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Theorem 6 — the clipped frontier

For the second-order surrogate, the optimal coupling is the clipped inverse of t(γ) = −V₀′/G′. If t sweeps (0,∞) the optimizer sweeps the whole bridge (Family A). If t reaches a finite limit t₀ at the HRP end, then for s = τ² ≥ t₀ the surrogate parks at HRP (Family B, with t₀ = 27/800).

Left: t(γ) for both families (log scale), computed from the recursion. Right: the resulting surrogate optimizer γ₂*(s); Family A sweeps the bridge, Family B clips to 0 at s = 27/800.
What to look for: Family A's t diverges at the left end and vanishes at the right (no clipping); Family B's t hits a ceiling at 27/800 ≈ 0.034, and its optimizer path drops to zero there.

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