Geometric explanation of the gap between determinant crossing and integrability boundary

Determine the geometric mechanism responsible for the limiting gap of approximately 0.2018714780 between the determinant crossing dimension n_N and the integrability boundary 2N+16 for the degree-eight Weyl profiles as N tends to infinity.

Background

The noncompactness construction for the constant Q_{2N}-curvature problem uses degree-eight Weyl profiles, whose integrability requires n>2N+16. Theorem~\ref{thm:cross} identifies a determinant crossing at n_N=2N+a_+c_*N{-1}+O(N{-2}), where a_≈16.2018714780. Consequently, the crossing lies asymptotically about 0.2018714780 dimensions above the integrability threshold.

Below the crossing, the scale-derivative matrix governing the restricted quadratic reduced energy is negative definite; above it, the matrix becomes indefinite, enabling the choice of perturbations needed for the noncompactness construction. The paper explicitly asks for a geometric explanation of why this local variational transition does not coincide exactly with the analytic integrability boundary.

References

The degree-eight Weyl profiles satisfy the integrability condition $n>2N+16$, whereas Theorem~\ref{thm:cross} gives $n_N-(2N+16)\to a_*-16\approx0.2018714780$ as $N\to\infty$. What geometric mechanism explains this small positive limiting gap between the determinant crossing and the integrability boundary?

— Noncompactness for the constant $Q_{2N}$-curvature problem  (2610.07992 - Gong et al., 6 Oct 2026) in Question environment immediately following Theorem 3 (Theorem~\ref{thm:cross}), Section 1, Introduction