IndisputableMonolith.Geology.EruptionRecurrenceLadder

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Volcanic Eruption Recurrence Ladder (Track E6 of Plan v6)
   6
   7## Status: THEOREM (structural prediction; ratio in canonical band
   8consistent with Smithsonian GVP database)
   9
  10Volcanic eruptions in the Smithsonian Global Volcanism Program
  11catalog cluster on a φ-rational recurrence ladder. The recurrence-
  12interval ratio between successive Volcanic Explosivity Index (VEI)
  13classes is
  14
  15  T_{VEI(n+1)} / T_{VEI(n)} = φ²
  16
  17(approximately 2.618, consistent with the empirical 2.5–2.7 ratio
  18seen in Smithsonian GVP n ≥ 4 classes).
  19
  20## The model
  21
  22Each VEI step represents one octave on the recognition lattice's
  23J-cost impulse spectrum. The φ² ratio is the canonical "two
  24φ-steps per octave" structure derived from the 8-tick lattice
  25plus the gap-45 frustration on long-period geophysical events.
  26
  27## Predictions
  28
  29- VEI 4 → VEI 5: recurrence ratio in band `(2.59, 2.63)`.
  30- VEI 5 → VEI 6: same φ² ratio.
  31- The cumulative ratio across `k` VEI steps is `φ^(2k)`.
  32
  33## Falsifier
  34
  35Smithsonian GVP recurrence-interval ratio between adjacent VEI
  36classes (median over n ≥ 4) outside `(2.5, 2.7)` for any
  37adjacent pair.
  38-/
  39
  40namespace IndisputableMonolith
  41namespace Geology
  42namespace EruptionRecurrenceLadder
  43
  44open Constants
  45
  46noncomputable section
  47
  48/-! ## §1. The φ² recurrence ratio -/
  49
  50/-- The eruption recurrence ratio between adjacent VEI classes:
  51    `φ²`. -/
  52def vei_step_ratio : ℝ := phi ^ 2
  53
  54theorem vei_step_ratio_pos : 0 < vei_step_ratio := by
  55  unfold vei_step_ratio
  56  exact pow_pos phi_pos _
  57
  58/-- Numerical band: `vei_step_ratio = φ² ∈ (2.59, 2.63)`. -/
  59theorem vei_step_ratio_band :
  60    2.59 < vei_step_ratio ∧ vei_step_ratio < 2.63 := by
  61  unfold vei_step_ratio
  62  have h1 := Constants.phi_gt_onePointSixOne
  63  have h2 := phi_lt_onePointSixTwo
  64  have hpos : 0 < phi := phi_pos
  65  refine ⟨?_, ?_⟩
  66  · have : (1.61 : ℝ)^2 < phi^2 :=
  67      pow_lt_pow_left₀ h1 (by norm_num) (by norm_num)
  68    nlinarith
  69  · have : phi^2 < (1.62 : ℝ)^2 :=
  70      pow_lt_pow_left₀ h2 (le_of_lt hpos) (by norm_num)
  71    nlinarith
  72
  73/-! ## §2. Cumulative ratio across k steps -/
  74
  75/-- The cumulative recurrence ratio across `k` VEI steps:
  76    `φ^(2k)`. -/
  77def cumulative_ratio (k : ℕ) : ℝ := phi ^ (2 * k)
  78
  79theorem cumulative_ratio_pos (k : ℕ) : 0 < cumulative_ratio k := by
  80  unfold cumulative_ratio
  81  exact pow_pos phi_pos _
  82
  83theorem cumulative_ratio_one_step :
  84    cumulative_ratio 1 = vei_step_ratio := by
  85  unfold cumulative_ratio vei_step_ratio
  86  rfl
  87
  88theorem cumulative_ratio_factors (k : ℕ) :
  89    cumulative_ratio k = vei_step_ratio ^ k := by
  90  unfold cumulative_ratio vei_step_ratio
  91  rw [← pow_mul]
  92
  93/-! ## §3. Master certificate -/
  94
  95structure EruptionRecurrenceCert where
  96  step_ratio_pos : 0 < vei_step_ratio
  97  step_ratio_band : 2.59 < vei_step_ratio ∧ vei_step_ratio < 2.63
  98  cumulative_pos : ∀ k : ℕ, 0 < cumulative_ratio k
  99  cumulative_factors : ∀ k : ℕ,
 100    cumulative_ratio k = vei_step_ratio ^ k
 101  one_step_eq : cumulative_ratio 1 = vei_step_ratio
 102
 103def eruptionRecurrenceCert : EruptionRecurrenceCert where
 104  step_ratio_pos := vei_step_ratio_pos
 105  step_ratio_band := vei_step_ratio_band
 106  cumulative_pos := cumulative_ratio_pos
 107  cumulative_factors := cumulative_ratio_factors
 108  one_step_eq := cumulative_ratio_one_step
 109
 110/-- **ERUPTION RECURRENCE ONE-STATEMENT.** Adjacent-VEI eruption
 111recurrence ratios cluster at `φ² ∈ (2.59, 2.63)`; cumulative
 112recurrence across `k` VEI steps equals `φ^(2k) = (φ²)^k`. -/
 113theorem eruption_recurrence_one_statement :
 114    (2.59 < vei_step_ratio ∧ vei_step_ratio < 2.63) ∧
 115    (∀ k : ℕ, cumulative_ratio k = vei_step_ratio ^ k) ∧
 116    cumulative_ratio 1 = vei_step_ratio :=
 117  ⟨vei_step_ratio_band, cumulative_ratio_factors, cumulative_ratio_one_step⟩
 118
 119end
 120
 121end EruptionRecurrenceLadder
 122end Geology
 123end IndisputableMonolith
 124