misalignment_exists

formal source

  66
  67/-- For any tick that is not a multiple of lcm(8,45) = 360, at least
  68    one cadence is not at its period boundary. -/
  69theorem misalignment_exists (t : ℕ) (h : t % 360 ≠ 0) :
  70    t % 8 ≠ 0 ∨ t % 45 ≠ 0 := by
  71  by_contra h_neg
  72  push_neg at h_neg
  73  obtain ⟨h8, h45⟩ := h_neg
  74  have : 360 ∣ t := by
  75    have := Nat.Coprime.mul_dvd_of_dvd_of_dvd (by native_decide : Nat.Coprime 8 45)
  76      (Nat.dvd_of_mod_eq_zero h8) (Nat.dvd_of_mod_eq_zero h45)
  77    simpa using this
  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