Isomorphic Emergence of Fermionic Excitations: Unifying Gravitational Particle Creation and Analogue Solid-State Horizons via an N-Dimensional Matrix Framework

The simple harmonic oscillator is universally recognized as the local approximation for stable equilibria, yet its generalization to N-dimensional non-Euclidean manifolds reveals profound isomorphisms between condensed matter physics and quantum field theory in curved spacetime. In this paper, we propose a unified matrix framework that bridges the emergence of topological solid-state phenomena and gravitational particle creation. By rigorously defining the functional action of an N-dimensional coupled system via its dynamical Hessian matrix, we demonstrate that taking the continuum limit naturally generates covariant field equations. We extend this formalism using tetrad (vierbein) mappings to show that the Bogoliubov mixing responsible for Hawking and Unruh radiation is mathematically identical to local gauge deformations in variable-dispersion optical crystals. This unified perspective establishes that fermionic particle creation is not exclusively a cosmological phenomenon, but a fundamental, observable geometric property of N-dimensional driven topological systems.