Explain the observed vanishing of D_{⟨r,3⟩} for certain discrete boundaries

Characterize and prove the mechanism causing the boundary-channel structure constants D_{⟨r,3⟩} to vanish for specific disc two-point functions with discrete diagonal boundary parameters S = 2 or 3, and determine the general conditions under which such vanishing occurs.

Background

Numerical bootstrap results reveal additional vanishing boundary-channel coefficients in some examples, reducing the effective boundary spectrum beyond the predicted subset. Specifically, for certain diagonal bulk fields and S = 2,3, the authors find D_{⟨r,3⟩} = 0.

They highlight that understanding this phenomenon is currently lacking and deferred to future work. Explaining and proving the vanishing would refine the predicted boundary-channel spectra and deepen the correspondence between combinatorial interpretations and bootstrap solutions.

References

In some examples, we find more structure constants that vanish, and the boundary spectrum becomes even smaller. In the cases of \langle V_{(\frac12 , 0)} V_{(\frac12, 0)} \rangle, \langle V_{(\frac12 , 0)} V_{(\frac32, 0)} \rangle and \langle V_{(\frac32 , 0)} V_{(\frac32, 0)} \rangle with S=2,3, we indeed find D_{\langle r, 3\rangle}=0. Understanding this observation is left for future work.

— Diagonal boundary conditions in critical loop models  (2512.10400 - Downing et al., 11 Dec 2025) in Section 3.3 Numerical bootstrap results