Formal Consolidation of the Generalized Z-Axis Framework: Geometric Curvature, Rice-Distributed Phase Estimation, and Metric Monotonicity

Para la descripción en Zenodo, pondría esto: Descripción sugerida: This paper formally consolidates and corrects two prior preprints by the same author: State-Space Lift of the Smith Chart (Zenodo, https: //doi. org/10. 5281/zenodo. 19555357) and The Generalized Z-Axis of the Smith Chart (Zenodo, https: //doi. org/10. 5281/zenodo. 19571909). The Generalized Z-Axis framework proposes augmenting the classical Smith Chart with a third coordinate χz=∣∂Γ/∂z∣ᵦ = |/ z| χz=∣∂Γ/∂z∣ that captures the rate of change of the reflection coefficient with respect to any perturbation variable — frequency, temperature, time, pressure, concentration, or position. This consolidated paper introduces four corrections and extensions absent from the prior works: Geometric curvature vs. spectral acceleration: the prior works conflated az=∣∂2Γ/∂z2∣aᵦ = |²/ z²| az=∣∂2Γ/∂z2∣ (spectral acceleration, parametrization-dependent) with geometric curvature κzᵦ κz (Frenet-Serret curvature, reparametrization-invariant). This paper defines both rigorously and explains why κzᵦ κz is the correct quantity for comparing measurements across different sweep rates or experimental protocols. Rice-distributed phase estimation: the decision rules in the prior works implicitly assumed Gaussian phase estimation errors. This paper shows that phase estimation is governed by a Rice distribution, and that the Gaussian approximation fails precisely in the low-SNR regimes where the phase is most unreliable — strengthening, not weakening, the recommendation to use the scalar Z-Axis under those conditions. Metric monotonicity with explicit parametrization: Theorem G3* is restated with the correct condition that the H¹ inequality holds for each fixed weighting parameter λ>0 > 0 λ>0, with the lift metric defined explicitly inside the proof. Fisher Information normalization: the scalar and full-jet Fisher Information expressions are unified under a consistent normalization (2/σd22/d² 2/σd2), eliminating an implicit factor-of-two ambiguity in the prior treatment. Three structural theorems are formally proven (exact collapse G1*, strict non-redundancy G2*, parametrized H¹ monotonicity G3*), together with the Fisher Information Trade-off theorem in closed form. Three application domains are stated as conjectures awaiting experimental validation: relaxation-mechanism discrimination in biological tissue, detection of subtle physiological changes, and internal-defect characterization in materials. Keywords: Smith Chart, state-space lift, generalized Z-axis, reflection coefficient, Fisher information, geometric curvature, Rice distribution, Frenet-Serret frame, bioelectromagnetism, dielectric spectroscopy, Cole-Cole model.