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Physical interpretation of generalized complex delays

Develop a physical interpretation and practical applications for generalized complex delay quantities defined as derivatives of log S with respect to non-frequency parameters (e.g., phase-shifter length), clarifying their relation to time-domain observables or other measurable response metrics beyond their utility for locating singularities.

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Background

The paper extends complex time delay to generalized delays using derivatives with respect to non-frequency parameters and shows these quantities diverge at the corresponding singularities. However, the physical meaning of such generalized delays (without units of time) is not established.

References

A logical question then arises, what is the physical interpretation of a complex general delay that does not have units of seconds? As of yet we do not have a new application or understanding of generalized delay beyond its use to find singularities.

— Superuniversal Statistics with Topological Origins for non-Hermitian Scattering Singularities (2507.14373 - Shaibe et al., 18 Jul 2025) in Appendix F (Complex Time Delay – Generalizations)