misaligned_ticks_per_cycle

formal source

  78  exact h (Nat.mod_eq_zero_of_dvd this)
  79
  80/-- Between any two multiples of 360, there are 359 misaligned ticks. -/
  81theorem misaligned_ticks_per_cycle :
  82    ∀ k : ℕ, ∀ j : ℕ, 1 ≤ j → j ≤ 359 →
  83    (360 * k + j) % 360 ≠ 0 := by
  84  intro k j hj1 hj2
  85  omega
  86
  87/-- **THEOREM (Perpetual Complexity)**: The combination of Ω_Λ > 0 and
  88    gcd(8,45) = 1 guarantees perpetual local complexity generation.
  89
  90    The H-theorem drives F_R → 0 globally. But coprimality ensures that
  91    at every non-360-aligned tick, some voxels have misaligned cadences.
  92    These voxels have positive J-cost (structured excitations). Since the
  93    expansion (Ω_Λ > 0) creates new lattice at every epoch, new
  94    synchronization mismatches are perpetually generated.
  95
  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