GlobalCY I: A JAX Framework for Globally Defined and Symmetry-Aware Neural K\"ahler Potentials

Abstract

We present \emph{GlobalCY}, a JAX-based framework for globally defined and symmetry-aware neural K\"ahler-potential models on projective hypersurface Calabi--Yau geometries. The central problem is that local-input neural K\"ahler-potential models can train successfully while still failing the geometry-sensitive diagnostics that matter in hard quartic regimes, especially near singular and near-singular members of the Cefal\'u family. To study this, we compare three model families -- a local-input baseline, a globally defined invariant model, and a symmetry-aware global model -- on the hard Cefal\'u cases $\lambda=0.75$ and $\lambda=1.0$ using a fixed multi-seed protocol and a geometry-aware diagnostic suite. In this benchmark, the globally defined invariant model is the strongest overall family, outperforming the local baseline on the two clearest geometric comparison metrics, negative-eigenvalue frequency and projective-invariance drift, in both cases. The gains are strongest at $\lambda=0.75$, while $\lambda=1.0$ remains more difficult. The current symmetry-aware model improves projective-invariance drift relative to the local baseline, but does not yet surpass the plain global invariant model. These results show that global invariant structure is a meaningful architectural constraint for learned K\"ahler-potential modeling in hard quartic Calabi--Yau settings.