note: Lossy 20 ps delay line, 0.5 dB insertion loss. The baseline sane case.


note: Shunt resonator, Q=30 at 20 GHz. Sharp phase slope near resonance -- the case where a group-delay check without phase unwrapping fails.


note: All-zero S: perfectly matched, fully absorbing. A degenerate but legal network; checkers that divide by |S| must not blow up.


note: Lossless line with sigma_max = 1 - 1e-12. Sits on the passivity boundary; a checker with a too-tight tolerance false-alarms here.


note: Four-port with two independent thru paths. Exercises N>2 handling.


note: Delay line with 3x through-path gain. Creates energy: fails both the spectral-norm and the row-power tests.


note: |S11| > 1 gives Re(Z_in) < 0: negative resistance at the port. It unavoidably breaks energy conservation too -- a reflection coefficient above unity returns more power than arrives -- so this case cannot isolate a single law, and the label says so.


note: Phase advances with frequency: the output precedes the input. Passive and reciprocal, so ONLY the causality check should fire.


note: A ferrite isolator. NON-RECIPROCAL BY DESIGN and entirely realizable -- the medium is not reciprocal. The reciprocity check correctly fires, and that is a true positive for the law but NOT a defect in the device. Any tool reporting this must let the user say so.