Compact special-conformal-transformation extension

Prove the cap estimate Θ_{ij}(|κ|^{-1};κ)=o(κ^2) required to extend the noncompact special-conformal-transformation pair-positivity result to compact extremal surfaces.

Background

The bulk pair-positivity theorem is established only on the noncompact branch, where the remote cutoff is held fixed while differentiating with respect to the special conformal transformation parameter κ. For compact surfaces, the closing cap moves to a scale of order |κ|{-1}, producing an additional cap contribution Θ_{ij}.

The compact result would follow if this cap contribution were o(κ2), but the paper does not establish that estimate and notes that the null energy condition alone is insufficient to control the relevant diagonal limit.

References

The strict noncompact result would transfer to the compact surface if, in addition, \begin{equation} \Theta_{ij}(|\kappa|{-1};\kappa)=o(\kappa2). \end{equation} Equation~eq:compact-cap-estimate is not proved, and the NEC alone does not control the diagonal limit. The compact extension therefore remains open, while Theorem~\ref{thm:local-pair-strictness} applies only to the noncompact branch.

— Beyond Strong Subadditivity: Holographic Entropy Inequalities Along Renormalization Group Flows  (2608.25287 - Bao et al., 26 Aug 2026) in Appendix B, subsection “Compact surface and positivity”; Section 2.3, final paragraph