arXiv:2610.35920v1 Announce Type: new 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.
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