Minimal Axiomatic Ontology of the Universe: Quantum Cellular Automaton Objects and Multi-Layer Emergent Structures

Minimal Axiomatic Ontology of the Universe: Quantum Cellular Automaton Objects and Multi-Layer Emergent Structures Anonymous Author November 26, 2025 Abstract This paper proposes a minimal ontology of the universe within the quantum cellular automaton (QCA) framework. The universe is defined as a QCA object with initial state, satisfying three axioms: (A1) discrete–unitary–local quantum dynamical system on countable lattice points; (A2) finite signal velocity upper bound cin Lieb–Robinson sense; (A3) existence of Dirac-type effective mode with two-dimensional internal degrees of freedom in some low-energy one-particle sector. Based on these three axioms, we prove: (1) Event sets and causal partial order can be constructed from local algebras and their unitary evolution, embedding into Lorentz-type macroscopic spacetime geometry in coarse-graining limit; (2) In one-dimensional Dirac-type QCA model, dispersion relation of single-step update operator is |cosΩ(p)= cos(m∆t) cos(pa), from which external group velocity vext(p) and internal evolution velocity vint(p) are symmetrically defined from same update operator, and information rate circle identity is rigorously proved v2 ext(p) + v2 int(p) = c2, c =a ∆t, where cis maximal signal rate of QCA; (3) Proper time element is defined accordingly as |dτ=vint cdt, from which special relativity’s time dilation relation and four-velocity normalization gµν uµuν= −c2can be derived, and relativistic energy–momentum relation is reconstructed in lowenergy limit E2=p2c2+m2c4. Mass thus obtains a purely internal information-theoretic definition mc2=ℏωint(0), namely energy scale of internal evolution frequency in rest frame. This paper further formalizes definitions of observer objects and observer networks, pointing out that observers are selfreferential local subsystems within universe object, whose proper time and macroscopic “public reality” are both characterized by emergent patterns of same QCA ontology at different scales. Appendices provide detailed proofs of universe objects, causal partial order, Dirac–QCA dispersion relations and information rate circle theorem, laying reusable foundation for subsequent discussions of gravity, quantum fields and cosmology within this framework. Keywords: Universe ontology; Quantum cellular automaton; Lieb–Robinson bound; Dirac model; Information rate circle; Emergence of relativity; Observer network 1 1 Introduction & Historical Context Continuous spacetime and quantum states are two grand narratives of modern physics. On one hand, general relativity models universe as four-dimensional manifold (M, gµν ) with Lorentz signature, where metric satisfies Einstein field equations and curvature corresponds to gravity. On other hand, quantum theory centers on Hilbert space, unitary evolution and measurement axioms, with physical systems described by state vectors or density operators and observation statistics given by Born rule. The two can operate synergistically in effective theories such as quantum field theory and semiclassical gravity, but ontological tension remains: which is fundamental between spacetime geometry and quantum state? Do observers and measurements require additional external mechanisms? Quantum cellular automata provide rigorous operator framework for “universe as quantum computation” perspective. QCA discretizes spacetime into lattice network, with finitedimensional Hilbert space at each lattice point, and interactions between degrees of freedom given by local unitary update rules. Early work showed that free Dirac fields, Weyl fields and Maxwell fields can emerge from QCA under appropriate symmetry and continuum limit conditions. Meanwhile, deep connection between discrete-time quantum walks and Dirac equation has been systematically clarified, proving that through appropriate choice of coin rotation and translation structure, one can obtain one-dimensional or even higher-dimensional Dirac-type effective dynamics in long-wavelength limit. On other hand, Lieb–Robinson bound provides rigorous characterization of “finite signal velocity” in non-relativistic quantum spin systems: even in lattice models without explicit relativistic structure, influence of local perturbations is confined within certain effective light cone in spacetime, with commutator norm outside decaying exponentially. This means “light speed upper limit”-like causal structure does not belong exclusively to continuous relativistic field theory, but exists broadly in quantum lattice systems with local interactions. Against this background, a natural ontological question is: Does there exist a minimal axiom system, using QCA as sole primitive, from which finite signal velocity, special relativistic geometry, mass–energy relation and observer structure can emerge? This paper attempts to give affirmative constructive answer. We propose three minimal axioms: 1. Universe is discrete–unitary–local QCA object; 2. Universe evolution satisfies Lieb–Robinson type finite light cone bound, with maximal signal rate c; 3. Universe has Dirac-type effective mode with two-dimensional internal degrees of freedom in some low-energy one-particle sector. Under these three axioms, we prove:  Event sets and causal partial order of universe can embed into Lorentz-type spacetime geometry in coarse-graining limit;  Dispersion relation and internal Bloch structure of Dirac-type low-energy mode lead to “information rate circle”: external velocity and internal evolution velocity form circle with radius c;  Defining proper time by internal evolution rate can rebuild special relativity’s time dilation, four-velocity normalization and energy–momentum relation; 2  Observers can be characterized as local subsystems in universe object satisfying selfreference and memory conditions, whose proper time and perceived “public reality” are both uniformly described by same QCA ontology. Compared with existing work deriving Dirac equation and free quantum field theory from information processing principles, this paper further translates “what is universe” ontological question into axiomatic characterization of QCA universe object, emphasizing one-to-one correspondence between information rate geometry and relativistic geometry. 2 Model & Assumptions This section gives formal definition of universe object and states three minimal axioms A1–A3. 2.1 Definition of Universe Object Let Λ be countable unbounded graph, with vertex set V(Λ) representing spatial cells and edge set E(Λ) ⊂V(Λ)×V(Λ) representing direct interaction adjacency relations. Assume Λ is locally finite, i.e., for any x∈V(Λ), its degree deg(x) has uniform upper bound. Take finite-dimensional Hilbert space Hcell ∼ =Cdas unit cell space. For each lattice point x∈V(Λ), assign copy Hx∼ =Hcell. For any finite subset Λ0⊂Λ, define local Hilbert space HΛ0:= O x∈Λ0 Hx. Define quasilocal operator algebra A:= [ Λ0⋐Λ B(HΛ0) ∥·∥ , where Λ0⋐Λ denotes finite subset, B(HΛ0) is bounded operator algebra, closure taken in operator norm. Time evolution given by family of C∗-algebra automorphisms {αt}t∈Z, satisfying: 1. Group property: α0= id, αt+s=αt◦αs; 2. Quasilocality: exists constant R > 0 such that for any A∈ B(HΛ0), support of its evolution α1(A) is contained in radius Rneighborhood of Λ0. In concrete construction, usually exists unitary operator Urealizing single-step evolution, i.e., for all A∈ A have α1(A) = U†AU. Universe initial state is state on A, i.e., positive normalized linear functional ω0:A → C satisfying ω0(I) = 1, ω0(A†A)≥0. Thus we make following definition. Definition 2.1 (Universe object).Denote five-tuple UQCA := Λ,Hcell,A, α, ω0 as universe object (QCA universe), where αis discrete-time evolution realized by quasilocal unitary operator. 3 2.2 Axiom A1: Discrete–Unitary–Local Axiom 2.2 (Discrete–unitary–local).Physical universe is some universe object UQCA satisfying: 1. Λcountable and locally finite; 2. Unit cell Hilbert space Hcell finite-dimensional; 3. Exists quasilocal unitary operator Usuch that for all A∈ A have α1(A) = U†AU. This axiom constrains universe to be “finite information density + local interaction + unitary evolution” discrete quantum dynamical system, without presupposing any continuous spacetime structure or Lorentz symmetry. 2.3 Axiom A2: Finite Light Cone and Lieb–Robinson Velocity Axiom 2.3 (Finite light cone).Exists constant c > 0such that for any local operators A, B and any t∈Z, exist constants C, µ > 0such that [αt(A), B]≤C∥A∥ ∥B∥exph−µdist(suppA, suppB)−c|t|i, where suppAis minimal support region of A,dist is graph distance. This law is abstract form of Lieb–Robinson bound, indicating existence of effective light cone on spacetime graph with “slope” c, with commutator norm outside light cone exponentially suppressed. Physically, ccan be interpreted as maximal signal rate in universe, i.e., discrete version of “light speed upper limit”. 2.4 Axiom A3: Dirac-Type Low-Energy Effective Mode Axiom 2.4 (Dirac effective mode).In some low-energy one-particle sector of UQCA, exists translation-invariant submodel with two-dimensional internal degrees of freedom, whose singlestep update operator U(p)∈SU(2) in one-dimensional momentum representation satisfies: 1. Can be written in Bloch form U(p) = exp−i Ω(p) ˆn(p)·σ, where σ are Pauli matrices, ˆn(p)∈S2,Ω(p)∈[0, π]; 2. In neighborhood of some p0, effective Hamiltonian in continuum limit Heff (k) := Ω(p0+k) ∆t≈c k σz+mc2σx, where ∆tis time step, kis small momentum offset, constant m= 0 called mass parameter of this mode. This axiom requires universe to have at least one mode exhibiting one-dimensional massive Dirac equation in low-energy limit, consistent with existing results deriving Dirac equation from QCA principles. 3 Main Results (Theorems and Alignments) Under above axioms, this paper gives following three types of main results. 4 3.1 Universe Causal Structure and Macroscopic Geometry Theorem 3.1 (Causal partial order and macroscopic Lorentz structure).In universe object UQCA satisfying A1–A2, define event set E:= {(Λ0, t)|Λ0⋐Λ, t ∈Z} and local algebra A(Λ0, t) := αtB(HΛ0)⊂ A. If for all A∈ A(Λ0, t), B ∈ A(Λ1, s)with t<shave [A, B] = 0, then denote (Λ0, t)⪯(Λ1, s). Then (E,⪯)forms directed acyclic partial order structure, and under appropriate coarse-graining can embed into continuous Lorentz-type spacetime manifold with ⪯compatible with causal partial order on that manifold. This result shows causal cones and macroscopic spacetime geometry can completely emerge internally from local algebras and their unitary evolution, without taking continuous spacetime as axiom. Proof relies on exponential suppression estimates of Lieb–Robinson bound on perturbation propagation range. 3.2 Dirac–QCA Dispersion and Information Rate Circle In one-dimensional Dirac-type QCA model satisfying A3, we select concrete quantum walk realization and obtain explicit dispersion relation and velocity definitions. Theorem 3.2 (Dirac–QCA dispersion relation).Consider one-dimensional lattice Λ = Z, local Hilbert space as two-component spin Hcell ∼ =C2. Define conditional translation operator T:= S+⊗ |↑⟩ ⟨↑| +S−⊗ |↓⟩ ⟨↓| , where S+|x⟩=|x+ 1⟩,S−|x⟩=|x−1⟩, and local “mass rotation” C(m) := exp−im∆t σx. Single-step update operator defined as U:= C(m)T. In momentum representation, characteristic angle Ω(p)of U(p)∈SU(2) satisfies dispersion relation cosΩ(p)= cos(m∆t) cos(pa), where ais lattice spacing, p∈[−π/a, π/a]. This dispersion structure consistent with results in Dirac QCA literature, reflecting symmetric role of coin angle m∆tand lattice momentum pa. Theorem 3.3 (Information rate circle).Let c:= a/∆tbe maximal signal rate of QCA. For each momentum mode in above Dirac–QCA model, define external group velocity vext(p) := adω(p) dp, ω(p) := Ω(p) ∆t, and internal evolution velocity vint(p) := csin(m∆t) cos(pa) sinΩ(p). Then for all allowed momentum phave identity v2 ext(p) + v2 int(p) = c2. 5 This theorem shows: in Dirac–QCA mode, total information update rate cis orthogonally decomposed into “external displacement” and “internal evolution” parts, both forming circle with radius cin velocity-squared sense, hence called “information rate circle”. 3.3 Emergence of Relativistic Structure and Mass–Energy Relation Corollary 3.4 (Time dilation and proper time).Let v:= vext(p)be particle velocity observed in some inertial frame, from information rate circle obtain vint(p) = pc2−v2. Define proper time element by internal evolution velocity dτ:= vint cdt=r1−v2 c2dt, obtaining time dilation relation consistent with special relativity. Corollary 3.5 (Four-velocity normalization and Minkowski line element).Let spacetime coordinates x0=ct, x1=x, four-velocity defined as uµ:= dxµ dτ. Then have u0=γc, u1=γv, γ =1 p1−v2/c2, satisfying normalization under Minkowski metric gµν = diag(−1,1) gµνuµuν=−c2. Line element can be written as ds2:= gµν dxµdxν=−c2dτ2. Corollary 3.6 (Internal definition of mass and energy–momentum relation).In rest frame p= 0, dispersion relation gives cosω(0)∆t= cos(m∆t). In m∆t≪1limit have ω(0) ≈m. Define rest energy E0:= ℏω(0) := mc2, then mass mis defined as energy scaling factor needed to maintain internal evolution, i.e., measure of “internal information oscillation density”. In long-wavelength limit pa ≪1, dispersion relation leads to ω2(p)≈m2+p2c2 ℏ2, total energy E(p) = ℏω(p)satisfies E2(p)≈p2c2+m2c4. In summary, special relativity’s time dilation, four-velocity normalization and relativistic energy–momentum relation can all be viewed as rewriting of information rate circle theorem and Dirac–QCA dispersion structure in macroscopic limit. 6 4 Proofs This section gives proof ideas of above main theorems, with detailed technical calculations in appendices. 4.1 Proof Idea of Theorem 1 By axiom A1, support of any local operator expands at most to finite radius neighborhood in single-step evolution; by Lieb–Robinson bound of A2, for operator pairs with support distance satisfying dist(suppA, suppB)> c|t|, norm of [αt(A), B] is exponentially suppressed. Accordingly define event set Eand local algebra A(Λ0, t), using commutativity to define partial order ⪯. Reflexivity self-evident from commutativity; transitivity can be proved using Jacobi identity and linear structure of commutators; directed acyclicity comes from time parameter monotonicity and property that perturbations cannot “trace back causally”, else would conflict with Lieb–Robinson bound and response theory of stable states. In coarse-graining limit, by selecting appropriate large-scale blocks, map Λ to approximately continuous spatial coordinates and integer time tto continuous time tcont; discrete light cone with slope cconverges to continuous Minkowski light cone, thus constructing compatible Lorentz-type spacetime manifold embedding. Related construction and technical details in Appendix A. 4.2 Proof Idea of Theorem 2 On one-dimensional lattice, introduce momentum eigenstates |p⟩:= 1 p2π/a X x∈Z eipax |x⟩, p ∈[−π/a, π/a]. Translation operators act as S+|p⟩= e−ipa |p⟩, S−|p⟩= eipa |p⟩. Thus conditional translation in momentum–spin subspace Hp∼ =C2is represented as T(p) = exp−ipa σz, single-step update operator is U(p) = C(m)T(p) = exp−im∆t σxexp−ipa σz. Using SU(2) group element multiplication formula exp−iαˆa·σexp−iβˆ b·σ= exp−iγˆc·σ, where cos γ= cos αcos β−(ˆa·ˆ b) sin αsin β, taking ˆa= (1,0,0),ˆ b= (0,0,1), α =m∆t, β =pa, obtain cosΩ(p)= cos(m∆t) cos(pa), proving Theorem 2. Explicit expressions of Bloch vector components can also be obtained from same formula, see Appendix B. 7 4.3 Proof Idea of Theorem 3 and Relativistic Corollaries Differentiating dispersion relation with respect to pobtains group velocity vext(p) = adω dp=ccos(m∆t) sin(pa) sinΩ(p). On other hand, SU(2) Bloch vector components give ˆn(p) sinΩ(p)=sin(m∆t) cos(pa),sin(m∆t) sin(pa),sin(pa) cos(m∆t). Accordingly define internal evolution velocity symmetrically vint(p) := csin(m∆t) cos(pa) sinΩ(p). Squaring and adding both using trigonometric identities gives v2 ext(p) + v2 int(p) = c2cos2(m∆t) sin2(pa) + sin2(m∆t) cos2(pa) sin2Ω(p). On other hand, sin2Ω(p)= 1 −cos2Ω(p)= 1 −cos2(m∆t) cos2(pa), direct algebraic calculation (see Appendix C) shows cos2(m∆t) sin2(pa) + sin2(m∆t) cos2(pa) = sin2Ω(p), therefore v2 ext(p) + v2 int(p) = c2, proving Theorem 3. Subsequently defining proper time by vint, constructing four-velocity uµ, obtaining gµν uµuν= −c2under Minkowski metric, and reconstructing E2=p2c2+m2c4through long-wavelength limit expansion of dispersion relation. Detailed calculations in Appendix C. 5 Model Apply This section explains meaning of above structures in “what is universe” question from ontological perspective. 5.1 Universe as QCA Object In this framework, universe is not “matter + spacetime” superposition, but QCA object UQCA with initial state. So-called “matter”, “field”, “particle”, “spacetime” concepts are all stable excitations or geometric encodings in different scales and different subsectors: 1. Geometric layer: Causal partial order and Lieb–Robinson light cone determined by A1– A2 manifest as Minkowski or more general Lorentz-type geometry after coarse-graining; 2. Matter quantity layer: Dirac modes satisfying A3 correspond to massive particles, with rest energy directly related to internal evolution frequency; 3. Field theory layer: In multi-particle and multi-mode extensions, free quantum field theory can be viewed as continuum limit description of QCA; introduction of interactions corresponds to more complex local structures in QCA update operators. In this sense, answer to “what is universe” is: Universe is one concrete instance among family of QCA universe objects satisfying minimal axioms A1–A3, whose entire unitary history carries spacetime, matter and interactions we observe. 8 5.2 Unified Explanation of Mass and Inertia Information rate circle gives unified explanation of inertia and time dilation: total information update rate cis fixed, and for massive excitations, there exists competition between internal evolution and external propagation. At high velocity, external group velocity approaches c, internal evolution forced to slow down, corresponding to slower proper time; at rest, all information rate used for internal evolution, corresponding to maximal proper time passage. Mass mis defined by rest frame internal evolution frequency ωint(0), manifesting as information oscillation density needed to maintain local mode stability. This perspective unifies relativistic “time dilation”, “inertia”, “mass–energy relation” into one geometric identity internal to QCA, no longer needing to treat them as independent empirical laws. 5.3 Observer Objects and Public Reality Observers in this ontology are not external entities, but class of local subsystems in universe object satisfying self-reference and memory properties. Definition 5.1 (Observer object).An observer object Ois triple O=Aloc, ωmem,M, where: 1. Aloc ⊂ A is some local subalgebra, corresponding to degrees of freedom accessible to observer; 2. ωmem(t)is family of states on Aloc with respect to discrete time t, corresponding to observer’s internal memory; 3. M={Mθ}is family of “world models”, each Mθmapping historical observation records to probability predictions of future observables, with parameter θ(t)varying over time according to some update rule θ(t+ 1) = Fθ(t),{ωmem(s)}s≤t. Observer object corresponds to worldline γO={(Λt, t)}t∈Z⊂ E in event set, whose proper time is defined by internal information rate, having same form as proper time of Dirac modes. Intersection of local algebras of multiple observer objects and consistency category of their states constitute so-called “public reality”, i.e., consensus different observers achieve on universe state in overlapping accessible regions. 6 Engineering Proposals This section proposes several engineering schemes that can test and utilize this ontological structure on experimental platforms or numerical simulations. 6.1 Quantum Simulation of Dirac–QCA and Experimental Verification of Information Rate Circle One-dimensional and higher-dimensional discrete-time quantum walks have been realized on ion trap, superconducting quantum circuit and optical platforms, and effective Hamiltonians can be precisely controlled by adjusting coin rotation and translation operations. These platforms can be used for following purposes: 9 References [1] G. M. D’Ariano, P. Perinotti, “Quantum cellular automata and free quantum field theory”, Phys. Rev. A 90, 062106 (2014). [2] A. Bisio, G. M. D’Ariano, P. Perinotti, “Quantum cellular automaton theory of light”, Ann. Phys. 368, 177–190 (2016). [3] T. A. Brun, et al., “Quantum cellular automata and quantum field theory in two spatial dimensions”, Phys. Rev. A 102, 062222 (2020). [4] L. S. Trezzini, “Renormalisation of quantum cellular automata”, Quantum 9, 1756 (2025). [5] F. W. Strauch, “Relativistic quantum walks”, Phys. Rev. A 73, 054302 (2006). [6] C. M. Chandrashekar, “Two-component Dirac-like Hamiltonian for generating quantum walk on one-, twoand three-dimensional lattices”, Sci. Rep. 3, 2829 (2013). [7] A. P´erez, “Asymptotic properties of the Dirac quantum cellular automaton”, Phys. Rev. A93, 012328 (2016). [8] A. Mallick, C. M. Chandrashekar, “Dirac cellular automaton from split-step quantum walk”, Sci. Rep. 6, 25779 (2016). [9] P. C. Costa, et al., “Quantum walks via quantum cellular automata”, Quantum Inf. Process. 17, 198 (2018). [10] C. H. Alderete, et al., “Quantum walks and Dirac cellular automata on a trapped-ion quantum computer”, npj Quantum Inf. 6, 89 (2020). [11] N. P. Kumar, C. M. Chandrashekar, “Bounds on the dynamics of periodic quantum walks and emergence of the gapless and gapped Dirac equation”, Phys. Rev. A 97, 012116 (2018). [12] B. Nachtergaele, R. Sims, “Much ado about something: why Lieb–Robinson bounds are useful”, J. Stat. Phys. 124, 1–13 (2006). [13] E. H. Lieb, D. W. Robinson, “The finite group velocity of quantum spin systems”, Commun. Math. Phys. 28, 251–257 (1972). [14] M. Cheneau, “Experimental tests of Lieb–Robinson bounds”, in Quantum Systems in and out of Equilibrium, EMS Press (2022). [15] C. Gupta, A. Perez, “The Dirac vacuum in discrete spacetime”, Quantum 9, 1845 (2025). [16] T. A. Brun, et al., “Quantum electrodynamics from quantum cellular automata”, Entropy 27, 492 (2025). 16