Phase Ic proof for 0 < α < 1/4

Prove the Phase Ic loss-curve characterization for the PLRF model—that P(r) \asymp Fpp(r) + F0(r) and the associated compute-optimal tradeoff—in the regime 2β > 1 and 0 < α < 1/4 by developing sharp bounds on the kernel function in this region.

Background

Phase Ic mirrors the simple dynamics of Phases Ia and Ib with dominance of the pure-point forcing and capacity terms. The authors present the Phase Ic result contingent on an assumption (Theorem 5.1 holding for α < 1/4) and note that kernel-function behavior changes in this regime.

They explicitly state they cannot prove the Phase Ic statement due to the lack of sharp kernel bounds when α < 1/4. Establishing such bounds would complete the analysis and confirm the phase behavior and compute-optimal exponents in this region.