A recent theoretical analysis has revisited Type IIB axion-dilaton Euclidean saddles, focusing on a specified axion charge sector. In this context, the solution with energy E=0 corresponds to a BPS instanton, while solutions with E>0 describe non-BPS wormholes with a smooth throat. Although both cases satisfy the same radial equations, their fluctuation problems are distinct, underscoring the complexity of these structures in string theory.

For the BPS instanton (E=0), the study details how the quadratic action is reduced to a physical Hessian after considering the Hamiltonian constraint, gauge quotient, charge-sector boundary condition, and the removal of collective zero modes. This Hessian, denoted as H_ν, factorizes into the form Q_ν†Q_ν. This result is interpreted as an endpoint theorem, extending beyond a simple stability theorem for the full E>0 wormhole. This finding provides a firmer foundation for understanding the spectra of wormholes in Type IIB string theory.

The work also separates the connected two-ended wormhole throat from its long-distance two-end multipole operator term. Once the coefficient matrix Cij is derived, the different-component and same-component placements of the two end insertions appear as terms in the same quadratic expression. Removing either term requires a genuine projection or explicit cancellation, highlighting the interconnectedness of these theoretical structures and their impact on understanding spacetime geometry in string theory.