Toward Mathematical Formalization¶

The Four-Model Theory is specified at a level of precision that makes mathematical formalization tractable — and the ConCrit framework, dynamical systems theory, and information geometry provide the starting points.

The theory's free-compute requirement is currently specified qualitatively: the substrate must have Wolfram Class-4 (universal-computation) capability and actually deploy it on open-ended self-modeling, with criticality as the dynamical signature this leaves. Its predictions are stated in qualitative terms: "criticality increases," "permeability changes," "the ESM redirects." A full mathematical treatment — defining the four models formally, specifying the criticality signature in measurable quantities, deriving predictions as formal consequences — remains to be developed; a formalization roadmap sets out the recommended approach (Gruber, 2026b).

Why Formalization Matters¶

Mathematical formalization transforms three categories of theoretical content:

Qualitative claims become quantitative predictions. "The substrate is in the Class 4 regime" becomes a specification in terms of branching parameters, DFA exponents, and avalanche power-law exponents. "Variable permeability" becomes a transfer function with measurable parameters. "The ESM redirects" becomes a dynamical systems attractor shift with calculable basin properties.

Derived phenomena become formal consequences. The theory's explanatory range — from psychedelic phenomenology to split-brain phenomena to sleep states — currently relies on verbal reasoning. Formalization would allow these phenomena to be derived from the axioms, making the theory's internal consistency checkable and its predictions more precise.

The architecture becomes simulatable. A formal specification of the four models, their interactions, and the free-compute requirement enables computational simulation of the architecture's component mechanisms. An in-silico research program has begun this in model systems (Gruber, 2026d); it tests mechanisms and does not attempt to build consciousness.

Starting Points¶

Three established mathematical frameworks offer natural entry points:

The ConCrit Framework¶

The Consciousness and Criticality (ConCrit) framework (Algom & Shriki, 2026, Neuroscience & Biobehavioral Reviews) provides mathematical tools developed specifically for the criticality-consciousness relationship: power-law exponents for neuronal avalanche distributions, detrended fluctuation analysis (DFA) for long-range temporal correlations, and branching parameters for propagation dynamics; Lempel-Ziv complexity adds a measure of the richness of the dynamics (see Information-Theoretic Measures). These tools can quantify the criticality signature — specifying the parameter ranges of the regime's measurable signatures. Which class a given automaton belongs to is formally undecidable, so the commitment is to those signatures rather than to a proof of class membership.

Dynamical Systems Theory¶

The four models can be formalized as coupled dynamical systems. The implicit models (IWM, ISM) are slow-timescale systems (changing over days to years through learning), while the explicit models (EWM, ESM) are fast-timescale systems (updating at perceptual frame rates, typically 7–13 Hz and up to about 20 Hz). The implicit-explicit boundary becomes a transfer function. Variable permeability becomes a parameter controlling information flow between slow and fast systems. The self-referential closure of the ESM becomes a fixed-point property of the coupled system.

Information Geometry¶

The four models can be represented as points or distributions in an information-geometric space. The real/virtual split maps to a distinction between structural parameters (which define the manifold) and dynamic trajectories (which move along it). The graduated levels of consciousness map to the dimensionality of the self-referential subspace. This approach connects naturally to IIT's mathematical framework while avoiding its panpsychist commitments.

Current Status¶

A formalization roadmap translating the theory's structural commitments into mathematical notation exists (Gruber, 2026b). Quantitative operationalization — effect sizes, measurement protocols, falsification thresholds — requires collaboration with empirical laboratories, and the model systems of the in-silico program supply a concrete setting in which the formal quantities can be operationalized.

The Four-Model Theory is specified precisely enough that formalization is tractable. The four models are defined by two axes (scope and mode). The free-compute requirement invokes a well-characterized dynamical regime. The predictions specify measurable quantities.

Figure¶

graph TD
    subgraph "Current State: Conceptual"
        A["Four-Model Architecture<br/>(qualitative)"] --> B["Class-4 Computation<br/>(criticality = signature)"]
        B --> C["Verbal Predictions<br/>(qualitative)"]
    end

    subgraph "Formalization Path"
        D["Dynamical Systems<br/>Coupled slow/fast systems<br/>Transfer functions"] --> G["Formal FMT<br/>Quantitative predictions<br/>Simulatable architecture"]
        E["ConCrit Tools<br/>Power-law exponents<br/>DFA, branching params"] --> G
        F["Information Geometry<br/>Model space topology<br/>Self-referential subspace"] --> G
    end

    C -.->|"formalize"| D
    C -.->|"operationalize"| E
    C -.->|"geometrize"| F

    style G fill:#e74c3c,stroke:#333,color:#fff
    style A fill:#3498db,stroke:#333,color:#fff
    style B fill:#3498db,stroke:#333,color:#fff
    style C fill:#3498db,stroke:#333,color:#fff

The path from conceptual to formal theory. Three mathematical frameworks provide complementary entry points: dynamical systems (for the four-model interactions), ConCrit tools (for operationalizing criticality), and information geometry (for the topology of model space). The target is a unified formal treatment that generates quantitative, simulatable predictions.

Key Takeaway¶

Mathematical formalization of the Four-Model Theory is tractable, and a roadmap exists. The ConCrit framework, dynamical systems theory, and information geometry provide the tools; quantitative operationalization is the work that remains.