Determine the resonant solution selected by parameter-space limits

Determine which nonzero element of the two-dimensional smooth horizon solution space is selected when the nonresonant ingoing solution normalized by h_0=1 is taken to an exact integer resonance along a specified curve in parameter space.

Background

At an exact integer resonance, the Frobenius recurrence develops a compatibility condition. When the solvability function vanishes, the smaller-root logarithmic obstruction disappears and the smooth horizon solution space becomes two-dimensional. Consequently, horizon smoothness alone does not select a unique resonant solution.

The paper distinguishes the regularized ingoing family, whose value at the resonant point is the zero solution, from the nonresonant ingoing solution normalized by h_0=1, whose limit along an approach curve can be a nonzero resonant solution. The subsequent analysis derives a formula for the selected Frobenius coefficient in terms of first derivatives of the solvability function and the resonant recurrence coefficient under a transversality assumption, but the quoted passage explicitly identifies the selection question as unresolved at that point. More generally, when the first derivative of the recurrence coefficient vanishes, higher-order parameter jets must be analyzed.