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

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.