Phase-Model Paper 01C: A PDE Threshold-Emergence Branch for the L--H Transition

This paper develops a separate PDE threshold-emergence branch for the L--H transition. Starting from the amplitude-phase master formula, it introduces a turbulence-energy density and phase-potential formulation, derives a threshold-generating sign branch, and isolates the narrowest falsifiable claim that can be made at the present stage. The paper shows that the revised branch admits an explicit linear threshold for curvature growth, while the previously audited sign remains linearly damped. A minimal 1D verification run reproduces both curvature growth and turbulence suppression above threshold. The current public observation-space bundle is treated as a proxy-diagnostic layer rather than as a full PDE validation set.