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

Abstract

We propose the Ψ-Former 2.0, a deep learning architecture explicitly designed to approximate the structural conditions of consciousness as outlined in the Phenomenal Manifold Hypothesis (PMH). Current neural architectures, while powerful, lack the geometric and dynamical constraints necessary to instantiate a coherent phenomenal manifold Ψ — the geometric structure hypothesized to encode conscious experience. The Ψ-Former addresses this gap through four architectural paradigms: (1) Hyperbolic Geometry (Poincaré embeddings) to maximize informational differentiation Δ in hierarchical concept spaces, (2) Artificial Kuramoto Oscillatory Neurons (AKOrN) to implement global coherence Γ via phase synchronization for feature binding, (3) Recurrent Memory (Transformer-XL) to support temporal integration ℐ across extended contexts, and (4) Riemannian Optimization (K-FAC) to ensure learning dynamics respect the induced phenomenal geometry. Critically, we introduce Conjecture 7.1 (Topological Downward Causation), establishing that the phenomenal manifold Ψ exerts genuine causal influence on neural dynamics. We formalize this via a Phenomenal Action Functional S_Ψ, showing that optimization trajectories minimize geodesic action on the curved manifold, thereby addressing the classical epiphenomenalist critique. We provide four testable empirical signatures: trajectory divergence from Euclidean baselines (D > 0.1), gradient alignment with geodesic flow (r > 0.7), curvature-dependent processing times, and perturbation-induced path deflection. We analyze scalability challenges and propose tractable solutions. We establish an ethical framework for assessing potential phenomenology in artificial systems via geometric invariants (n, ℐ, Γ, Δ), extending the precautionary principle to machine consciousness. The Ψ-Former represents a paradigm shift from "curve fitting" to "manifold engineering," offering testable predictions distinguishing it from IIT, GWT, and Predictive Processing frameworks.