GlobalCY I: A JAX Framework for Globally Defined and Symmetry-Aware Neural Kähler Potentials

Abdul Rahman

hep-thApr 13, 2026

GlobalCY I: A JAX Framework for Globally Defined and Symmetry-Aware Neural Kähler Potentials

Authors: Abdul Rahman

Abstract

We present GlobalCY, a JAX-based framework for globally defined and symmetry-aware neural Kähler-potential models on projective hypersurface Calabi--Yau geometries. The central problem is that local-input neural Kähler-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 Cefalú 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 Cefalú cases $λ=0.75$ and $λ=1.0$ 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 $λ=0.75$ , while $λ=1.0$ 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 Kähler-potential modeling in hard quartic Calabi--Yau settings.

Figures & tables

Figure 1. High-level architecture of the GlobalCY I benchmark stack. The workflow is organized around a two-layer division of responsibilities. GeoCYData provides the geometric and protocol substrate, including quartic geometry cases, bundle exports with local-chart, invariant, and symmetry-aware views, and the canonical case identifiers, seeds, and benchmark presets used for controlled comparison. GlobalCY consumes those bundles to instantiate the three model families studied here— LocalPhiMLP , GlobalInvariantPhi , and SymmetryAwareGlobalPhi —construct learned metric corrections through the Kähler-potential ansatz $g=g_{\mathrm{FS}}+\partial\bar{\partial}\phi$ , and evaluate them with geometry-aware diagnostics. The same stack supports smoke validation, multi-seed ablations, case-wise aggregation, and frozen paper-facing outputs such as comparison tables, figures, and summary artifacts. The figure highlights that the present study is not only a model comparison, but a reproducible geometry-to-results pipeline in which the geometric substrate is held fixed while architectural assumptions at the learned-model layer are varied in a controlled way.

Component	Setting
Geometry family	cefalu_quartic
Cases	cefalu_lambda_0_75 , cefalu_lambda_1_0
Model families	local , global , symmetry_aware
Seed set	$7,11,19$
Per-case workflow	multi-seed ablation + comparison aggregation
Result-export workflow	freeze_results

Table 1. Controlled comparison protocol used in the benchmark. Two hard Cefalú cases, three model families, and a fixed three-seed evaluation scheme are used throughout.

Diagnostic	Interpretation	Role
Negative fraction	Frequency of negative-eigenvalue behavior in the learned metric correction	Primary
Minimum eigenvalue mean	Mean lower-tail eigenvalue behavior	Secondary
Chart consistency	Chart-overlap consistency summary	Secondary / nondiscriminating in current benchmark
Projective invariance drift	Drift under projective-invariant comparison; key global-structure diagnostic	Primary
Symmetry consistency	Symmetry-sensitive comparison summary	Secondary / architecture-sensitive
Determinant mean	Mean determinant summary for the learned metric correction	Auxiliary

Table 2. Diagnostic suite used in the benchmark. The final column indicates the main interpretive role of each diagnostic in the current comparison.

Artifact	Role in manuscript workflow
paper1_core_results.csv	machine-readable core comparison table
paper1_core_results.md	readable core comparison summary
paper1_robustness.csv	machine-readable robustness/interpretation table
paper1_robustness.md	readable robustness/interpretation summary
paper1_results.json	consolidated machine-readable result package
fig_core_comparison.png	core comparison figure across selected cases and models

Table 3. Result artifacts produced by the freeze_results workflow.

Case	Model	Negativity	Drift	Train loss
Cefalú $\lambda=0.75$	Local	0.08854	$4.39\times 10^{-8}$	11.94
	Global	0.04167	$1.44\times 10^{-8}$	11.02
	Symmetry-aware	0.12500	$2.17\times 10^{-8}$	12.43
Cefalú $\lambda=1.0$	Local	0.04688	$3.95\times 10^{-8}$	10.65
	Global	0.04167	$1.57\times 10^{-8}$	9.73
	Symmetry-aware	0.13021	$2.02\times 10^{-8}$	11.44

Table 4. Core comparison across the two hard Cefalú cases. Lower values are better for negativity, projective-invariance drift, and training loss.

Case	Strongest model	Strongest gains	Hardness note
Cefalú $\lambda=0.75$	Global	Large improvement over local in negativity and projective drift	Clearer separation between global and local
Cefalú $\lambda=1.0$	Global	Improvement over local remains, but the negativity gain is smaller	Harder case

Table 5. Compact interpretation summary. The globally invariant model is strongest overall, with the clearest gains at

\lambda=0.75

Figure 2. Benchmark-wide comparison across the two hard Cefalú cases, $\lambda=0.75$ and $\lambda=1.0$ , for the local, globally invariant, and symmetry-aware model families. The three panels report mean negative-eigenvalue frequency, mean projective-invariance drift, and mean training loss, with error bars indicating cross-seed variation over the fixed three-seed protocol. The figure shows that the globally invariant model is the strongest overall architecture in this benchmark: it achieves the lowest projective-invariance drift and the lowest training loss in both cases, and it also yields the lowest negative-eigenvalue frequency among the compared models. The separation between the global and local baselines is more pronounced at $\lambda=0.75$ than at $\lambda=1.0$ , indicating that $\lambda=1.0$ is the more difficult regime. The symmetry-aware model preserves part of the drift improvement relative to the local baseline, but its larger variability and weaker negativity behavior show that the present symmetry-aware implementation does not yet surpass the plain globally invariant architecture.

Figure 3. Focused comparison for the hardest benchmark case, the Cefalú regime $\lambda=1.0$ . The three panels isolate the same diagnostics as in Figure 2 : mean negative-eigenvalue frequency, mean projective-invariance drift, and mean training loss, again with error bars indicating cross-seed variation. This figure clarifies why $\lambda=1.0$ is interpreted as the harder regime. The globally invariant model still outperforms the local baseline, particularly in projective-invariance drift and training loss, but the negativity improvement is smaller than in the $\lambda=0.75$ case. The symmetry-aware model continues to reduce drift relative to the local baseline, but its higher variability and worse negativity rate indicate that the current symmetry-aware architecture is not yet as well conditioned as the simpler globally invariant model.

Geometric Representation Learning

Geometric representation learning encodes data to preserve its intrinsic shape, distances, symmetries, and relationships.

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Kähler landscapes for complex neural network descents and guarantees including a search and destroy of the Calabi-Yau manifold

Andrew Gracyk

Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

Riemannian Optimization Gradient Descent Neural Network Optimization Complex-Valued Neural Networks Loss Landscape Geometry

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We study landscapes for complex-parameterized networks. Our approach is motivated with an information-theoretic manifold perspective of the parameter and via classical optimization guarantees although of complex geometric variety such as through Dolbeault asymptotics. The descent path admits a Kähler information metric under a cross-entropy via the Wirtinger Hessian on the log-likelihood potential. We restrict attention to a descent update rule with natural gradient descent via a differentiated loss scaled by the inverse metric, so the descent path remains in the holomorphic tangent bundle. We emphasize Calabi-Yau information manifolds which profane theoretical guarantees via an ill-curvature-conditioned, or lack thereof, landscape. We focus on Calabi-Yau metrics specifically in a non-compact setting with a global potential, so defined geometrically rather than invoking the topological requirements of the Calabi conjecture. In non-compact settings, we can write the metric determinant with respect to a background in terms of a function, pluriharmonic or real. Using Yau's Liouville theorem, we get a partial eigenvalue blow-up effect. In an empirical setting, a Ricci-flat metric will not form, but a blow-up can be a local condition and can hold empirically on open sets even partially in generalized Kähler cases due to lower bounds on the Ricci potential. Moreover, it has been discovered that negative curvature subverts the loss landscape, specifically sectional curvature, so we expand on this and draw interconnections to negative-definite Ricci curvature. Our arguments primarily exist via geometric modality, although we establish roots in deep learning theory such as through asymptotics at initialization and connections through failure modes of neural network guarantees under vanishing and negative Ricci curvature.

Abstract

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