Every Spectral Switch Costs Memory: Sharp Robust Wigner—Smith Speed Limits for Passive Quantum Networks

Exact interpolation can force the internal degree of a passive network, but laboratory calibrations are approximate. We prove a sharp frequency-domain speed limit requiring neither exact zeros nor analytic continuation away from the measured frequency axis. Let (S(e^{itheta})) be an absolutely continuous unitary scattering path with positive Wigner—Smith generator (Q(theta)=-iS(e^{itheta})^astpartial_theta S(e^{itheta})succeq0). If a fixed (k)-dimensional input subspace is routed approximately between complementary output sectors with amplitude leakages (varepsilon_p,varepsilon_s), define (alpha=[pi/2-arcsinvarepsilon_p-arcsinvarepsilon_s]_+). Every transition then requires Wigner—Smith trace action at least (2kalpha) and largest-proper-delay action at least (2alpha). Costs add over disjoint frequency arcs. For a rational inner network of McMillan degree (n), (M) alternating pass/stop pairs imply (nge 2Mkalpha/pi), recovering (nge Mk) at zero error. An explicit (2k)-port interferometric family attains the bounds for every admissible error pair. A tomography-error corollary converts finite scattering measurements directly into certified degree and delay lower bounds. Reproducible certificates audit equality cases, positive-block inequalities and random Blaschke—Potapov products.