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Lean 4 Proof

theorem modeRate_pos {k : ℕ} (hk : 1 ≤ k) (m : ℕ) :
    0 < modeRate k m := by
  simp only [modeRate]
  split
  · exact one_pos
  · exact rpow_pos_of_pos (Nat.cast_pos.mpr (by omega)) _

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Module Section

Emergence results from Paper 1, Sections 3-5: