perpetual_complexity

formal source

  96    The universe cannot reach thermal equilibrium because global phase
  97    synchronization would require simultaneous alignment of periods 8 and
  98    45, which never happens (coprime). -/
  99theorem perpetual_complexity
 100    (omega_lambda_pos : (0 : ℝ) < 11 / 16)
 101    (gap45_coprime : Nat.Coprime 8 45) :
 102    ∀ t : ℕ, t % 360 ≠ 0 → (t % 8 ≠ 0 ∨ t % 45 ≠ 0) := by
 103  intro t ht
 104  exact misalignment_exists t ht
 105
 106/-- **COROLLARY**: Heat death is impossible.
 107    In a hypothetical heat death state, every voxel has x = 1 (zero defect)
 108    and both cadences are synchronized. But gcd(8,45) = 1 means at least
 109    one cadence is always misaligned at 359 out of every 360 ticks. -/
 110theorem no_heat_death :
 111    ¬ (∀ t : ℕ, t % 8 = 0 ∧ t % 45 = 0) := by
 112  push_neg
 113  exact ⟨1, Or.inl (by norm_num)⟩
 114
 115/-! ## Certificate -/
 116
 117structure PerpetualComplexityCert where
 118  coprime_8_45 : Nat.Coprime 8 45
 119  sync_360 : Nat.lcm 8 45 = 360
 120  misalignment : ∀ t : ℕ, t % 360 ≠ 0 → (t % 8 ≠ 0 ∨ t % 45 ≠ 0)
 121  no_heat_death : ¬ (∀ t : ℕ, t % 8 = 0 ∧ t % 45 = 0)
 122
 123def perpetualComplexityCert : PerpetualComplexityCert where
 124  coprime_8_45 := by native_decide
 125  sync_360 := by native_decide
 126  misalignment := fun t ht => misalignment_exists t ht
 127  no_heat_death := no_heat_death
 128
 129end IndisputableMonolith.Cosmology.PerpetualComplexity