Provide a rigorous justification of the replica calculation used in this model

Develop a mathematically rigorous justification of the replica-based computation presented in Section 3 for the Gaussian random least-squares problem with a spherical constraint, including the saddle-point approximation, the replica-symmetric ansatz, and the replica-limit manipulations leading to the average minimal loss and large-deviation results.

Background

The derivations of the average minimal loss and the large-deviation rate function rely on the replica method, which is heuristic. Although similar problems have been rigorously analyzed in related contexts, the specific steps employed here—such as exchanging limits, the saddle-point evaluation in the replicated order-parameter space, and the replica-symmetric ansatz—lack a full mathematical justification in this model.

The author explicitly highlights the absence of a rigorous foundation for these steps and points to recent rigorous advances in related spin-glass models as potential guidance for establishing such a foundation here.