First-order resistance expansion at zero resistance

Derive a first-order expansion of the directional phase and arrival-time difference as the interface resistance tends to zero, by obtaining quantitative control of the transmission perturbation throughout the critical-moment time integral.

Background

The paper proves joint continuity and, under concavity, strict ordering of the directional amplitude and arrival-time difference with respect to resistance. These results do not determine the derivative or first-order asymptotic behavior at the fully permeable interface.

A solution would require quantitative estimates strong enough to justify differentiating the critical moment with respect to the resistance parameter, including control of its long-time tail.

References

A second question concerns differentiability at zero resistance. Under concavity, joint continuity and strict ordering determine how the phase and arrival-time difference vary with resistance, but they do not yield a first-order expansion of $\xi_\beta$ or $\tau_\beta$ as $\beta\downarrow0$.

— Directional delays and singular strong-barrier limits for Fisher--KPP propagation across a semipermeable interface  (2610.21328 - Wang, 15 Oct 2026) in Section 6, Further questions