⚠️ WARNING (ВОО): THIS VERSION IS DEPRECATEDWe introduce the Resolvent Hankel Discriminant (RHD, denoted κ), a spectral invariant designed to detect and quantify quasi-degeneracies in the eigenvalue distributions of differential operators on compact Kähler manifolds. The method constructs Hankel matrices from resolvent moments and uses determinant collapse as a signature of spectral clustering. We validate this approach through numerical tests on synthetic models, random matrix ensembles, and quantum geometric operators. Results demonstrate that κ reliably identifies artificial degeneracies and integrability signatures in toy systems, but does not universally discriminate realistic algebraic geometries from transcendental ones without additional constraints. Specifically, Riemann zeta zeros and GUE ensembles yield similar κ values to certain Calabi-Yau spectra, suggesting sensitivity to spectral rigidity rather than algebraicity per se. We propose RHD as a computational tool for exploring spectral structure in geometry and mathematical physics.Version 2 Update:This revision substantially updates the interpretation of the method based on extended numerical validation. Previous strong claims linking RHD directly to the Hodge Conjecture as a "universal criterion" (in v1) have been retracted, as they overestimated the method's applicability to generic algebraic manifolds. The method is now correctly framed as a heuristic for detecting spectral clustering and quasi-integrability. V1 is deprecated.