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

Abstract

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 v4 This version is a substantial correction and strengthening pass, prompted by an adversarial review that found five errors serious enough to affect the argument. All five are repaired, the whole paper was then re-reviewed, and every reference in the paper has since been resolved against an external source. Five corrections to the physics Information impermeability (IB1) restated as a two-tier property. The earlier flat claim contradicted §8.2 and contradicted Raju (2022), which the paper cites in its own support. Horizons are now semiclassically impermeable but non-perturbatively boundary-available (Bahiru et al. 2024; Raju 2022), and §8.2 is cross-linked so the two sections agree. The tier structure is the substrate/simulation split, so the repair sharpens IB1 rather than weakening it. The Bekenstein bound was misattributed throughout, and it carried one of the three properties the unification rests on. §5.2 Step 1 is rewritten to separate two distinct results: the Bekenstein bound (1981) is S ≤ 2πkER/ℏc — energy × radius, with the rigorous form a relative-entropy statement (Casini 2008) — while the area law is 't Hooft/Susskind, in covariant form Bousso (1999). The two coincide only at the collapse threshold, which is where horizons sit. The property is renamed holographic saturation throughout (26 sites). §8.1 now uses the Bekenstein bound for what it actually says (energy ↔ information), which strengthens that section. §5.6's particle-spectrum argument was self-refuting — an O(1)-bit boundary was asked to encode ≳60 Standard Model states. The bound A ≥ 4ℓP² ln N is now stated transparently, one Planck area is conceded to be 0.36 bits, and "Planck-scale" is scoped to of order. The capacity argument now yields bounds rather than the multiplicity. The §5.6/§5.7 no-hair conflict is resolved in the direction physics points. Baryon and lepton number have no boundary home — and that is the model agreeing with sphaleron violation (Klinkhamer & Manton 1984) and with no-global-symmetries-in-quantum-gravity (Harlow & Ooguri 2021). The paper no longer derives B/L conservation. The heat-death mechanism was wrong by roughly eighteen orders of magnitude. Interior entropy does not rise to meet the horizon: Sobs ≈ 3×10104 k against SCEH ≈ 2.6×10122 k (Egan & Lineweaver 2010). The interior empties instead, and the de Sitter horizon saturates by construction. Two silences a referee would have named DESI DR2 (Phys. Rev. D 112, 083515) added: the Big Rip branch is disfavoured at 3.1σ, not excluded — and the paper now says so rather than treating the three endgames as equally open. Tolman (1934) stated as an objection to cyclic cosmology and answered from the holographic ceiling, with two limits of that answer recorded. Axiom 1 is now argued rather than asserted (§3.1) The claim that nothingness is impossible had rested on calling it "a Platonic abstraction" — a bare assertion carrying a contested thesis. It now runs on an exhaustive dilemma. Any assertion that nothingness is possible must either locate it or not. Located: fixing it relative to what exists implies a separation, hence a dimension, hence a position, hence a property — which meets the Cambridge-property objection head-on, and of which vacuum decay and the empty possible world are instances. Unlocated: the claim becomes "nothing, everywhere and everywhen", refuted by the existence of whoever asserts it. No third form exists. An independent epistemic line is added (nothingness is unobservable in principle, since observing it requires standing in a spacetime relation to it), together with the deflationary last exit. Krauss and Albert are corrected — they reach opposite conclusions — and Albert's objection is granted as an instance of the first horn. A new conditional result in §6.5 The paper's stated open obligation was to derive the Tsirelson bound. §6.5 no longer calls this a candidate direction: it states a conditional derivation. Information causality's inequality and the holographic bound on the shared locus turn out to be the same inequality once m is read as locus capacity rather than a transmitted-bit count, so Tsirelson follows via Pawłowski and Uffink — given a single-locus decoding postulate. That postulate is §5.2's single-surface ontology, adopted for reasons having nothing to do with Bell. The Oughton & Timpson measure-dependence objection is answered from the α = 1 relative-entropy form of the Bekenstein bound. This is a conditional result with its condition named in the text, not an unconditional claim; the remaining gap is formalizing the decoding postulate in entanglement-wedge language. Every reference now names the source that confirms it Nine defective references were found and repaired. Six were citation chimeras or wrong metadata — Rubio & Dunningham (the work is Tsang's), Bisio & Tosini (Perinotti), Konopka & Smolin (Severini), an invented Rowland title, a wrong Boyle & Turok year, and Easson & Brandenberger dated 1999 when the paper is 2001 (JHEP 2001(06) 024). Three more were found in this final pass: the Boyle, Finn & Turok (2018) Physical Review Letter had lost its middle author; an Elze entry welded the title of the 2022 Universe paper onto a 2020 Foundations of Physics slot with page numbers belonging to neither; and a 1968 Gruber monograph that appears in no bibliographic index was replaced with the 1968 chapter that does. All 145 references have now been resolved individually against Crossref, arXiv, DBLP, OpenLibrary or the publisher of record, and each carries the identifier of the record that confirms it. Approximately eight further references were added in support of the corrections above.