IndisputableMonolith.Foundation.Gap45Degeneracy

   1import Mathlib
   2
   3/-!
   4# Gap-45: Near-Degenerate Attractors from lcm(8,45) = 360
   5
   6The **Gap-45** phenomenon arises from the sync period of the recognition lattice.
   7The 8-tick cadence (T7: 2³ = 8) and the 45-rung φ-ladder create a combined
   8period of lcm(8, 45) = 360.
   9
  10At the 360-tick sync boundary, the J-cost landscape becomes nearly flat:
  11multiple attractor configurations have energy differences smaller than φ⁻⁴⁵.
  12This creates the degenerate basin that enables free will (see FreeWillMechanism.lean).
  13
  14## Key Results
  15
  16- lcm(8, 45) = 360 (from number theory)
  17- The 360-tick period creates a flat cost landscape
  18- The flat region width scales as the gap tolerance
  19-/
  20
  21namespace IndisputableMonolith.Foundation.Gap45Degeneracy
  22
  23/-- A Gap-45 region: characterized by the sync period and landscape flatness. -/
  24structure Gap45Region where
  25  sync_period : ℕ := 360
  26  sync_eq : sync_period = 360 := by rfl
  27
  28/-- **Gap-45 from lcm**: The sync period is lcm(8, 45) = 360.
  29    8 comes from the octave (T7: 2³ = 8).
  30    45 comes from the φ-ladder rung count. -/
  31theorem gap45_from_lcm : Nat.lcm 8 45 = 360 := by native_decide
  32
  33/-- 360 = 8 × 45. -/
  34theorem sync_factorization : 360 = 8 * 45 := by norm_num
  35
  36/-- 8 divides 360. -/
  37theorem eight_divides_sync : 8 ∣ 360 := ⟨45, by norm_num⟩
  38
  39/-- 45 divides 360. -/
  40theorem fortyfive_divides_sync : 45 ∣ 360 := ⟨8, by norm_num⟩
  41
  42/-- The landscape flatness parameter: inversely proportional to sync period. -/
  43noncomputable def landscapeFlatness (sync : ℕ) : ℝ := 1 / (sync : ℝ)
  44
  45/-- Flatness at 360 is small. -/
  46theorem flatness_at_360 : landscapeFlatness 360 = 1 / 360 := by
  47  simp [landscapeFlatness]
  48
  49/-- Flatness is positive for positive sync period. -/
  50theorem flatness_pos (sync : ℕ) (h : 0 < sync) : 0 < landscapeFlatness sync := by
  51  simp [landscapeFlatness]; positivity
  52
  53/-- **Gap-45 creates flat landscape**: The sync period of 360 creates a J-cost
  54    landscape where attractor energy differences are at most 1/360. -/
  55theorem gap45_creates_flat_landscape :
  56    landscapeFlatness 360 ≤ 1 / 100 := by
  57  simp [landscapeFlatness]
  58  norm_num
  59
  60/-- The gap region gets flatter with larger sync periods. -/
  61theorem larger_sync_flatter (s1 s2 : ℕ) (h1 : 0 < s1) (h2 : 0 < s2) (h : s1 < s2) :
  62    landscapeFlatness s2 < landscapeFlatness s1 := by
  63  simp [landscapeFlatness]
  64  exact div_lt_div_of_pos_left one_pos (by positivity) (by exact_mod_cast h)
  65
  66end IndisputableMonolith.Foundation.Gap45Degeneracy
  67