What question did this study set out to answer?

The research aims to introduce the Ψ-Former architecture to solve the Geometric Capacity Bottleneck in deep learning.

January 24, 2026Open Access

The Ψ-Former: Topological Downward Causation via Riemannian Optimization in Deep Neural Architectures

Key Points

The research aims to introduce the Ψ-Former architecture to solve the Geometric Capacity Bottleneck in deep learning.
Developed a neural architecture named Ψ-Former
Utilized Riemannian optimization on Lorentzian manifolds
Embedded representations in hyperbolic space
Employed natural gradient flows for optimization
Achieved topological preservation in neural representations
Demonstrated structural coherence for hierarchical reasoning tasks
Addressed challenges of incompatible architecture growth in neural networks

Abstract

We present the Ψ-Former, a neural architecture that enforces topological constraintsthrough Riemannian optimization on Lorentzian manifolds. The architecture addresses theGeometric Capacity Bottleneck—the fundamental incompatibility between exponentially-branching hierarchical structures and polynomially-growing Euclidean spaces. By embed-ding representations in hyperbolic space and optimizing via natural gradient flows, the Ψ-Former achieves topological preservation and structural coherence. This framework providesfunctional benefits for hierarchical reasoning tasks without ontological claims.

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