recognition_zero_cost (e : LedgerForcing.RecognitionEvent) :
  26    e.ratio = 1 → recognition_cost e = 0 := by
  27  intro h
  28  simp only [recognition_cost, h, LedgerForcing.J]
  29  norm_num
  30
  31/-- Non-trivial recognition has positive cost.
  32    Uses the fact that J(x) = (x + 1/x)/2 - 1 ≥ 0, with = 0 iff x = 1. -/
  33theorem nontrivial_recognition_positive_cost (e : LedgerForcing.RecognitionEvent)
  34    (h : e.ratio ≠ 1) : recognition_cost e > 0 := by
  35  simp only [recognition_cost, LedgerForcing.J]
  36  have hpos := e.ratio_pos
  37  have h0 : e.ratio ≠ 0 := hpos.ne'
  38  -- (x - 1)² > 0 when x ≠ 1
  39  have hne : (e.ratio - 1)^2 > 0 := by
  40    have hsq : (e.ratio - 1)^2 ≥ 0 := sq_nonneg _
  41    have hne2 : (e.ratio - 1)^2 ≠ 0 := by
  42      intro heq
  43      have heq2 : e.ratio - 1 = 0 := sq_eq_zero_iff.mp heq
  44      have : e.ratio = 1 := by linarith
  45      exact h this
  46    exact lt_of_le_of_ne hsq (Ne.symm hne2)
  47  -- Expand: x² - 2x + 1 > 0
  48  -- So: x² + 1 > 2x
  49  -- So: (x² + 1)/x > 2 (since x > 0)
  50  -- So: x + 1/x > 2
  51  have h2 : e.ratio^2 + 1 > 2*e.ratio := by nlinarith [sq_nonneg (e.ratio - 1)]
  52  have h3 : e.ratio + e.ratio⁻¹ > 2 := by
  53    have heq : e.ratio + e.ratio⁻¹ = (e.ratio^2 + 1) / e.ratio := by field_simp
  54    rw [heq, gt_iff_lt, lt_div_iff₀ hpos]
  55    linarith
  56  linarith
  57
  58/-- Recognition is cost structure. -/
  59theorem recognition_is_cost_structure :
  60    ∀ (e : LedgerForcing.RecognitionEvent),
  61    (e.ratio = 1 ↔ recognition_cost e = 0) ∧
  62    (e.ratio ≠ 1 → recognition_cost e > 0) := by
  63  intro e
  64  refine ⟨⟨self_recognition_zero_cost e, ?_⟩, nontrivial_recognition_positive_cost e⟩
  65  intro h
  66  simp only [recognition_cost, LedgerForcing.J] at h
  67  have hpos := e.ratio_pos
  68  have h0 : e.ratio ≠ 0 := hpos.ne'
  69  -- h says (e.ratio + e.ratio⁻¹)/2 - 1 = 0
  70  -- So e.ratio + e.ratio⁻¹ = 2
  71  have h1 : e.ratio + e.ratio⁻¹ = 2 := by linarith
  72  -- This means (e.ratio - 1)² = 0
  73  have heq : e.ratio + e.ratio⁻¹ = (e.ratio^2 + 1) / e.ratio := by field_simp
  74  have h2 : (e.ratio^2 + 1) / e.ratio = 2 := by rw [← heq]; exact h1
  75  have h3 : e.ratio^2 + 1 = 2 * e.ratio := by
  76    have := congrArg (· * e.ratio) h2
  77    simp only [div_mul_cancel₀ _ h0] at this
  78    linarith
  79  have h4 : (e.ratio - 1)^2 = 0 := by nlinarith [sq_nonneg (e.ratio - 1)]
  80  exact sub_eq_zero.mp (sq_eq_zero_iff.mp h4)
  81
  82/-! ## Part 1: Observable Extraction = Recognition -/
  83
  84structure Observable (S : Type) where
  85  value : S → ℝ
  86
  87structure ObservableExtractionMechanism (S : Type) where
  88  extract : S → ℝ
  89  nonconstant : ∃ s₁ s₂ : S, extract s₁ ≠ extract s₂
  90
  91structure RecognitionStructure (S : Type) where
  92  recognizes : S → S → Prop
  93  self_recognition : ∀ s, recognizes s s
  94  symmetric : ∀ s₁ s₂, recognizes s₁ s₂ → recognizes s₂ s₁
  95
  96def recognition_from_extraction {S : Type} (M : ObservableExtractionMechanism S) :
  97    RecognitionStructure S := {
  98  recognizes := fun s₁ s₂ => M.extract s₁ = M.extract s₂
  99  self_recognition := fun _ => rfl
 100  symmetric := fun _ _ h => h.symm
 101}
 102
 103/-- Recognition is unique extraction mechanism. -/
 104theorem recognition_unique {S : Type} (M : ObservableExtractionMechanism S) :
 105    ∃ R : RecognitionStructure S,
 106    (∀ s₁ s₂, M.extract s₁ = M.extract s₂ ↔ R.recognizes s₁ s₂) :=
 107  ⟨recognition_from_extraction M, fun _ _ => Iff.rfl⟩
 108
 109/-! ## Part 2: Cost Minima = Recognition -/
 110
 111structure Configuration where
 112  value : ℝ
 113  pos : 0 < value
 114
 115def config_to_recognition (c : Configuration) : LedgerForcing.RecognitionEvent :=
 116  { source := 0, target := 0, ratio := c.value, ratio_pos := c.pos }
 117
 118theorem cost_minima_are_recognition (c : Configuration) :
 119    ∃ (e : LedgerForcing.RecognitionEvent), e.ratio = c.value :=
 120  ⟨config_to_recognition c, rfl⟩
 121
 122theorem global_minimum_is_self_recognition :
 123    ∃ (e : LedgerForcing.RecognitionEvent), e.ratio = 1 ∧ recognition_cost e = 0 := by
 124  use { source := 0, target := 0, ratio := 1, ratio_pos := one_pos }
 125  simp only [recognition_cost, LedgerForcing.J]
 126  norm_num
 127
 128/-! ## Part 3: Stability = Recognition -/
 129
 130structure JStableStructure where
 131  carrier : Type
 132  cost : carrier → ℝ
 133  cost_bounded : ∃ m : ℝ, ∀ x, m ≤ cost x
 134
 135structure RecognitionLikeStructure where
 136  carrier : Type
 137  rel : carrier → carrier → Prop
 138  refl : ∀ x, rel x x
 139  symm : ∀ x y, rel x y → rel y x
 140
 141def stable_to_recognition (S : JStableStructure) : RecognitionLikeStructure := {
 142  carrier := S.carrier
 143  rel := fun x y => S.cost x = S.cost y
 144  refl := fun _ => rfl
 145  symm := fun _ _ h => h.symm
 146}
 147
 148theorem stability_forces_recognition (S : JStableStructure) :
 149    ∃ (R : RecognitionLikeStructure), R.carrier = S.carrier :=
 150  ⟨stable_to_recognition S, rfl⟩
 151
 152/-! ## Part 4: Master Theorem -/
 153
 154theorem recognition_necessary (S : Type) (obs : Observable S)
 155    (h : ∃ s₁ s₂, obs.value s₁ ≠ obs.value s₂) :
 156    ∃ (R₁ R₂ : Type), Nonempty (Recognition.Recognize R₁ R₂) := by
 157  obtain ⟨s₁, s₂, _⟩ := h
 158  exact ⟨S, S, ⟨⟨s₁, s₂⟩⟩⟩
 159
 160/-- **MASTER THEOREM: Recognition Forcing Complete** -/
 161theorem recognition_forcing_complete :
 162    (∀ (S : Type) (obs : Observable S),
 163      (∃ s₁ s₂, obs.value s₁ ≠ obs.value s₂) →
 164      ∃ (R₁ R₂ : Type), Nonempty (Recognition.Recognize R₁ R₂)) ∧
 165    (∀ (S : Type) (M : ObservableExtractionMechanism S),
 166      ∃ R : RecognitionStructure S, True) ∧
 167    (∀ (e : LedgerForcing.RecognitionEvent),
 168      (e.ratio = 1 ↔ recognition_cost e = 0) ∧
 169      (e.ratio ≠ 1 → recognition_cost e > 0)) ∧
 170    (∀ (c : Configuration),
 171      ∃ (e : LedgerForcing.RecognitionEvent), e.ratio = c.value) ∧
 172    (∀ (S : JStableStructure),
 173      ∃ (R : RecognitionLikeStructure), R.carrier = S.carrier) :=
 174  ⟨recognition_necessary,
 175   fun _ M => ⟨recognition_from_extraction M, trivial⟩,
 176   recognition_is_cost_structure,
 177   cost_minima_are_recognition,
 178   stability_forces_recognition⟩
 179
 180/-! ## Part 5: Ledger Forcing -/
 181
 182structure RecognitionTracker where
 183  events : List LedgerForcing.RecognitionEvent
 184
 185def PreservesJSymmetry (T : RecognitionTracker) : Prop :=
 186  LedgerForcing.balanced_list T.events
 187
 188theorem ledger_is_minimal_recognition_tracker (T : RecognitionTracker) (hSymm : PreservesJSymmetry T) :
 189    ∃ (L : LedgerForcing.Ledger), L.events = T.events :=
 190  ⟨{ events := T.events, double_entry := hSymm }, rfl⟩
 191
 192/-! ## Part 6: Complete Bridge -/
 193
 194theorem cost_to_recognition_bridge :
 195    (∀ x : ℝ, x ≠ 0 → LedgerForcing.J x = LedgerForcing.J x⁻¹) ∧
 196    (∃ e : LedgerForcing.RecognitionEvent, e.ratio = 1 ∧ recognition_cost e = 0) ∧
 197    (∀ (S : Type) (M : ObservableExtractionMechanism S), ∃ R : RecognitionStructure S, True) ∧
 198    (∀ (S : JStableStructure), ∃ (R : RecognitionLikeStructure), R.carrier = S.carrier) :=
 199  ⟨fun x hx => LedgerForcing.J_symmetric hx,
 200   global_minimum_is_self_recognition,
 201   fun _ M => ⟨recognition_from_extraction M, trivial⟩,
 202   stability_forces_recognition⟩
 203
 204end RecognitionForcing
 205end Foundation
 206end IndisputableMonolith
 207