Geometric Fingerprints of Neutrinoless Double Beta Decay from Multi-Isotope Half-Life Ratios

The interpretation of a future observation of neutrinoless double betadecay ($0\nu\beta\beta$) will be limited by nuclear matrix elementuncertainties, which hinder the use of absolute half-life measurementsto identify the underlying decay mechanism.Ratios of half-lives across different isotopes offer a natural way tosuppress normalization effects, motivating multi-isotope analyses. We develop a geometric framework for multi-isotope $0\nu\beta\beta$based on logarithmic half-life ratios.For each isotope triplet, a two-dimensional fingerprint space isconstructed and an empirical light-neutrino reference manifold isdefined using principal component analysis.Deviations from this structure are quantified by a simple,non-parametric separation metric. The robustness of this diagnostic is evaluated by propagating half-lifemeasurement uncertainties through the full analysis using a Monte Carloprocedure with log-normal fluctuations.We find that the separation power depends strongly on the choice ofisotope triplet.While certain combinations retain near-maximal discrimination even forrelative uncertainties at the $30\%$ level, others exhibit a gradualdegradation as experimental precision decreases. These results show that geometric fingerprints constructed fromhalf-life ratios provide a transparent and conservative diagnostic forassessing the compatibility of multi-isotope $0\nu\beta\beta$ data withpure light-neutrino exchange.