theorem modeRate_eq_rpow_neg_scaling {k : ℕ} (_hk : 1 ≤ k) (m : ℕ) :
modeRate k m = (↑k : ℝ) ^ (-scalingDimension m) := by
by_cases hm : m ≤ 2
· simp [modeRate, scalingDimension, if_pos hm, neg_zero, rpow_zero]
· have hm3 : ¬ m ≤ 2 := by omega
simp only [modeRate, scalingDimension, if_neg hm3]
congr 1; ringRenormalization Group for CES: