Universality Classes of Saturating Oscillators: Asymptotic Depletion Near Saturation Separatrices

We classify the asymptotic behaviour of the time-averaged mixing scalar f (r) = ⟨M⟩V (r), M = 1 − w²ₑff, near the saturation separatrix ε = ½ − r → 0⁺ for symmetric oscillators with saturating potential V. The main result (Classification Theorem) establishes that the depletion exponent α in f (ε) ~ C·εᵅ (up to logarithmic corrections) is determined entirely by the asymptotic tail Vₘax − V (φ) as φ → ∞. For the exponential class (Vₘax − V ~ exp (−aφᵖ) ): f ~ Lₚ·√ε·|ln ε|^ ( (p−1) /p), verified for specific potentials (tanh² gives C = 2√2 exactly; V = ½ (1−e^−φ) gives Q₁ ≈ 2. 40 numerically). The exponent is class-universal; the prefactor Lₚ is potential-specific. For the algebraic class (Vₘax − V ~ φ^−n): α (n) = min (1, ½ + 1/n) with logarithmic correction at n = 2, proved rigorously via the Beta function identity ID = φA·ε^−1/2·βₙ where βₙ = √π·Γ (½+1/n) / n·Γ (1+1/n). The fixed point f (0) = ½ holds for all symmetric potentials with quadratic minimum. All results are stated with explicit epistemic levels (Layer A: proved; Layer B: analytical argument with error bounds). The paper includes a complete Watson's lemma error analysis for the exponential class (Appendix B), giving a relative error of O (1/|ln ε|).