Geometric priors for single-patch neural Calabi--Yau metrics on the quintic: a conservative fixed-$\psi$ Ablation study

Abstract

We investigate how geometric priors shape neural approximations of the complex Monge--Amp`ere equation on the quintic Calabi--Yau threefold. This fixed-$\psi$ diagnostic compares four single-patch ans"atze: a coordinate SIREN potential model, a direct-metric Cholesky baseline, an algebraic section ansatz, and a section-plus-neural hybrid. In a 96-run learning-rate and frequency archive, section-based models reach validation $\sigma =4.14$--$4.16$, whereas SIREN and Cholesky remain at $41.29$ and $44.27$. Multi-seed controls use a shared affine-$\eta$ evaluator and five fresh validation clouds. Stable section settings give $3.84$--$4.77$, compared with $28.46$--$30.59$ for Cholesky and $34.73$--$38.13$ for SIREN. Hybrid projective features do not improve best-case accuracy, but they stabilise the stressed regime. They limit $\sigma$ to $4.26$--$4.41$, versus $4.45$--$205.45$ for raw coordinates. Within this protocol, section-based geometric structure is the main driver of accuracy; coordinate-only models plateau at substantially higher error, and raw hybrid flexibility can amplify instability. We provide a projective-scaling derivation, document protocol asymmetries, and identify projectively consistent hybrid corrections as the next step.