Variational modulus of convexity under Moreau envelopes: attenuation, recovery, and oracle tuning

<div> The variational modulus of convexity at a pair in the graph of the subgradient mapping for a generally nonconvex extended-real-valued function captures in a single number the degree of variational convexity, uniform quadratic growth, and subdifferential monotonicity seeded at that location. It has fundamental implications for understanding local optimality and variational stability, but its behavior with respect to taking Moreau envelopes is explored here for the first time, the motivation being the importance of those envelopes in smoothing and proximal-point methodology. This exploration results in an exact second-order transmission rule reflecting a rational attenuation together with recovery in the limit of vanishing smoothing. The attenuation law is applied to produce a diagnostic, computable from proximal-oracle schemes, for selecting a smoothing parameter while maintaining explicit guarantees of admissibility and inner accuracy. </div>