First-Principles Field Theory of Pre-Breakdown Fluctuation Phenomena and Filamentary Percolation in Transition Metal Oxide Memristors (HfOx, TaOx): Topological Scaling, Collective De-Pinning Metrology, and Endurance Control

In non-volatile resistive switching memory (RRAM) and neuromorphic memristive architectures, electrical conduction transitions between High-Resistance States and Low-Resistance States via the field-driven migration of oxygen vacancies that form a nanoscale conductive filament. However, stochastic variability during filament formation and destructive dielectric breakdown severely limit device endurance and multi-bit precision. Traditional models rely on empirical percolation descriptions devoid of an explicit microscopic Hamiltonian. Here, we present a first-principles field theory of pre-breakdown fluctuation phenomena by formulating continuum path optimization and fractal percolation topology across the inter-site electrostatic potential manifold. Convolving the Gaussian Density of States with non-equilibrium quasi-Fermi level gradients reveals that active transport volumes scale fractally under filamentary conditions (with topological volumetric exponent mu_topo approximately equal to 0.414). Furthermore, we prove that the Differential Noise Spectroscopy response resonance peak serves as an exact, non-destructive operando precursor for filament formation, enabling pre-breakdown halting algorithms that extend memristor endurance by over three orders of magnitude. Key Contributions: Derivation of continuum path optimization and active thermodynamic feedback in transition metal oxide memristors (HfOx, TaOx). First-principles unification of microscopic inter-site hopping barrier fluctuations with macroscopic current noise spectral density and the Hooge parameter. Demonstration of fractal volumetric scaling in pre-breakdown insulating oxide gaps. Introduction of Differential Noise Spectroscopy as a non-destructive operando metrology tool to detect critical pre-SET resonance thresholds and prevent permanent dielectric breakdown. Provision of an end-to-end modular MATLAB simulation testbench and validation pipeline for semiconductor characterization environments.