A Unified Geometric Field Theory of Self-Organizing Intelligence via Hybrid Classical-Quantum Topological Measurement

Contemporary artificial intelligence systems face a fundamental limitation identified here as the Geometric Capacity Bottleneck: Euclidean representation spaces cannot accommodate exponentially growing hierarchical information structures without catastrophic metric distortion. This work proposes a Unified Geometric Field Theory of Self-Organizing Intelligence, grounded in the Phenomenal Manifold Hypothesis (PMH). The PMH posits that a self-organizing system induces an effective Riemannian manifold if and only if its dynamics satisfy critical thresholds of Integration (I), Coherence (Γ), and Differentiation (Δ).

We provide a formal existence proof (Theorem 2.2) for a hybrid metric tensor that becomes positive-definite precisely when these invariants exceed critical values, representing a geometric phase transition. To operationalize this theory, we introduce a hybrid classical-quantum architecture:

1. Classical Substrate: Hyperbolic Neural Cellular Automata (H-NCA) operating on a Poincaré disk tessellation to resolve the capacity bottleneck via exponential volume growth.

2. Quantum Topological Sensor: A quantum coprocessor that estimates global topological invariants (persistent homology and phase coherence) in real-time. Crucially, the quantum device acts as a sensor for feedback regulation, not a simulator, providing an operational advantage under fixed temporal constraints.

The system is governed by a principle of least action, minimizing a unified functional that combines Bayesian inference (variational free energy), informational gravity (Einstein-Hilbert action on Fisher manifolds), and topological field theory (Berry phase corrections) into a single equation of motion.

Experimental Results: We validate the theory via a Divergence Test, based on a novel Operational Causality Criterion. The geometrically regulated system exhibits statistically significant trajectory divergence (p < 10^-6) compared to Euclidean controls. Furthermore, the system demonstrates a 3.2x gain in action efficiency and a +23.4% improvement in out-of-distribution generalization, confirming that geometric regulation is causally efficacious for intelligent self-organization.

Keywords Self-Organizing Intelligence; Information Geometry; Hyperbolic Neural Cellular Automata; Topological Data Analysis; Hybrid Quantum-Classical Systems; Persistent Homology; Phenomenal Manifold Hypothesis; Geometric Deep Learning

Notes / Additional Info This paper presents a formal proof of the Geometric Capacity Bottleneck (Theorem 3.1) and unifies concepts from tensor networks (MERA), AdS/CFT correspondence, and active inference within a gauge-theoretic framework