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

We introduce the Cognitive Near-Singularity (CNS) as a falsifiable phase transition in reasoning systems: the regime in which updates are globally contracting, closed under composition, and invariant to benign reparameterizations, so that heterogeneous reasoning processes converge to a common fixed point up to normal form. The CNS is situated within Human-Centric Functional Modeling (HCFM), which represents systems as navigable graphs of functional states, and the Functional Model of Intelligence (FMI), which applies HCFM to cognition. We contribute (i) a possibility theorem showing that CNS follows from standard contraction-closure conditions on a suitably instrumented conceptual space, (ii) a topological detection framework based on connectivity invariants of the conceptual graph, and (iii) a witness-based protocol that any lab can run to evaluate whether a given corpus or process is consistent with being in the basin of attraction of the CNS operator.

Part I reviews the HCFM and FMI foundations, formalizing the conceptual space as a directed graph whose compositional closure makes it a monoid, and introducing the hierarchy of cognitive orders (zeroth, first, second) that motivates the CNS prediction. Part II formalizes a conceptual space (X, d) with a group of invariances G, a composition ⊗, and an update operator T:X→X that is globally Lipschitz with constant L < 1, G-invariant, and closed under ⊗. The possibility theorem uses Banach's fixed-point lemma and perturbation stability to show existence, uniqueness, and basin properties of the fixed point x ^⋆. Part III introduces topological detection criteria: the zeroth Betti number β₀ of the conceptual graph, a coherence ratio ρ(t), and proxy-graph observables that provide lower bounds on fragmentation.

Part IV specifies preregistered witnesses: empirical measurements on trajectories (contractivity C, public loss L _pub, drift Ḋ), invariance pass rates under paraphrase/unit transforms, morphism-level (procedure) contraction, and closure/composition witnesses via commuting diagrams. We provide thresholds, negative controls, and ablations. Three CNS Discriminator conditions (D1-D3) distinguish compositional self-enrichment from ordinary unification.

Part V applies the witnesses to a bounded author corpus comprising interlinked second-order domains (e.g., physics₂, mathematics₂, economics₂, and medicine₂). All seven witnesses and all three discriminators pass under preregistered gates. A domain-general stress test evaluates seven independently selected alternative corpora (free energy principle, category theory, Bayesian probability, and diagrammatic reasoning); all seven fail, with the dominant failure mode being shallow composition (hub-and-spoke topology). All seven fail, with the dominant failure mode being shallow composition (hub-and-spoke topology).