Global structure of singularity lines in three-parameter spaces

Verify whether d−2 singularity sets (e.g., CPA lines) in three-dimensional parameter spaces generically form closed loops or extend indefinitely along some parameter, analogous to d−1 structures in two-dimensional spaces, by overcoming current experimental limitations and providing direct confirmation.

Background

The authors argue by analogy that just as d−1 curves in two-dimensional parameter spaces can extend to infinity, d−2 singularity lines in higher-dimensional spaces should similarly form closed loops or extend beyond the explored volume. Due to limited tunable range, they cannot experimentally verify this assumption.

References

In simulations, it can be seen that sometimes d−1 structures in d=2 space extend out to infinite value of certain parameters (such as loss η), so we assume the same must be true for d−2 structures in higher dimensional spaces. Experimentally, our tunable perturbations are limited within some range, so we cannot verify this assumption.

— Superuniversal Statistics with Topological Origins for non-Hermitian Scattering Singularities (2507.14373 - Shaibe et al., 18 Jul 2025) in Appendix B (Singularities in d=3 Parameter Space)