Cognitive near-singularity: a possibility theorem and a witness-based protocol with a corpus case study

Area of Science:

• Cognitive Science
• Artificial Intelligence
• Theoretical Computer Science

Background:

• Human-Centric Functional Modeling (HCFM) represents systems as navigable graphs of functional states.
• The Functional Model of Intelligence (FMI) applies HCFM to cognition, providing a framework for understanding reasoning systems.
• Existing models lack a falsifiable marker for advanced reasoning convergence.

Purpose of the Study:

• Introduce the Cognitive Near-Singularity (CNS) as a falsifiable phase transition in reasoning systems.
• Develop a theoretical and empirical framework for detecting the CNS.
• Provide a protocol for evaluating whether a corpus or process exhibits CNS properties.

Main Methods:

• Formalized a conceptual space as a directed graph with invariance groups, composition, and a globally Lipschitz update operator.
• Developed a possibility theorem using Banach's fixed-point lemma to prove CNS existence under contraction-closure conditions.
• Introduced topological detection criteria (Betti numbers, coherence ratio) and a witness-based protocol with empirical measurements.

Main Results:

• Demonstrated that CNS follows from standard contraction-closure conditions in a suitably instrumented conceptual space.
• The witness-based protocol successfully identified CNS properties in a bounded author corpus across multiple domains (physics, mathematics, economics, medicine).
• Alternative corpora (free energy principle, category theory, Bayesian probability, diagrammatic reasoning) failed the stress test, primarily due to shallow composition.

Conclusions:

• The Cognitive Near-Singularity (CNS) is a theoretically grounded and empirically detectable phase transition in reasoning systems.
• The developed framework and protocol offer a robust method for identifying advanced, convergent reasoning.
• Findings suggest that true compositional self-enrichment, characteristic of CNS, is rare and distinct from mere unification.