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 v2 Version 2 changes (2026-05-19): New §5.2 Step 4: Singularity unification argued via Leibniz's Identity of Indiscernibles — burden of proof flipped, boundary-vs-interior distinction formalized New §5.2: ER=EPR (Maldacena & Susskind 2013) and Complexity=Action (Brown et al. 2016) support for singularity unification New §5.7: Black holes, particles, and the topology of spin — Carter g=2 result, Burinskii Kerr-Newman electron, spin as topological circumnavigation number, singularity interiors reframed as unconnectable regions of one computation (not baby universes), Big Rip didactic bridge, Einstein-Cartan torsion support Rewritten §6.3: Operational description of self-referential closure Φ(U)=U — standing wave analogy, circularity objection addressed Rewritten §6.4: Inexpressibility argument strengthened via Gödel, constructor theory parallel (Deutsch & Marletto 2015) Fixed §5.4: Heat death → Bekenstein saturation pathway now physically justified (black hole mergers, Hawking evaporation, cosmological horizon entropy) Fixed §5.6: Universality claim corrected — simulation ≠ physical equivalence; open problem reframed as finding the most direct automaton Tegmark (2008) Mathematical Universe differentiated in §3 Van Raamsdonk (2010) emergent spacetime support in §6.3 Ruggiero (2020) independent Big Rip → CCC convergence in §5.4 Boyle, Finn & Turok (2022a) CPT + neutrino dark matter prediction added Jow et al. (2022) CCC negative CMB result noted in §9.6 Vopson (2025) gravity as information entropy reduction in §9.1 Elze (2020, 2024) CA-QFT extensions in §5.6 Wolfram, Gorard & Peaslee (2020) differentiation sharpened in §11.1 Abstract updated with Leibniz argument and Carter/Burinskii correspondence References fully re-sorted alphabetically, Boyle 2022a/2022b disambiguated 23 new references added (total now ~70)