Symbolic Parameter Classification of Order Parameter Linear Response Spectra — Exhaustive Enumeration Based on the Damped Harmonic Oscillator Model

This paper starts from the most universal linear response dynamical equation for an order parameter in the long-wavelength limit, and systematically investigates all parameter configurations that the spectral peaks of its Green's function exhibit on the complex plane when the effective inertial mass and the dissipation coefficient traverse all physically possible signs and magnitudes. We establish a classification matrix based on 18 fundamental parameter types---with the sign and limiting behavior of the effective inertial mass as rows and the strength of the dissipation coefficient as columns. After an anisotropic generalization, this matrix can be expanded into 250 anisotropic response modes, among which 170 configurations satisfy the spectral peak distinguishability criterion. Of the 18 fundamental configurations, 9 distinguishable oscillatory configurations and 1 critical closure state have been covered by experimental observations, leaving 160 anisotropic subtypes as theoretical candidate configurations. This work establishes a complete pole-symbol classification system for the quasi-particle dynamics of condensed matter linear response, providing a standardized theoretical framework for the identification of anomalous spectral signals and the prediction of novel quasi-particles. In addition, starting from the inhomogeneity of the order parameter space metric, we derive a falsifiable dynamical prediction---the metric gradient force, whose acceleration direction is determined by the metric gradient and is independent of the velocity direction. It is the only even-function response among all known classical forces and can be directly tested on hyperbolic polariton near-field optical platforms. This paper is the eighth in a series. The foundational axiomatic framework is provided in a companion paper (DOI: 10.5281/zenodo.21325460), and the evolutionary time field theory is established in another (DOI: 10.5281/zenodo.21436542).