Strict model-threshold dimension condition for general sigma-k curvature

Establish whether every complete conformal metric with admissible Schouten tensor and constant positive sigma_k-curvature on the complement of a smooth p-dimensional singular submanifold of the sphere must satisfy the strict dimension inequality p<p_k(n) for general k.

Background

The paper defines the model threshold p_k(n) using the positivity of the elementary symmetric functions associated with the product model H^{p+1} × S^{n-p-1}. The main theorem proves only the non-strict bound p≤p_k(n) for complete admissible conformal metrics with constant positive sigma_k-curvature.

A cited March 2026 revision by Espinal and Gonzalez is described as formulating the strict inequality p<p_k(n) as the conjectured necessary condition for general k. The paper constructs equality examples for k=2 at integral thresholds, showing that a universal strict inequality cannot hold without additional hypotheses; it proves strictness only under a finite positive linear-contact condition.