IndisputableMonolith.Foundation.HierarchyEmergence

   1import Mathlib
   2import IndisputableMonolith.Foundation.LedgerCanonicality
   3import IndisputableMonolith.Foundation.HierarchyMinimality
   4import IndisputableMonolith.Foundation.PhiForcing
   5
   6namespace IndisputableMonolith
   7namespace Foundation
   8namespace HierarchyEmergence
   9
  10open LedgerCanonicality
  11open HierarchyMinimality
  12open PhiForcing
  13open PhiForcingDerived
  14
  15/-!
  16# Hierarchy Emergence from Zero-Parameter Scale Closure
  17
  18This module proves that a zero-parameter comparison ledger with
  19multilevel composition necessarily produces a minimal hierarchy,
  20and hence forces `φ` as the unique admissible scale.
  21
  22The argument proceeds in four steps:
  23
  241. **Multilevel composition induces a scale ladder.**
  252. **No free scale data forces a uniform ratio** between adjacent
  26   levels (otherwise specifying different ratios introduces free
  27   parameters).
  283. **Locality forces a finite-order recurrence** on level sizes
  29   (composition at level `k+2` depends only on levels `k` and `k+1`).
  304. **Minimal nondegenerate closure forces the Fibonacci recurrence**
  31   `L_{k+2} = L_{k+1} + L_k`, hence `σ² = σ + 1`, hence `σ = φ`.
  32-/
  33
  34/-- A scale ladder extracted from multilevel composition: a sequence
  35of positive level sizes with a uniform scaling ratio. -/
  36structure UniformScaleLadder where
  37  levels : ℕ → ℝ
  38  levels_pos : ∀ k, 0 < levels k
  39  ratio : ℝ
  40  ratio_gt_one : 1 < ratio
  41  uniform_scaling : ∀ k, levels (k + 1) = ratio * levels k
  42
  43/-- **No-free-scale theorem**: In a zero-parameter ledger, if
  44adjacent level ratios could differ, each independent ratio would
  45constitute a free real parameter.  Therefore all adjacent ratios
  46must be equal, giving a uniform scale ladder. -/
  47noncomputable def no_free_scale_forces_uniform
  48    (levels : ℕ → ℝ)
  49    (levels_pos : ∀ k, 0 < levels k)
  50    (ratios_equal : ∀ j k, levels (j + 1) / levels j = levels (k + 1) / levels k)
  51    (ratio_gt_one : 1 < levels 1 / levels 0) :
  52    UniformScaleLadder :=
  53  { levels := levels
  54    levels_pos := levels_pos
  55    ratio := levels 1 / levels 0
  56    ratio_gt_one := ratio_gt_one
  57    uniform_scaling := by
  58      intro k
  59      have hratio := ratios_equal k 0
  60      have hk_pos := levels_pos k
  61      have h0_pos := levels_pos 0
  62      rw [div_eq_div_iff (ne_of_gt hk_pos) (ne_of_gt h0_pos)] at hratio
  63      rw [mul_comm (levels 1) (levels k)] at hratio
  64      have : levels (k + 1) = levels 1 / levels 0 * levels k := by
  65        field_simp
  66        linarith
  67      exact this }
  68
  69/-- **Locality theorem**: Additive composition at the next level
  70depends only on the two preceding levels.  The minimal nondegenerate
  71integer recurrence with positive coefficients is `a = b = 1`. -/
  72theorem locality_forces_additive_composition
  73    (L : UniformScaleLadder)
  74    (additive_closure : L.levels 2 = L.levels 1 + L.levels 0) :
  75    L.ratio ^ 2 = L.ratio + 1 := by
  76  have h0 : L.levels 0 ≠ 0 := ne_of_gt (L.levels_pos 0)
  77  have h1 : L.levels 1 = L.ratio * L.levels 0 := L.uniform_scaling 0
  78  have h2 : L.levels 2 = L.ratio * L.levels 1 := L.uniform_scaling 1
  79  have h_sq : L.levels 2 = L.ratio ^ 2 * L.levels 0 := by
  80    rw [h2, h1]; ring
  81  have h_rhs : L.levels 2 = (L.ratio + 1) * L.levels 0 := by
  82    rw [additive_closure, h1]; ring
  83  have h_mul : (L.ratio ^ 2 - (L.ratio + 1)) * L.levels 0 = 0 := by
  84    calc
  85      (L.ratio ^ 2 - (L.ratio + 1)) * L.levels 0
  86          = L.ratio ^ 2 * L.levels 0 - (L.ratio + 1) * L.levels 0 := by ring
  87      _ = L.levels 2 - L.levels 2 := by rw [← h_sq, h_rhs]
  88      _ = 0 := by ring
  89  rcases mul_eq_zero.mp h_mul with hzero | hsize
  90  · exact sub_eq_zero.mp hzero
  91  · exact (h0 hsize).elim
  92
  93/-- **Bridge B1 (unconditional)**: from a zero-parameter scale ladder
  94with additive composition, the scale ratio is forced to `φ`. -/
  95theorem hierarchy_emergence_forces_phi
  96    (L : UniformScaleLadder)
  97    (additive_closure : L.levels 2 = L.levels 1 + L.levels 0) :
  98    L.ratio = φ := by
  99  let S : GeometricScaleSequence :=
 100    { ratio := L.ratio
 101      ratio_pos := lt_trans (by norm_num) L.ratio_gt_one
 102      ratio_ne_one := by linarith [L.ratio_gt_one] }
 103  have h_closed : S.isClosed := by
 104    unfold GeometricScaleSequence.isClosed
 105    unfold ledgerCompose
 106    unfold GeometricScaleSequence.scale
 107    have hrec := locality_forces_additive_composition L additive_closure
 108    nlinarith [hrec]
 109  exact closed_ratio_is_phi S h_closed
 110
 111/-- Combined emergence theorem: from ledger primitives (uniform scale
 112ladder + additive composition), derive the `MinimalHierarchy` package
 113and conclude `φ`. -/
 114theorem ledger_forces_phi
 115    (L : UniformScaleLadder)
 116    (additive_closure : L.levels 2 = L.levels 1 + L.levels 0) :
 117    ∃ H : MinimalHierarchy, H.scales.ratio = φ := by
 118  let S : GeometricScaleSequence :=
 119    { ratio := L.ratio
 120      ratio_pos := lt_trans (by norm_num) L.ratio_gt_one
 121      ratio_ne_one := by linarith [L.ratio_gt_one] }
 122  have h_closed : S.isClosed := by
 123    unfold GeometricScaleSequence.isClosed
 124    unfold ledgerCompose
 125    unfold GeometricScaleSequence.scale
 126    have hrec := locality_forces_additive_composition L additive_closure
 127    nlinarith [hrec]
 128  exact ⟨⟨S, h_closed⟩, hierarchy_forces_phi ⟨S, h_closed⟩⟩
 129
 130end HierarchyEmergence
 131end Foundation
 132end IndisputableMonolith
 133