Emergent Spacetime from Self-Referential Computation: A Hierarchical Cellular Automaton Framework

This paper proposes a cosmological model — the Singularity-Bounded Holographic Class 4 Automaton (SB-HC4A) — derived from the convergence of four independently motivated frameworks: a five-class computational taxonomy that refines Wolfram's (2002) classification by separating fractal from random dynamics, a theoretical framework for self-referential computation in self-modeling systems (Gruber, 2015, 2026a, 2026b) which identifies self-referential simulation at criticality as a universal computational pattern, and 't Hooft's (1993, 2016) holographic automaton interpretation of quantum mechanics. The model proceeds by elimination: Classes 1–3 cannot sustain the universal computation the universe demonstrably supports; Class 5 (genuine randomness) makes physics fundamentally impossible; therefore the universe operates at Class 4 — the edge of chaos. Combined with the information-theoretic observation that singularities at every physical scale (Planck regime, particle interiors, event horizons, cosmological horizons, temporal endpoints) share the property of information impermeability and Bekenstein saturation, the model proposes that these singularities are structurally identical — scale-invariant instances of the same information boundary. The resulting architecture is a self-referential holographic Class 4 automaton bounded at every scale by singularity surfaces, where the observable interior is the "simulation" and the singularity boundary is the "substrate." All singularities — including temporal endpoints — are shown to be asymptotically unreachable from within the computational domain, strengthening the unification claim. Because singularities transform rather than destroy information, heat death constitutes a singularity transition that triggers cyclic renewal, with potential CPT signature alternation across cycles — connecting to Penrose's Conformal Cyclic Cosmology and Boyle and Turok's CPT-symmetric universe. All three cosmological endgames — heat death, Big Crunch, and Big Rip (Caldwell, 2002) — drive the computational domain to Bekenstein saturation, with the Big Rip uniquely producing a branching tree of daughter universes rather than a linear successor. This architecture is structurally identical to self-referential computational systems that operate at criticality, where implicit knowledge (substrate) is separated from explicit representation (simulation) by an information-opaque boundary. Self-modeling cognitive systems are thus local, scale-reduced instances of the same computational pattern the universe implements globally. Six weak points are identified, including the fundamental epistemological objection that Class 4 observers may be constitutionally incapable of determining whether this model describes the universe or merely the ceiling of their own computational capacity. Changelog v3 Major soundness-and-rigor revision in two passes (Fable 5-assisted), plus an author-driven reframe of the unreachability and observer material. Round 1 — soundness corrections (C1–C6, NEW-1–4): Taxonomy (§2.3/§3.2): the cellular-automaton classification now rests on computational reducibility (Rule 90 = reducible fractal, Class 3; Rule 30 and Rule 110 = computationally irreducible, Class 4); the undecidability of CA classification (Culík substrate determinism is now an explicit, stated-once assumption ('t Hooft, 2016), and the Class-4 elimination is conditional on it. Singularity unification (§5.2): the Identity-of-Indiscernibles argument is replaced by a single-surface ontology (one encoding surface; each singularity a local reflection), with an operational indiscernibility razor retained as a scoped secondary line. Kerr–Newman (§5.7): the naked-singularity / Compton-vs-Planck scale tension is named explicitly and addressed (conjecturally) via Einstein–Cartan torsion; "structural identity" is demoted to "striking correspondence carrying an unresolved tension." Entanglement and Bell (§6.5): an explicit Bell/CHSH treatment via holographic non-separability (entangled pair = one boundary locus; interior locality denied; ER=EPR and Van Raamsdonk wired in; Bohmian existence proof; entanglement-monogamy answer to superdeterminism; no-signalling). The James–Stein argument is demoted to a heuristic pending a discrete/CAT(0) formalization, and the genuine open obligation is reframed as deriving the Tsirelson bound (information causality flagged as a candidate route, not a proof). Reversibility and time (§8.4, new): a reversible/unitary substrate with an emergent thermodynamic arrow (coarse-graining + the Past Hypothesis); playback-reversal and the forward/boundary-ward asymmetry; the CPT theorem and block-universe as confirmation. Genericity (§9.6): Class-4 genericity rises with dimension, softening the fine-tuning worry; a seventh weak point (§9.7) on the saturation trigger. Round 2 — unreachability, the observer, and the saturation mechanism (author-driven): §5.3 reframed as "Unreachability Along Three Axes" — recession (horizons), scale-shielding (the Planck floor; interactions never resolve zero separation), and termination-without-arrival (the temporal termini are boundaries, not events) — with the BKL/Mixmaster observation that finite proper time need not bound computational depth, the realistic-Crunch causal fragmentation (asymptotic silence; the merged endpoint in no observer's past light cone), and the past's informational shrouding (Borde–Guth–Vilenkin). Black-hole complementarity (§8.2): added as the established local instance of the substrate/simulation duality, with the firewall problem flagged and no side taken. Saturation trigger (§5.4/§9.7): the honest status expanded — the ingredients the saturate-and-decompress mechanism needs (complexity sustained at high density, exact on/off symmetry, reversibility) each exist in known cellular automata (e.g. Day the abstract and introduction were reconciled to all of the above.