MisalignmentWitness

formal source

  60/-- At every 360-tick boundary, there exist voxels where the recognition
  61    and phase cadences are misaligned. These voxels have positive local
  62    J-cost (they are not at x=1) and constitute structured excitations. -/
  63structure MisalignmentWitness where
  64  tick : ℕ
  65  misaligned : tick % 8 ≠ 0 ∨ tick % 45 ≠ 0
  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