IndisputableMonolith.Verification.Exclusivity.HierarchyTheorem

   1/-
   2  HierarchyTheorem.lean — Bridge B1
   3
   4  Proves: (A2) + Hierarchical Structure (HS) ⟹ (A3) Self-similarity with φ.
   5-/
   6
   7import Mathlib
   8import IndisputableMonolith.Verification.Exclusivity.Framework
   9import IndisputableMonolith.Foundation.HierarchyMinimality
  10import IndisputableMonolith.Foundation.PhiForcing
  11
  12namespace IndisputableMonolith.Verification.Exclusivity
  13
  14open IndisputableMonolith.Foundation.HierarchyMinimality
  15open IndisputableMonolith.Foundation.PhiForcing
  16
  17structure HierarchicalLedger where
  18  scale : ℝ
  19  scale_gt_one : 1 < scale
  20  level_size : ℕ → ℝ
  21  level_size_pos : ∀ k, 0 < level_size k
  22  uniform_scaling : ∀ k, level_size (k + 1) = scale * level_size k
  23  composition : level_size 2 = level_size 1 + level_size 0
  24
  25theorem hierarchy_forces_fibonacci_recurrence (L : HierarchicalLedger) :
  26    L.scale ^ 2 = L.scale + 1 := by
  27  have h0 : L.level_size 0 ≠ 0 := ne_of_gt (L.level_size_pos 0)
  28  have h_s1 : L.level_size 1 = L.scale * L.level_size 0 := L.uniform_scaling 0
  29  have h_s2 : L.level_size 2 = L.scale * L.level_size 1 := L.uniform_scaling 1
  30  have h_sq : L.level_size 2 = L.scale ^ 2 * L.level_size 0 := by
  31    rw [h_s2, h_s1]
  32    ring
  33  have h_rhs : L.level_size 2 = (L.scale + 1) * L.level_size 0 := by
  34    rw [L.composition, h_s1]
  35    ring
  36  have h_mul : (L.scale ^ 2 - (L.scale + 1)) * L.level_size 0 = 0 := by
  37    calc
  38      (L.scale ^ 2 - (L.scale + 1)) * L.level_size 0
  39          = L.scale ^ 2 * L.level_size 0 - (L.scale + 1) * L.level_size 0 := by ring
  40      _ = L.level_size 2 - L.level_size 2 := by rw [← h_sq, h_rhs]
  41      _ = 0 := by ring
  42  rcases mul_eq_zero.mp h_mul with hzero | hsize
  43  · exact sub_eq_zero.mp hzero
  44  · exact (h0 hsize).elim
  45
  46/-- Bridge B1: Hierarchical structure ⟹ scale = φ.
  47
  48    The Fibonacci recurrence σ² = σ + 1 has unique positive root > 1 at φ.
  49    The root uniqueness uses the existing phi_forced infrastructure. -/
  50theorem bridge_B1_hierarchy_implies_phi (L : HierarchicalLedger) :
  51    L.scale = φ := by
  52  let S : IndisputableMonolith.Foundation.PhiForcingDerived.GeometricScaleSequence :=
  53    { ratio := L.scale
  54      ratio_pos := lt_trans (by norm_num) L.scale_gt_one
  55      ratio_ne_one := by linarith [L.scale_gt_one] }
  56  have h_closed : S.isClosed := by
  57    unfold IndisputableMonolith.Foundation.PhiForcingDerived.GeometricScaleSequence.isClosed
  58    unfold IndisputableMonolith.Foundation.PhiForcingDerived.ledgerCompose
  59    unfold IndisputableMonolith.Foundation.PhiForcingDerived.GeometricScaleSequence.scale
  60    have hrec := hierarchy_forces_fibonacci_recurrence L
  61    nlinarith [hrec]
  62  exact hierarchy_forces_phi ⟨S, h_closed⟩
  63
  64end IndisputableMonolith.Verification.Exclusivity
  65