Determine the domain of validity of the large-deviation rate function for E_min

Determine the precise range of scaled energies x for which the large-deviation rate function L(x), derived via the replica method for the distribution of the minimal loss in the Gaussian random least-squares problem with spherical constraint, is valid away from its minimum x*, and provide a mathematically rigorous justification of this validity domain.

Background

Using a double-scaling replica method, the notes derive a large-deviation rate function L(x) for the minimal loss E_min in the constrained random least-squares problem. While the method yields explicit expressions and a Gaussian expansion near the typical value x*, it does not rigorously establish for which values of x this rate function is valid.

The author explicitly notes that identifying the exact validity domain of L(x) away from its minimum remains largely unresolved and calls for a rigorous treatment, reflecting a significant open issue in the probabilistic characterization of the optimization landscape.