Toward a Mathematical Formalization of the Four-Model Theory: A Recommended Approach

Abstract

The Four-Model Theory of Consciousness (FMT; Gruber, 2015, 2026) proposes that consciousness is constituted by real-time self-simulation across four nested models — Implicit World Model (IWM), Implicit Self Model (ISM), Explicit World Model (EWM), and Explicit Self Model (ESM) — operating at the edge of chaos. The theory currently operates in natural language. This paper outlines a recommended mathematical formalization strategy. The four models are understood as a minimum sufficient set for human-level consciousness, not an exhaustive enumeration: the biological substrate — built from spiking neurons atop proteomic networks with their own intracellular learning — implements an uncountable number of overlapping models on both sides of the implicit/explicit divide. The formalization must therefore treat models not as discrete, countable objects but as a continuous density over a model space, with the virtual/non-virtual split as the one hard ontological boundary. Six formalization modules are proposed: (1) a continuous model-space framework replacing the discrete 2×2 taxonomy, formalizing the multiple-generator architecture; (2) permeability as a family of channel-specific information-theoretic gating mechanisms; (3) criticality as a conjectured operating-regime prerequisite — an empirically-motivated hypothesis resting on two open lemmas, not a derived consequence of self-referential closure; (4) ESM redirection dynamics; (5) self-referential closure including the observability constraint (O_ESM ⊆ S_EWM); and (6) category-theoretic architecture. A phased build order prioritizes empirically testable components. The formalization is offered as a research program specification for mathematically trained collaborators; verification of the formal apparatus is explicitly deferred to domain experts. Changelog v2 **v2 (2026-08-12) — theory alignment, citation repair, and notation hygiene** This version brings the roadmap into line with the parent theory as currently published, repairs its reference list, and removes several notation collisions. No formalization module is withdrawn, and no proposal is retracted; two claims are weakened deliberately, and one is now attributed correctly. *Principle structure.* Section 1.1 stated the theory as resting on five principles (criticality, virtual qualia, a redirectable Explicit Self Model, variable implicit-explicit permeability, virtual model forking). The parent theory settled on **three** — open-ended computation permitting free modelling, a second computational level hosted by recurrent non-linear dynamics, and phenomenality arising where the modelling closes on itself. The four model kinds, redirection, permeability and forking are consequences rather than axioms, and Section 1.1 now says so, because a formalization that axiomatized the consequences would be formalizing the wrong objects. *Section 1.1, framing.* The formalization gap was argued from the possession of notation: IIT has Φ, Predictive Processing has free energy, and the Four-Model Theory has prose. That argument is now rejected in the text rather than made by it. Possession of a formalism cannot be the standard when Φ is the standing counterexample — fully specified, never computed for a real brain, provably intractable for realistic systems (Aaronson, 2014). The gap is re-measured as constraints accepted: commitments to functional forms that can be empirically wrong. *Criticality.* Demoted throughout from a requirement to an operating-regime hypothesis, matching the posture Sections 4.4 and 4.5 already held. The σ band is presented as an empirical generalization calibrated to the recordings it was estimated from, with the caution that nominal σ = 1 is a normalization convention rather than a substrate-independent edge. The three candidate criticality measures in Section 4.3 are now stated as options for which observable indexes the regime, not as requirements on consciousness. *Prediction 1.* Restated from a categorical behavioural difference to a graded, capacity-relative cost, motivated theory-side: a finite closed loop unrolled over a finite horizon is a feedforward computation with memory, so a clean binary loop-cut test cannot exist on a deterministic, episode-stationary task, and predicting one would contradict the framework's own commitments. The falsifier is now a magnitude at matched capacity. *Citations.* Kauffman's *Self-reference and recursive forms* is **1987**, not 2005. The third author of Bhatt, Zhang & Gan (2009) is **Gan, W.-B.**; the entry carried a corrupted name. The Hengen & Shew (2025) entry was a placeholder with an invented-format title and is now the real paper, *Is criticality a unified setpoint of brain function?*, *Neuron*. Smithe (2024) has its venue completed (arXiv:2406.07577). Two references present in the list but never cited — Baars (1988) and Carhart-Harris et al. (2014) — are now cited where they belong. Bhatt et al. supported a claim about intracellular computation that it does not make, and now sits behind Bhalla (2014) for that claim. The companion cosmology paper, previously cited five times as a manuscript, has been published and is cited by DOI. *Cross-references.* Every reference to the parent paper by internal draft version has been re-pinned to the published version. Two were wrong on substance rather than form: the controllability-of-ego-dissolution result is the parent's **Prediction 2**, not Prediction 3; and the time-dilation claim was attributed to the parent paper, which does not make it. It is now presented as developed here, with the parent's temporal-echo mechanism cited for what it does say. *Notation.* ρ served simultaneously as the model density, the recursion map and a group representation; it is now reserved for the model density, with the recursion map written μ_n and the group representation π(g). The self-representation map is renamed Φ → **Ψ**, with a note stating that no relationship to integrated information theory's Φ is intended — the two appeared within pages of each other. The extent fraction is renamed f_crit for the same reason. Permeability's transfer entropy is now defined between time series, as transfer entropy requires, rather than between regions of a density. *Also.* The seizure case is stated route-agnostically, as an exit from Class 4 rather than as a specific departure direction, the ictal route being phase- and scale-dependent and unsettled. The graded implicit-explicit boundary is explicitly reconciled with the hard threshold ν_crit. Section 2.3's necessity claims are marked as definitional. A rendering defect is corrected: Section 6.4's central equation previously appeared as the identity map, the fixed-point notation having been consumed as markup.