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 v5 **v5 (2026-08-27) — a five-agent adversarial review, plus three revisions that never reached the v4 artifact** Two independent things are corrected here, and it is worth separating them. **First, v4's file was three commits behind its own source.** Work that had been written, reviewed and committed never reached the deposited PDF: the terminology note that governs how the word "simulation" is used throughout the paper, the definition of Φ as the composition of holographic encoding with decompression, and the jointness and minimality clauses that qualify the entanglement argument. A reader of v4 was reading a paper the repository had already superseded — the word "simulation" appeared twenty-two times in the published file against nine in the source. All three are present now. The companion formalization paper was unaffected and is unchanged. **Second, a five-agent review of the whole paper found defects the stale file does not explain.** They are corrected below. Nothing in the architecture has changed. What has changed is that several claims now say what the paper's own sections earn, and three citations now say what their sources say. *Citations that misrepresented their primaries.* Bahiru et al. (2024) was cited in support of the holography-of-information position; the paper constructs approximately local bulk observables and argues that it **is** possible to localize information in perturbative quantum gravity, which is a qualification of that position rather than support for it. The passage now distinguishes the semiclassical and non-perturbative tiers explicitly and represents the source correctly. Uniqueness for the rotating **charged** black hole was credited to Carter (1971) and Robinson (1975), who proved the vacuum rotating case; the charged result is Mazur (1982) and Bunting (1983), and both now appear. The Gutenberg–Richter law is no longer described as documenting self-organized criticality, a framework proposed three decades after it: it is cited as the power law that Bak later read as an SOC signature. Salmon et al. (2024) was glossed as reporting James–Stein advantages in entangled Gaussian sensing; the paper finds that entanglement *diminishes* the shrinkage advantage in the noiseless case and that noise restores it, and it is now described that way. Wetterich's spinor-gravity automaton was presented as a model of quantum gravity "not a metaphor", dropping both conditions the source attaches — the continuum limit in question is the naive one, and the author's own claim is that the setting *could serve* as such a model *if* diffeomorphism symmetry is realized in the true limit; both conditions are now carried. And a torsion result Poplawski establishes for the cosmological singularity was stated as preventing classical point singularities generally; the extension to the rest of the inventory is now marked as this paper's conjecture rather than as his result. *Two mathematical errors in the entanglement section.* The James–Stein theorem was given a hypothesis it does not have — that the parameters are drawn from a shared distribution — when dominance holds for arbitrary fixed parameters, which is the theorem's notorious feature and the one the argument actually needs. The shrinkage coefficient was printed as a function of the unknown parameter rather than of the data. Both are corrected. Separately, the Bell factorizability condition omitted the measurement settings and no longer does. *A claim the paper argued nowhere.* Section 11.2 said the emergence of self-modeling systems in a Class 4 universe is "structurally guaranteed". No section argues guaranteed emergence: the nesting result of Section 2.5 establishes possibility. The claim is now an efficiency claim — where such systems arise, the closed architecture is the economical one — and the paper says plainly that the argument which rescues the biological version of the claim is unavailable at cosmological scale, because nothing in the model bounds the universe's own compute. *Definitional and cross-section inconsistencies.* The defining criterion of Class 4 was stated two incompatible ways, once as irreducibility with universality serving as evidence and once as irreducibility conjoined with universality. The first is now used throughout; under the second, the paper's re-filing of Rule 30 would have rested on two independent conjectures rather than one. Further contradictions have been resolved by deciding which side the paper actually holds: the observable universe's horizon is now treated as observer-relative rather than as "plainly an artifact", reconciling Sections 4.1 and 5.6; the claim that Big Rip fragments never exchange information is scoped to the terminal configuration, where Section 5.6 holds the opposite of interacting particles; the derivation of quantum-number discreteness from boundary labels is narrowed to the quantities that have a boundary home, since the paper's own next subsection denies one to colour, weak isospin, baryon number and lepton number; and the timeless fixed-point reading of Φ(U) = U is now explicitly distinguished from the interior, ongoing-process reading that the saturation-trigger argument requires. *Evidential grading.* Holographic saturation was listed among three properties described as rigorous and shared by every member of the singularity inventory. It is established for the horizons, is this model's own prediction for particles, follows for the Big Bang from a mechanism the paper bars from support duty, and is not established for the Planck floor. The unification argument is now stated as resting on impermeability and domain-bounding, with saturation entering as a shared expectation. A conservation row in the cross-scale mapping table asserted an information-conservation claim across the implicit/explicit split that the paper contradicts twice elsewhere and defends nowhere; it has been removed. *The physics gaps are now stated as gaps.* The energy-information hypothesis is not supplied with an exchange rate by either of the results that motivate it — Landauer's cost is temperature-dependent, the Bekenstein bound radius-dependent — and no universal conversion factor follows; this is now named as a distinct and prior gap. The same hypothesis, read globally, would entail a conserved energy that general relativity does not supply in a Friedmann–Robertson–Walker spacetime, so conservation is now located where relativity defines it: exactly, as a substrate property, and locally and approximately as an interior bookkeeping quantity. The asymptotic heat-death limit is distinguished from the finite-time configuration on which the cyclic trigger actually fires, which the paper had elsewhere described as never arriving. And the substitution of a static capacity bound for information causality's transmitted-bit count is now named as an assumption carried by that argument, with the Tsirelson recovery stated as conditional on it. *The falsification section.* Two resolution criteria offered for the criticality weak point — a power law in the microwave background and scale-free galaxy structure — can