Neural Geometrization: Learning Riemannian Metrics on Compact 3-Manifolds and What a Data Field Theory Would Require of Them

We ask whether a variational autoencoder whose latent space is given the differentiable structure of the 3-sphere can, under curvature-driven regularization, learn a metric of constant sectional curvature, and we report a study designed to make that question operationally well posed. Three ingredients are developed. First, a fully differentiable Riemannian pipeline (Christoffel symbols, Riemann tensor, Ricci tensor, Ricci scalar, and their scale-invariant combinations) together with a numerical test suite that validates it against the eight Thurston model geometries, for which the exact scalar curvature, Ricci endomorphism spectrum, and Einstein defect are recovered to or better. Second, a diagnostic hierarchy: we prove that the coefficient of variation of the Ricci scalar, the quantity used as the primary criterion in earlier versions of this programmer, vanishes identically for all eight homogeneous geometries and therefore cannot identify any of them; we replace it by the spectrum of the Ricci endomorphism, which is a complete, scale-invariant signature of a model geometry, and by the dimensionless Einstein defect, which vanishes if and only if the metric has constant sectional curvature (Schur’s lemma in three dimensions). Third, a well-posedness analysis: on the stereographic chart that the latent space actually occupies, a complete metric of constant curvature can only be or ; requires an explicit compactification. The implemented model imposes no such condition, and we show that its metric network does not enter the likelihood at all, so the geometry is not identified by the data. Recomputing all geometric quantities from the released checkpoints in float64 with validated derivatives changes the earlier conclusions qualitatively: the median Einstein defect of the learned metric is (baseline) and (Hessian penalty), comparable to the anisotropic model geometries (1.22 to 4.90) and far from the space forms (); the signature matches none of the eight, whatever the tolerance, fewer than 1% of latent points lie within 0.25 of a model signature, and the Hessian penalty has no statistically resolvable effect (its confidence interval for the median defect overlaps the baseline’s). The Ricci-variance regulariser diverges within four epochs; we identify the mechanism as a scale degeneracy of the functional combined with third-order derivative amplification, and show that gradient clipping cannot prevent it. We conclude that the hypothesis, as stated for a fixed topology, is falsifiable but was not adequately tested by this architecture, and we specify the conditions under which a decisive test becomes possible. We also implement the first three of those conditions in a pilot study: in the round conformal class, with a scale-invariant loss and a volume coupling, the same latent cloud admits a metric that satisfies the geometrisation criterion to ten per cent at a cost of nats per point of density fit, whereas the freely trained metric satisfies no tolerance and is anti-correlated with the density of the data. The obstruction documented here therefore lies in the formulation rather than in the spherical topology.