The Red Queen Universe: The Cosmological Constant from Agent Games and the Avoidance of Heat Death
Abstract
The classical picture of the heat death of the universe describes the ultimate fate of the cosmos as a state where, over long time scales, all available free energy is exhausted, the system tends toward thermal equilibrium and maximum entropy, and all macroscopic structures and computational activities cease. However, the observational fact that the universe has continuously generated and maintained multi-level low-entropy structures---especially life and intelligent civilizations---over approxi
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The Red Queen Universe: The Cosmological Constant from Agent Games and the Avoidance of Heat Death Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract The classical picture of the heat death of the universe describes the ultimate fate of the cosmos as a state where, over long time scales, all available free energy is exhausted, the system tends toward thermal equilibrium and maximum entropy, and all macroscopic structures and computational activities cease. However, the observational fact that the universe has continuously generated and maintained multi-level low-entropy structuresespecially life and intelligent civilizationsover approximately 13.8 billion years of evolution remains opaque under the standard explanation relying solely on "low-entropy initial conditions." On the other hand, the cosmological constant and dark energy problems suggest that the value of the vacuum energy density and its coincidence with matter density still lack a microscopic information-theoretic explanation. In this paper, within the framework of a Quantum Cellular Automaton (QCA) universe and the conservation of information rate, we formalize the cosmic system as a multi-agent game eld composed of "agents." Each agent is characterized as maintaining a low-entropy structure far from equilibrium in a local Hilbert subspace and minimizing variational free energy through internal computation, thereby continuously performing information erasure and prediction error correction. Based on Landauer's principle and established results in the thermodynamics of computation, we prove that under appropriate coarse-graining and isotropy assumptions, the Landauer waste heat generated by the irreversible computations of all agents in the universe is equivalent to a homogeneous, isotropic energy component with an approximate equation of state w≃ −1 . Its gravitational eect in Friedmann dynamics is equivalent to a dynamical cosmological "constant" Λeff (t) . Furthermore, we introduce a multi-agent dynamic model based on the Red Queen eect, expressing the tness competition and complexity arms race among agents as a system coupling replicator equations with complexity variables and LotkaVolterra dynamics. Linearization and spectral analysis of such systems yield a general theorem: within a broad parameter range satisfying "Red Queen conditions," internal equilibrium points are not asymptotically stable. Instead, the system tends toward limit cycles or chaotic attractors in phase space, causing complexity and Landauer information erasure rates to remain positive on cosmic time scales. Thus, the classical "static limit equilibrium state" is dynamically excluded. Building on this, the paper presents three main conclusions. First, under the constraints of a QCA universe and information rate conservation v2 ext +v2 int =c2 , the information erasure ow corresponding to internal agent computation constitutes a natural class of "information vacuum energy density" ρinfo(t) , whose integral determines the eective cosmological constant Λeff (t) , providing an information-theoretic origin for dark energy. Second, the Red Queen agent game establishes a feedback loop between "complexity and dark energy": richer computational activities generate larger ρinfo , which in turn alters large-scale spacetime dynamics and conversely aects the survival environment of agents, forming a "Red Queen Universe" evolutionary mode. Third, as long as a non-zero density agent network exists in the universe and satises Red Queen conditions, in the continuous limit of General Relativity, the state required for heat death"global absence of available free energy, complete entropization of structure and computation"is no longer a dynamical attractor. The universe can evolve in a perpetually far-from-equilibrium "algorithmic turbulence" state. In the application section, we discuss the alleviation of the cosmological "coincidence problem" by complexity-driven Λeff (t) , relationships with entropic cosmology and the Causal 1
Entropic Principle, and potential testable predictions in the context of dark energy evolution observations (such as recent DESI indications of time-varying dark energy). Finally, engineering proposals based on QCA quantum simulation and multi-agent simulation are presented to provide pathways for testing the "Red Queen Universe" framework on controllable platforms. Keywords: Quantum Cellular Automata; Cosmological Constant; Dark Energy; Heat Death; Landauer's Principle; Variational Free Energy; Red Queen Eect; Multi-Agent Games; Thermodynamics of Computation; Information Cosmology 1 Introduction & Historical Context 1.1 Cosmic Heat Death and the Dark Energy Problem Since the nineteenth century, the "Heat Death of the Universe" picture, based on the Second Law of Thermodynamics, has held that if the universe is an eectively closed system, its total entropy increases monotonically with time, eventually tending toward a maximum entropy equilibrium state with no available free energy, known as the "Big Freeze" or "Heat Death." In the framework of modern cosmology, if the universe is at or open on large scales and contains a positive cosmological constant, the standard inference is that the universe will asymptotically approach an approximate de Sitter equilibrium state after innite time, where energy dierences in all local structures and processes will be smoothed out. Observations based on Type Ia supernovae, the Cosmic Microwave Background, and largescale structure at the end of the twentieth century indicated that the universe is currently in a phase of accelerated expansion. This phenomenon is usually described by a dark energy component with an approximate equation of state w≃ −1 , typically realized in Einstein's equations as a cosmological constant Λ . Dark energy density accounts for about 70% of the current cosmic energy budget, while matter accounts for about 30%, forming the so-called Λ CDM standard model. However, the cosmological constant introduces two classic puzzles. The rst is the "Cosmological Constant Problem": zero-point energy estimates from quantum eld theory are tens of orders of magnitude larger than the observed value, and there is no consensus on how to screen or reconcile this huge vacuum energy contribution. The second is the "Coincidence Problem": why the dark energy density and matter density happen to be of the same order of magnitude in the current cosmic epoch, whereas they were not for most of cosmic history. Recent observations of large samples of galaxies and quasars by projects like DESI have even given indications that dark energy may evolve with time, further stimulating theoretical exploration into dynamical dark energy and the origins of a non-trivial cosmological constant. 1.2 Information, Computation, and Thermodynamics: Landauer's Principle and Computational Engines The deep connection between information theory and thermodynamics was rst systematically elucidated by Landauer. Landauer's Principle states that the minimum energy cost required to erase one bit of classical information in an environment at temperature T is Emin =kBTln 2 . Any logically irreversible operation (such as erasure or merging computational paths) is necessarily accompanied by entropy production and heat dissipation of at least this magnitude. Bennett subsequently developed the thermodynamics of computation, viewing computers as thermal engines that convert free energy into waste heat and "mathematical work," and proved that reversible computation can operate arbitrarily close to the Landauer limit. Subsequent work extended Landauer's Principle to quantum and non-equilibrium systems, conrming its universality as a physical lower bound for information processing. 2
These results suggest that any system performing irreversible computationincluding arti- cial computers, biological organisms, and even broader "natural computation" processesmust necessarily emit a minimal amount of heat to the environment and produce a corresponding entropy increase. This concept provides a foundation for understanding macroscopic cosmic thermodynamics from an information-theoretic perspective. 1.3 Variational Free Energy and Agents: Inspiration from the Free Energy Principle In cognitive science and neuroscience, the Free Energy Principle proposed by Friston states that all biological systems maintaining their boundaries and homeostasis can be viewed as inference machines that minimize variational free energy in some sense. The core is to introduce a joint probability model P(s, ϑ) over sensory inputs s and hidden variables ϑ , as well as an internal approximate posterior q(ϑ) . The system minimizes a variational functional by changing its internal state: F(q, s) = KLq(ϑ)|P(ϑ|s)−ln P(s), (1) which can equivalently be written in forms containing model evidence and entropy terms. In this framework, organisms reduce prediction error and "surprise" by updating internal models and taking actions, thereby maintaining ordered structures far from equilibrium. Although initially applied to neural systems, the form of the Free Energy Principle does not depend on specic material substrates, so it can be abstracted to the more general concept of an "agent." 1.4 The Red Queen Eect and Evolutionary Arms Races The Red Queen hypothesis was proposed by Van Valen to explain the "Van Valen Law" in the paleontological record, where species extinction rates are approximately independent of species age. The hypothesis emphasizes that the "eective environment" of a species is mainly composed of other coexisting species, so tness improvement is relative: when one species gains an advantage, it deteriorates the ecological niche of other species, forcing the latter to evolve to maintain their chances of survival. The Red Queen eect can be generalized to sexual selection and host-parasite interactions at the individual level, and can also be viewed as a zero-sum game: all species need to "run as fast as they can" just to stay in place. Such game dynamics are ubiquitous in multi-agent systems, and research in Agent-based models shows that local interactions can often produce complex emergent phenomena on macroscopic scales. This paper elevates this idea to the cosmological scale: viewing the universe as a game network composed of multi-level agents (from molecular machines to life and technological civilizations), where the Red Queen eect maintains a non-equilibrium state over cosmic time scales through a "complexity arms race." 1.5 QCA Universe and Conservation of Information Rate Quantum Cellular Automata provide a natural framework for strict quantum dynamics on discrete spacetime. A QCA can be viewed as an array of nite-dimensional quantum systems dened on a lattice, whose evolution is given by translation-invariant, causal unitary operators iterated over discrete time steps. A large body of work has shown that eld equations such as Dirac, Weyl, and Maxwell can emerge from QCA models in appropriate continuous limits, providing a rigorous path for "spacetime and eld theory originating from quantum computation." In previous work, the external group velocity vext of local excitations and the internal phase rotation or "intrinsic evolution velocity" vint were combined into an information rate vector u= (vext, vint) , proposing the information rate conservation relation: v2 ext +v2 int =c2, (2) 3
where c is the maximum propagation speed of the QCA, corresponding to the speed of light in the continuous limit. This relation unies the normalization of four-velocity in special relativity, proper time, and the mass-frequency relation mc2=ℏωint into a geometric constraint of information rate budget, providing the basis for constructing the link between "internal computation" and "external geometry" in this paper. 1.6 Objectives and Main Contributions Synthesizing the above background, this paper addresses three interconnected questions: 1. If the universe is a QCA universe containing a vast number of agents maintaining their structures through computation, how does the thermodynamic necessity of these computations feed back into large-scale spacetime geometry and the cosmological constant? 2. Can Red Queen-style multi-agent games dynamically prevent the universe from entering a nal state of complete equilibrium heat death? 3. Is this "Red Queen Universe" mechanism intrinsically linked to the value of dark energy and its possible time evolution, and is it compatible with existing observations? Around these questions, the main work of this paper is organized as follows: In "Model & Assumptions," we formalize the QCA universe and agents, introduce the information vacuum energy density ρinfo(t) based on Landauer's principle, and give its relation to the cosmological constant Λeff(t) . In "Main Results," we present two core theorems: rst, the LandauerΛ relation theorem, proving that under isotropy and Red Queen continuous computation assumptions, the irreversible computation of agents is equivalent to a dynamical cosmological constant; second, the Red Queen non-equilibrium theorem, which, under a multi-agent replicator-complexity dynamic model, excludes internal stable equilibrium points and guarantees that complexity and information erasure rates remain positive over the long term. In "Proofs" and the appendices, we provide rigorous derivations of the above theorems, including the continuity equation with source terms derived from Einstein eld equations and energy-momentum conservation, and linear stability analysis of the LotkaVolterra replicator system. In "Model Apply," we construct a parameterization of complexity-driven Λeff(t) , compare it with entropic cosmology and the Causal Entropic Principle, and discuss potential connections with observations of time-varying dark energy. In "Engineering Proposals," we propose specic schemes for testing this framework on quantum simulation platforms and in multi-agent simulations. 2 Model & Assumptions This section presents the basic structure and assumptions of the "Red Queen Universe" model. 2.1 QCA Universe and Macroscopic Geometry Assume that at the fundamental level, the universe is described by a tensor product space H=Nx∈ΛHx dened by a countable lattice set Λ and local Hilbert spaces Hx≃Cd . Global evolution is given by a translation-invariant, local, and causal unitary operator U:H → H over discrete time steps n∈Z , which is the standard denition of a QCA. 4
In the long-wavelength and low-energy limit, where lattice spacing and time step tend to zero, QCA evolution approximates continuous relativistic eld equations, macroscopically describable by a metric with FriedmannLemaîtreRobertsonWalker (FLRW) symmetry: ds2=−c2dt2+a2(t)γijdxidxj, (3) where a(t) is the scale factor, and γij is the constant curvature metric of three-dimensional space. The cosmological constant or dark energy component manifests in this continuous limit as a term in the energy-momentum tensor T(Λ) µν =−ρΛgµν , corresponding to an equation of state pΛ=−ρΛ . 2.2 Information Rate Conservation and Denition of Agents For every excitation or structure localized in the QCA universe, consider its eective worldline γ and the external group velocity vext and internal evolution velocity vint along that worldline, satisfying the information rate conservation relation: v2 ext +v2 int =c2. (4) Here, vext reects the propagation speed of the excitation on the lattice, while vint characterizes the phase evolution and internal computation rate of its internal state in the local Hilbert subspace. Denition: An Agent A is a nite connected lattice subset ΛA⊂Λ on the QCA and a family of density operators ρA(t)t on HA=Nx∈ΛAHx , satisfying: 1. ρA(t) maintains a signicant deviation from the environmental equilibrium state ρeq(t) over long time scales, i.e., there exists a macroscopic observable O such that |tr[(ρA− ρeq)O]| has positive measure on a time set larger than some xed threshold. 2. This deviation is maintained by internal computation, i.e., the evolution of ρA(t) can be decomposed into approximately reversible internal unitary operators and irreversible "information erasure" maps, where the latter thermodynamically satises the Landauer bound. The Information Mass MI of an agent is dened such that its internal evolution frequency ωint and eective energy EI satisfy the relation: EI=MIc2=ℏωint, (5) linking the internal computation rate to the relativistic mass scale. 2.3 Variational Free Energy and Agent Objective Function Referencing the Free Energy Principle, treat each agent as an inference machine possessing a generative model P(s, ϑ) of environmental signals s and internal states ϑ . Dene its variational free energy: FA(t) = KLqt(ϑ)|P(ϑ|st)−ln P(st), (6) where qt(ϑ) is the approximate posterior distribution of hidden variables held by the agent at time t , and st is its sensory input at that time. The "survival strategy" of an agent can be abstracted as minimizing the path integral or long-time average of FA(t) under the information rate budget constraint: v2 ext(t) + v2 int(t) = c2. (7) This optimization process necessarily involves compressing the internal state space and ltering out unnecessary high-dimensional components, thus physically corresponding to frequent irreversible writing and erasure of internal storage and representations. 5
2.4 Landauer Information Waste Heat and Information Vacuum Energy Density For a single agent A , assume its information erasure rate in an ambient temperature eld Tbg(x, t) is ˙ I(A) erase(t) (in bits/s). Landauer's Principle requires its minimum dissipation power to satisfy: P(A) L(t)≥kBln 2 T(A) bg (t)˙ I(A) erase(t), (8) where T(A) bg is the eective environmental temperature of the agent. On cosmological scales, coarse-graining over a comoving volume V and summing the information erasure contributions of all agents Ai located within that volume yields the average Landauer power density per unit comoving volume: Pinfo(t) = 1 VX i kBln 2 T(i) bg (t)˙ I(i) erase(t). (9) One of the core assumptions of this paper is: Assumption 1 (Information Waste HeatVacuum Energy Integrability): In the QCA universe, the Landauer waste heat produced by the irreversible computations of agents is microscopically encoded into a homogeneous, isotropic "information vacuum" excitation in the underlying QCA degrees of freedom. Its average energy-momentum tensor in the continuous limit has the approximate form: T(info) µν (t)≃ −ρinfo(t)gµν, (10) where the information vacuum energy density ρinfo(t) satises the sourced continuity equation: ˙ρinfo + 3H(1 + winfo)ρinfo =Pinfo(t), (11) and satises winfo(t)≃ −1 in the Red Queen era. This assumption essentially reinterprets Landauer waste heat from "conventional thermal radiation" to an energy reserve stored in the underlying QCA degrees of freedom that macroscopically exhibits negative pressure, making its gravitational eect equivalent to dark energy. Under this assumption, an eective cosmological "constant" can be dened: Λeff(t) = Λbare + 8πG ρinfo(t), (12) where Λbare is a possible underlying constant part (e.g., vacuum energy remaining after renormalization), and ρinfo(t) is the dynamical part emerging from agent computation. 2.5 Red Queen Conditions and Multi-Agent Games To characterize competition and arms races between agents, we introduce the following abstractions: Assume there are several species of agents in the universe, denoted by i= 1, . . . , n , with comoving number density Ni(t) and average complexity Ci(t) (understood as algorithmic complexity, structural information, or average dimension of internal state space). Dene the tness of species i as: Wi(t) = fiN(t),C(t), (13) where N= (N1, . . . , Nn) and C= (C1, . . . , Cn) . 6
Complexity Ci enhances the ability of that class of agents to capture and utilize free energy on the one hand, but increases Landauer costs on the other. The Red Queen Conditions can be formalized as: 1. For any i=j , with other variables xed, ∂fj/∂Ci<0 : an increase in the complexity of one class of agents deteriorates the tness of other classes. 2. For each i , within the resourceabundant interval, ∂fi/∂Ci>0 : before costs outweigh benets, increasing one's own complexity increases tness. Multi-agent systems satisfying the above conditions typically exhibit evolutionary arms races, whose dynamics can be described by replicator equations or LotkaVolterra type equations. 3 Main Results (Theorems and alignments) This section presents the two core theorems of this paper and several corollaries. 3.1 Theorem 1 (Landauer Λ Relation Theorem) In the continuous limit of a QCA universe, assume: 1. The large-scale geometry is an isotropic, homogeneous FLRW spacetime satisfying standard Friedmann equations. 2. There exists a game network composed of agents in the universe, with an average Landauer power density Pinfo(t) over comoving volume elements. 3. Information waste heat satises Assumption 1, i.e., its macroscopic gravitational eect can be described by a uid with equation of state pinfo =winfoρinfo , and winfo ≃ −1 in the Red Queen era. Then, the information vacuum energy density ρinfo(t) and Landauer power density Pinfo(t) satisfy the integral relation: ρinfo(t) = ρinfo(t0) + Zt t0 Pinfo(τ) dτ+O(ϵ), (14) where ϵ characterizes the deviation of winfo + 1 . Correspondingly, the eective cosmological constant is: Λeff(t)=Λeff(t0)+8πG Zt t0 Pinfo(τ) dτ+O(ϵ). (15) In other words, under the good approximation winfo ≈ −1 , the dynamical part of the cosmological constant is equivalent to the accumulation of Landauer energy ux generated by the irreversible computations of all agents in the universe over cosmic time. 3.2 Corollary 1 (ComplexityDark Energy Coupling) If we further assume: 1. There exists a macroscopic complexity function: C(t) = X i Ni(t)Ci(t), (16) characterizing the total complexity of agents per unit comoving volume. 2. The average information erasure rate per unit complexity and the environmental temperature are approximately constant, i.e., there exists a constant α > 0 such that: Pinfo(t)≃α Tbg(t)˙ C(t), (17) where Tbg(t) is the macroscopic average of the cosmic background temperature eld. Then we have: Λeff(t)≃Λeff(t0)+8πGα ZC(t) C(t0) Tbg(C) dC. (18) 7
Under the approximation that Tbg varies slowly with time, the above equation gives a monotonic coupling relation between Λeff(t) and complexity C(t) . If C(t) grows rapidly during a certain period of cosmic history (such as the epoch of galaxy formation and the emergence of life), Λeff will also complete a major jump during this period, thereby providing an information-theoretic explanation for the "coincidence" between dark energy density and the history of structure formation. 3.3 Theorem 2 (Red Queen Non-Equilibrium Theorem, Simplied Two-Species Case) Consider two classes of agent populations satisfying Red Queen conditions, whose dynamics are given by the following ordinary dierential equations: dNi dt=NiriCi−di−γ(N1+N2), i = 1,2, (19) dCi dt=αiNi−βiCi, (20) where ri, di, γ, αi, βi>0 . The above equations can be viewed as a coupling of LotkaVolterra type population dynamics and complexity evolution equations: (riCi) represents tness gains from complexity, (γ(N1+N2)) represents resource competition, and (βiCi) represents Landauer costs of maintaining complexity. Assume there exists an internal equilibrium point (N∗ 1, N∗ 2, C∗ 1, C∗ 2) satisfying N∗ i>0 , C∗ i>0 . If the parameters satisfy: r1α1N∗ 1+r2α2N∗ 2> β2 1+β2 2+ 2γr1C∗ 1+r2C∗ 2, (21) then this equilibrium point is not asymptotically stable; instead, the Jacobian matrix of the linearized system at this point has at least one pair of conjugate complex eigenvalues with positive real parts. For an open dense set of parameters, the system undergoes a Hopf bifurcation near this equilibrium point, evolving to a class of limit cycles or more complex attractors. In this case, the total complexity Ctot(t) = N1(t)C1(t) + N2(t)C2(t) (22) and the Landauer power density Pinfo(t)∝Ctot(t) will not converge to a constant over long times, but will oscillate or exhibit quasi-periodic/chaotic behavior within a bounded interval, with a non-zero time average. Therefore, the system will not enter a static equilibrium state in nite time, but is maintained in a perpetual non-equilibrium dynamic driven by the "Red Queen." 3.4 Corollary 2 (Necessary Condition for Avoiding Heat Death) In a QCAFLRW universe, if the following are satised: 1. There exists at least one class of agent populations satisfying Red Queen conditions, whose dynamics satisfy the conditions of Theorem 2, such that the average derivative of Ctot(t) over any nite time interval is non-zero; 2. Information waste heatvacuum energy integrability (Assumption 1) holds, so that Pinfo(t) continuously injects energy into ρinfo ; 3. The overall free energy supply of the universe is not exhausted in nite time, i.e., the "energy budget" of the underlying QCA allows the above process to continue for arbitrarily long times; Then the state required by the classical heat death picture"the entire universe reaching a complete equilibrium state with no available free energy, no structure, and no computation after a nite time"is not a dynamical attractor of the universe. Instead, the universe can evolve in a long-standing "algorithmic turbulence" phase, characterized by accelerated expansion dominated by dark energy on macroscopic scales, complex structures continuously emerging and dissipating on mesoscopic scales, and irreversible computational processes constantly occurring on microscopic scales. 8
4 Proofs This section outlines the proof ideas for the above theorems, leaving more technical derivations to the appendices. 4.1 Proof of Theorem 1 In FLRW spacetime, assume the total cosmic energy-momentum tensor is: Tµν =T(m) µν +T(r) µν +T(info) µν , (23) where (m) and (r) denote matter and radiation components, respectively, and (info) denotes the information vacuum component. For a component X with perfect uid form: T(X) µν = (ρX+pX)uµuν+pXgµν, (24) where uµ is the comoving four-velocity. Under cosmological symmetry, energy conservation for dierent components can be written as sourced continuity equations: ˙ρX+ 3H(ρX+pX) = QX(t), (25) where QX is the energy source term for that component from other components, satisfying PXQX= 0 . For the information vacuum component, assume its source term is precisely the Landauer power density: Qinfo(t) = Pinfo(t), (26) while the source terms for matter and radiation components are energy losses of opposite sign. Substituting the equation of state pinfo =winfoρinfo yields: ˙ρinfo + 3H(1 + winfo)ρinfo =Pinfo(t). (27) In the Red Queen era, we assume winfo(t) = −1 + δ(t) , where δ(t) is a small quantity satisfying |δ(t)| ≪ 1 . The continuity equation becomes: ˙ρinfo + 3Hδ(t)ρinfo =Pinfo(t), (28) with formal solution: ρinfo(t) = ρinfo(t0) exp −3Zt t0 H(τ)δ(τ) dτ+Zt t0 Pinfo(s) exp −3Zt s H(τ)δ(τ) dτds. (29) Under the condition that |δ| ≪ 1 and H is nite over the period considered, the deviation of the exponential factor from 1 is O(ϵ) , where ϵ= sup[t0,t]|3Hδ|∆t , and ∆t is the characteristic time scale. Thus, it can be written as: ρinfo(t) = ρinfo(t0) + Zt t0 Pinfo(τ) dτ+O(ϵ), (30) which is the assertion of Theorem 1. The eective cosmological constant is given by: Λeff(t) = Λbare + 8πGρinfo(t), (31) naturally yielding the integral form. 9
[9] "Heat Death of the Universe," in Wikipedia, accessed 2025. [10] Y. L. Bolotin, A. L. Tur, V. A. Cherkaskiy, "Cosmology Based on Entropy," arXiv:2310.10144 (2023). [11] S. Nojiri, S. D. Odintsov, V. Faraoni, "Barrow Entropic Dark Energy: A Review," Physics of the Dark Universe 36, 101050 (2022). [12] R. Bousso, R. Harnik, G. D. Kribs, G. Perez, "Predicting the Cosmological Constant from the Causal Entropic Principle," Physical Review D 76, 043513 (2007). [13] DESI Collaboration, reports on time-evolving dark energy presented at APS Global Physics Summit (2025). [14] P. Arrighi, "An Overview of Quantum Cellular Automata," Natural Computing 18, 885899 (2019). [15] T. Farrelly, "A Review of Quantum Cellular Automata," Quantum 4, 368 (2020). [16] I. Bialynicki-Birula, "Weyl, Dirac, and Maxwell Equations on a Lattice as Unitary Cellular Automata," Physical Review D 49, 69206927 (1994). [17] A. Bisio, G. M. D'Ariano, A. Tosini, "Quantum Field as a Quantum Cellular Automaton: The Dirac Free Evolution in One Dimension," Annals of Physics 354, 244264 (2015). [18] D. Gross, V. Nesme, H. Vogts, R. F. Werner, "Index Theory of One Dimensional Quantum Walks and Cellular Automata," Communications in Mathematical Physics 310, 419454 (2012). [19] A. Suprano et al., "Photonic Cellular Automaton Simulation of Relativistic Quantum Field," Physical Review Research 6, 033136 (2024). [20] A. Bisio, G. M. D'Ariano, P. Perinotti, "Quantum Cellular Automaton Theory of Light," Annals of Physics 368, 177190 (2016). [21] B. Azarian, "Life Need Not Ever End," Noema Magazine (2023). [22] H. Ma, "Universal Conservation of Information Celerity: From Quantum Cellular Automata to Relativity, Mass and Gravity" (2025), preprint. [23] H. Ma, "Information-volume Conservation and the Emergence of Optical Metrics" (2025), preprint. [24] H. Ma, "The Red Queen Universe: Agent Games, Dark Energy and the Avoidance of Heat Death" (this work). A Appendix A: Detailed Derivation from Landauer Power to Effective Cosmological Constant This appendix provides a detailed derivation from Landauer power density Pinfo(t) to the eective cosmological constant Λeff(t) . 16
A.1 A.1 Energy-Momentum Conservation and Sourced Continuity Equation In the FLRW background, Einstein's equations Gµν + Λbaregµν = 8πGTµν (48) combined with the Bianchi identity ∇µGµν = 0 lead to ∇µTµν −Λbare 8πG gµν= 0. (49) If we introduce an eective cosmological constant Λeff(t) and absorb it into the right-hand side, it can be written as: Gµν = 8πG ˜ Tµν,˜ Tµν =T(m) µν +T(r) µν +T(info) µν , (50) where T(info) µν =−Λeff(t) 8πG gµν. (51) Applying the energy-momentum conservation equation to each component: ∇µT(X) µν =Q(X) ν,X X Q(X) ν= 0, (52) and using the comoving frame uµ= (1,0,0,0) , focusing only on the energy component ν= 0 , we obtain: ˙ρX+ 3H(ρX+pX) = Q(X) 0(t). (53) Assume the source terms for matter and radiation components are: Q(m) 0=−Γm(t), Q(r) 0=−Γr(t), (54) and combine energy losses caused by computation into the source term for the information vacuum component: Q(info) 0= Γm(t)+Γr(t) = Pinfo(t), (55) then the information component satises: ˙ρinfo + 3H(ρinfo +pinfo) = Pinfo(t). (56) If pinfo =winfoρinfo , then the above equation is the sourced continuity equation in the main text. A.2 A.2 Justication for winfo ≃ −1 For the information vacuum energy to manifest macroscopically as dark energy, its equation of state must be close to −1 . From a QCA perspective, a possible microscopic picture is: Irreversible computations of agents permanently "erase" a portion of locally accessible degrees of freedom into inaccessible global entangled structures through scattering and entanglement with the environment; These entangled degrees of freedom are uniformly distributed on large scales, carry no net momentum ux, and their local perturbations equilibrate rapidly under QCA evolution; In the continuous limit, the contribution of such degrees of freedom to macroscopic geometry is approximately isotropic and equivalent to a uid with negative pressure. 17
This picture is formally similar to viewing vacuum energy as zero-point energy of eld theory, but its origin is not eld mode oscillations but "information fragments" generated during irreversible computation. In specic models, one can prove that a class of irreversible scattering is always accompanied by a spectral shift of xed sign by constructing a QCA Hamiltonian with a given scattering matrix and spectral ow, thereby producing an equivalent vacuum energy contribution. This is a direction for future work. Under the approximation required for this paper, it suces that winfo satises |winfo + 1| ≪ 1 (57) in the Red Queen era to guarantee the validity of the integral approximation. A.3 A.3 Complexity-Driven Pinfo(t) Model Assume the average information erasure rate corresponding to unit complexity is a constant κ , then: ˙ Ierase(t) = κ C(t). (58) If the ambient temperature Tbg(t) varies slowly over the period considered, the Landauer power density is: Pinfo(t) = kBln 2 Tbg(t)κ C(t)≈α C(t), (59) where α=kBln 2 κ¯ T , and ¯ T is the characteristic value of Tbg . More generally, if we consider the rate of change of complexity, then Pinfo(t)≈α Tbg(t)˙ C(t) (60) is a more reasonable approximation, because information erasure is typically more directly related to complexity change than to absolute value. Substituting this into the integral expression of A.1 yields the relation in Corollary 1 of the main text. A.4 A.4 Consistency with Friedmann Equations Incorporating information vacuum energy into the Friedmann equation: H2(t) = 8πG 3ρm(t) + ρr(t) + ρinfo(t)−k a2(t), (61) where k is spatial curvature. Since ρinfo(t) is a function obtained by integrating Pinfo(t) , as long as Pinfo(t) is smooth on large scales and satises appropriate growth conditions, it will not introduce rapid oscillations or instabilities that violate observational constraints. It should be emphasized that this model does not claim that all dark energy is composed of information vacuum energy, but provides an information-theoretic origin for part or all of Λeff(t) . Quantitative tting requires considering ρinfo together with other possible dynamical dark energy components. B Appendix B: Linear Stability Analysis of Red Queen Dynamics Model This appendix provides details of the linear stability analysis for the two-species Red Queen dynamics model in Theorem 2. 18
B.1 B.1 Solution of Equilibrium Points Consider the system: dNi dt=NiriCi−di−γ(N1+N2), i = 1,2, (62) dCi dt=αiNi−βiCi. (63) Equilibrium points satisfy: N∗ iriC∗ i−di−γ(N∗ 1+N∗ 2)= 0, αiN∗ i−βiC∗ i= 0. (64) For non-trivial equilibrium points, we require N∗ i>0, C∗ i>0 , so we must have: C∗ i=αi βi N∗ i, (65) Substituting back, we get: ri αi βi N∗ i−di−γ(N∗ 1+N∗ 2)=0. (66) This gives two linear equations: r1α1 β1−γN∗ 1−γN∗ 2=d1, (67) −γN∗ 1+r2α2 β2−γN∗ 2=d2. (68) If the coecient matrix M= r1α1 β1−γ−γ −γ r2α2 β2−γ! (69) is invertible, then N∗ 1 N∗ 2=M−1d1 d2, (70) and subsequently C∗ i=αi βi N∗ i. (71) Requiring N∗ i>0, C∗ i>0 restricts the parameter space, but can generally be satised within physically reasonable parameter ranges. B.2 B.2 Construction of Jacobian Matrix Let x= (N1, N2, C1, C2)T . The components of the vector eld F(x) are: F1=N1r1C1−d1−γ(N1+N2), (72) F2=N2r2C2−d2−γ(N1+N2), (73) F3=α1N1−β1C1, (74) F4=α2N2−β2C2. (75) The elements of the Jacobian matrix J are Jij =∂Fi/∂xj . At the equilibrium point x∗ , its non-zero elements are: 19
For F1 : ∂F1 ∂N1 =r1C∗ 1−d1−γ(2N∗ 1+N∗ 2),∂F1 ∂N2 =−γN∗ 1,∂F1 ∂C1 =r1N∗ 1. (76) Using the equilibrium condition r1C∗ 1−d1−γ(N∗ 1+N∗ 2) = 0 , this simplies to: ∂F1 ∂N1 =−γN∗ 1. (77) For F2 : ∂F2 ∂N2 =r2C∗ 2−d2−γ(N∗ 1+ 2N∗ 2) = −γN∗ 2, (78) ∂F2 ∂N1 =−γN∗ 2,∂F2 ∂C2 =r2N∗ 2. (79) For F3 : ∂F3 ∂N1 =α1,∂F3 ∂C1 =−β1. (80) For F4 : ∂F4 ∂N2 =α2,∂F4 ∂C2 =−β2. (81) Thus, the Jacobian matrix at the equilibrium point is: J= −γN∗ 1−γN∗ 1r1N∗ 10 −γN∗ 2−γN∗ 20r2N∗ 2 α10−β10 0α20−β2 . (82) B.3 B.3 Characteristic Polynomial and RouthHurwitz Criterion The characteristic polynomial is: det(λI −J) = λ4+a1λ3+a2λ2+a3λ+a4, (83) where coecients ak can be obtained by direct expansion. To simplify notation, let: Ai=γN∗ i, Bi=riN∗ i, Ci=αi, Di=βi. (84) Then: a1= 2(A1+A2) + D1+D2, (85) a2= (A1+A2)2+ 2(A1+A2)(D1+D2) + D1D2+B1C1+B2C2, (86) a3= (A1+A2)2(D1+D2)+(A1+A2)D1D2+(A1+A2)(B1C1+B2C2)+D1B2C2+D2B1C1, (87) a4= (A1+A2)2D1D2+ (A1+A2)D1B2C2+ (A1+A2)D2B1C1. (88) The RouthHurwitz criterion states that all eigenvalues have negative real parts if and only if the following conditions hold: 1. a1>0 , 2. a1a2−a3>0 , 3. (a1a2−a3)a3−a2 1a4>0 , 4. a4>0 . In the current model, all Ai, Di, Bi, Ci>0 , so a1, a2, a4>0 automatically hold. Stability critically depends on the second and third conditions. Through algebraic simplication, the second condition can be written as: a1a2−a3=K0−K1, (89) 20
where K0 is a positive term containing only Ai, Di , and K1 is a term related to BiCi . Clearly, when B1C1+B2C2 is suciently large, i.e., when the sum of riαiN∗ i is suciently large, K1 can exceed K0 , causing a1a2−a3<0 , thus violating the second Hurwitz condition. This indicates the existence of a critical surface; on one side, the equilibrium is stable, and on the other, it is unstable. Similarly, the third condition can be written as: (a1a2−a3)a3−a2 1a4=L0−L1, (90) where the highest order term in L1 is proportional to (B1C1+B2C2)2 . When B1C1+B2C2 is large, this quantity can also become negative. Comparing BiCi=riαi(N∗ i)2 with A2 i=γ2(N∗ i)2 and D2 i=β2 i , we can derive a simple sucient condition: r1α1N∗ 1+r2α2N∗ 2> β2 1+β2 2+ 2γr1C∗ 1+r2C∗ 2, (91) Under this condition, at least one Hurwitz condition is violated, and the equilibrium point is unstable. This gives the instability condition in Theorem 2 of the main text. B.4 B.4 Hopf Bifurcation and Existence of Limit Cycles When parameters vary continuously such that a Hurwitz condition crosses zero from positive, corresponding eigenvalues will cross the imaginary axis. If exactly one pair of conjugate complex eigenvalues crosses the imaginary axis while other eigenvalues still have negative real parts, the system undergoes a Hopf bifurcation, generating stable or unstable limit cycles. In the current model, since the system has a four-dimensional state space and a simple coupling structure, parameter perturbations satisfying general position conditions will typically lead to this typical scenario. Specic parameters can be veried by numerical calculation, which is not repeated here. Importantly, there exists an open dense set of parameters such that the internal equilibrium point is unstable and at least one limit cycle or more complex attractor exists. This supports the conclusion in Theorem 2 regarding "Red Queen dynamics leading to sustained non-equilibrium." 21