The Shape of Disagreement: Six Mathematical Lenses on the Structure -- Not the Magnitude -- of Meta-Analytic Heterogeneity

Abstract

Background: Between-study heterogeneity in meta-analysis is almost always reported as a single scalar -- I-squared or tau-squared. A scalar answers how much studies disagree, but is silent on the structure of that disagreement: whether it lives in the tails, drifts over time, breaks into clusters, or curves the parameter space. Methods: We assemble six advanced-mathematics lenses -- information theory (entropy), extreme value theory (EVT), spectral analysis, topological data analysis, information geometry, and a hyperbolic-geometry umbrella (HyperMeta) -- each computed from one common study-level feature matrix (effect, precision, time, risk-of-bias). Every lens was validated on simulated meta-analyses only. Topology and information geometry are presented as deep-dives of HyperMeta's homology and information-geometry modules, not as independent results. Results: On simulated data: entropy tracked prediction-interval (PI) width better than I-squared (r = 0.91, 95% CI 0.87-0.94, vs 0.73); GEV-tail detection recovered planted outliers at 94% sensitivity (95% CI 90-97; 3% false-positive rate) vs 78% for standardised residuals; spectral decomposition flagged a periodic component in 12% (95% CI 8-17) and trend greater than 30% of heterogeneity in 23%; persistent homology found distinct clusters in 34% (95% CI 28-41); Fisher-information distance detected outliers at 91% sensitivity (95% CI 86-95) vs 82% for Mahalanobis; and hyperbolic embedding preserved hierarchy with 15% lower distortion (95% CI 11-19) than Euclidean MDS. Conclusion: Six lenses, one argument: heterogeneity has a shape, and a scalar collapses it. The case is proof-of-concept -- entirely simulated, each lens bounded by a sample-size or modelling requirement -- and the necessary next step is validation on real meta-analytic corpora.