Uncertainty Quantification of the $^{76}$Ge Neutrinoless Double-Beta Decay Nuclear Matrix Element

The experimental pursuit of neutrinoless double-beta decay ($0\nu\beta\beta$) constitutes one of the most compelling avenues for probing lepton-number violation and exploring physics beyond the Standard Model. Within this landscape, $^{76}$Ge has consistently ranked among the most promising isotopes for current and next-generation bolometric and liquid-scintillator experiments, notably GERDA and LEGEND. In the present work, we adapt a rigorous statistical protocol previously established for $^{48}$Ca~\cite{Horoi-prc22} and $^{136}$Xe~\cite{Horoi-Xe-2023} to the $^{76}$Ge system, utilizing a valence configuration that aligns with our recent investigation of $^{82}$Se~\cite{Neacsu-Symmetry-2024}. Our methodology introduces systematic, bounded fluctuations to the two-body matrix elements of established effective interactions, subsequently monitoring how these perturbations propagate through a suite of low-energy nuclear observables. Special emphasis is placed on the $0\nu\beta\beta$ nuclear matrix element (NME), whose theoretical uncertainty currently dominates the interpretation of experimental half-life limits. By integrating these simulated variations into a Bayesian Model Averaging framework and benchmarking against empirical spectroscopic data, we derive a constrained probability distribution for the NME. The resulting analysis yields a central value of 2.46 with an associated standard deviation of 0.25, thereby quantifying the intrinsic theoretical spread within the interacting shell model approach. Furthermore, we perform a comprehensive correlation analysis across all computed observables to evaluate internal consistency, identify non-trivial structural dependencies, and establish benchmarks that may guide the refinement of future effective interactions.