scale_step_ratio

formal source

  41
  42/-- In any geometric scale sequence, each adjacent ratio equals the
  43base ratio. -/
  44theorem scale_step_ratio (S : GeometricScaleSequence) (k : ℕ) :
  45    S.scale (k + 1) / S.scale k = S.ratio := by
  46  unfold GeometricScaleSequence.scale
  47  rw [pow_succ]
  48  have hr : S.ratio ≠ 0 := ne_of_gt S.ratio_pos
  49  have hk : S.ratio ^ k ≠ 0 := pow_ne_zero k hr
  50  simpa using (mul_div_cancel_left₀ S.ratio hk)
  51
  52/-- The realized orbit has constant adjacent ratio. -/
  53theorem realized_closed_scale_ratio_step
  54    (F : ClosedObservableFramework) (H : RealizedClosedScaleModel F) (k : ℕ) :
  55    F.r (F.T^[k + 1] H.baseState) / F.r (F.T^[k] H.baseState) = H.scales.ratio := by
  56  rw [H.realize (k + 1), H.realize k]
  57  have ha : H.amplitude ≠ 0 := ne_of_gt H.amplitude_pos
  58  calc
  59    H.amplitude * H.scales.scale (k + 1) / (H.amplitude * H.scales.scale k)
  60      = H.scales.scale (k + 1) / H.scales.scale k := by
  61          rw [mul_div_mul_left _ _ ha]
  62    _ = H.scales.ratio := scale_step_ratio H.scales k
  63
  64/-- Therefore the realized orbit satisfies ratio self-similarity. -/
  65theorem ratio_self_similar_of_realized_closed_scale
  66    (F : ClosedObservableFramework) (H : RealizedClosedScaleModel F) :
  67    ∀ k,
  68      F.r (F.T^[k + 2] H.baseState) / F.r (F.T^[k + 1] H.baseState) =
  69        F.r (F.T^[k + 1] H.baseState) / F.r (F.T^[k] H.baseState) := by
  70  intro k
  71  rw [realized_closed_scale_ratio_step F H (k + 1), realized_closed_scale_ratio_step F H k]
  72
  73/-- Closure of the earlier geometric scale sequence yields additive
  74posting on the realized orbit. -/