The Generalized Z-Axis of the Smith Chart: χz = |∂Γ/∂z| as a Universal Auxiliary Observable for State Detection at Electromagnetic Interfaces

This preprint generalizes the framework introduced in the earlier work State-Space Lift of the Smith Chart, where the specific case was first developed. It introduces a universal family of lifted Smith Chart representations, , where captures the local sensitivity of the reflection state with respect to any admissible parametrizing variable . In this formulation, the previous log-frequency lift appears as the first specific instantiation of a broader theoretical structure. Three universal properties are established for any admissible : exact collapse to the classical 2D Smith Chart when (G1), non-redundancy, meaning that two states indistinguishable in the 2D chart may remain distinguishable in the lifted space through different derivative structure (G2), and H¹-monotonicity of the lifted discrimination metric, ensuring that the extended representation can never be less informative than the classical one (G3). These results formalize the Smith Chart as a projection that preserves instantaneous reflection state while discarding how that state changes as the system evolves. A central claim of this work is that the relevant information was always present in the measurement itself but remained structurally unexploited because it was not represented along a dimension explicitly accounting for variation of the reflection state with respect to a physical variable. In that sense, the contribution is not the introduction of a new measurement, but a new representation that recovers information already contained in the signal and makes it accessible in a form that the classical Smith Chart does not capture. The manuscript develops six further instantiations of the generalized Z-axis across temperature, time, pressure, ionic concentration, spatial position, and gas concentration, spanning biomedical sensing, structural monitoring, industrial non-destructive testing, and environmental detection. Because is obtained through post-processing derivatives from measurements already acquired by existing instrumentation, the framework requires no new hardware and proposes a general method for recovering informational structure that has long been present in broadband electromagnetic measurements but had not been formalized or represented in this way within the classical Smith Chart framework.