theorem mode_geometric_decay (k m L : ℕ) (a₀ : ℝ) : modeAfterL k m (L + 1) a₀ = modeRate k m * modeAfterL k m L a₀ := by simp only [modeAfterL, pow_succ]; ring
thesis/CESProofs/Foundations/Emergence.lean:140
Emergence results from Paper 1, Sections 3-5: