Phase Ib extension to α < 1/4

Prove the Phase Ib loss-curve characterization for the PLRF model—that P(r) \asymp Fpp(r) + F0(r)—in the regime 2β < 1, α < 1/4, and 2(α + β) > 1 by deriving the necessary kernel-function estimates to extend the result below α = 1/4.

Background

Phase Ib (2β < 1, 1/4 < α < 1/2, 2(α + β) > 1) is characterized by loss dynamics dominated by the pure-point forcing and the capacity term, with SGD noise negligible. The authors prove this behavior for α > 1/4.

They note that extending the proof to α < 1/4 is hindered by missing kernel-function bounds; resolving these would generalize the phase characterization and unify the analysis across the boundary at α = 1/4.