Test and prove the conjectured bulk-channel decomposition for continuous boundaries

Show that the conjectured bulk-channel decomposition (Equation (2pt-bulk)) holds for continuous diagonal boundaries by demonstrating crossing symmetry with the corresponding boundary-channel integral, thereby removing the current status of the bulk-channel formula as an untested conjecture in this case.

Background

While the decomposition is numerically tested and supported for discrete boundaries, the authors cannot presently test the continuous case due to unknown boundary structure constants and numerical truncation issues.

An analytic confirmation of crossing symmetry for continuous boundaries would settle the conjectured bulk-channel expansion in this regime and extend the applicability of the bootstrap framework for loop models with continuous diagonal boundary conditions.

References

With known numerical bootstrap methods, we cannot extract any information from the existence of the boundary channel decomposition. Therefore, the bulk channel decomposition remains an untested conjecture in this case.

— Diagonal boundary conditions in critical loop models  (2512.10400 - Downing et al., 11 Dec 2025) in Section 3.2 Boundary channel decomposition