Unified Constraints in Finite-Entropy-Density Quantum Cellular Automaton Universe: Deriving Cosmological Constant--Discrete Scale--Encoding Efficiency Relations from Black Hole Area Law and Cosmological Horizon Entropy

Unified Constraints in Finite-Entropy-Density Quantum Cellular Automaton Universe: Deriving Cosmological Constant–Discrete Scale–Encoding Efficiency Relations from Black Hole Area Law and Cosmological Horizon Entropy Anonymous Author November 26, 2025 Abstract The Bekenstein–Hawking area law for black hole horizons and the entropy formula for de Sitter cosmological horizon together suggest: gravitational systems obey “entropy scales with area rather than volume” holographic constraints at both infrared and ultraviolet extremes. However, in a discrete universe ontology described by quantum cellular automaton (QCA), the underlying is a computational network with fixed cell spacing and finite information capacity—how to simultaneously reproduce cosmological horizon entropy and black hole area law under the same set of microscopic parameters still lacks systematic axiomatic derivation. This paper introduces the “finite entropy density per cell” postulate within the QCA universe framework, i.e., assuming each lattice cell can carry a maximum von Neumann entropy as a constant upper bound ηcell, and with this as the sole information budget, establishes unified relations through the following two types of constraints: 1. On cosmological scale, the total entropy capacity Scap(VdS) that QCA network can accommodate in volume VdS enclosed by de Sitter radius RdS =p3/Λ must at least cover cosmological horizon entropy SdS =πR2 dS/ℓ2 P; 2. In local strong gravity limit, viewing black hole horizon as a discrete “screen” in QCA network, requiring when cells reach critical arrangement on this screen, degrees of freedom counting on horizon reproduces area law SBH =A/(4ℓ2 P). By relating “volume element entropy upper bound” to “surface element entropy upper bound” through a projection efficiency function ξ(Θ) dependent on microscopic update rule parameter vector Θ, we prove: under simplest isotropic approximation, cosmological constant Λ, discrete cell spacing ℓcell, single-cell entropy upper bound ηcell and encoding efficiency ξ(Θ) satisfy a set of unified constraint equations: ηcell =ℓ2 cell 4ξ(Θ) ℓ2 P ,Λℓ2 cell ≈9 64 1 ξ(Θ)2. This means: after giving projection efficiency ξ(Θ) determined by Θ, the effective magnitude of cosmological constant is directly determined by QCA’s discrete scale ℓcell and cell entropy capacity ηcell; conversely, observed Λ can inversely constrain allowed intervals of (ℓcell, ξ(Θ)). Our derivation provides an observation-driven constraint framework for “finite information universe” hypothesis: the same set of underlying discrete parameters is simultaneously constrained at ultraviolet (black hole) and infrared (cosmological constant) extremes, significantly narrowing feasible QCA universe parameter space. Keywords: Quantum cellular automaton; Finite information ontology; Black hole entropy; Cosmological constant; Holographic principle; Discrete spacetime; Horizon entropy; Planck length 1 1 Introduction Black hole Bekenstein–Hawking entropy formula SBH =A 4ℓ2 P and de Sitter universe horizon entropy formula SdS =πR2 dS ℓ2 P =3π Λℓ2 P are two of the most profound entropy–geometry relations in modern gravitational theory. They jointly point to a fundamental fact: under strong gravitational background, maximum entropy achievable by physical systems no longer scales with volume, but switches to scaling with boundary area. This “holographic” behavior suggests: there exist structural bonds not yet fully revealed between gravity and quantum information. On the other hand, quantum cellular automaton (QCA) provides a strictly unitary, strictly local, and discrete universe ontology picture. In QCA, universe is modeled as tensor product of local Hilbert spaces on lattice set, with time evolution given by unitary update rules preserving local causal structure. If further assuming each cell can carry finite information, then the usable state space dimension of entire universe is strongly constrained, potentially providing “microscopic information-theoretic” origins for cosmological constant, black hole entropy, and other macroscopic phenomena. This paper’s core idea is: in a finite information universe with QCA as underlying description, introduce “maximum entropy upper bound per cell” ηcell as the sole microscopic information budget parameter, requiring this budget simultaneously satisfy in two extreme cases: 1. On cosmological scale, entropy capacity of all cells contained within de Sitter horizon radius must at least realize cosmological horizon entropy SdS; 2. In black hole limit, when local region in QCA network forms horizon, degrees of freedom counting encodable on this horizon must reproduce standard black hole area law. We will prove these two seemingly different entropy constraints can be unified in QCA language as equation constraints on same set of three parameters (ℓcell, ηcell, ξ(Θ)), thus obtaining explicit relation between cosmological constant Λ and QCA discrete scale and encoding efficiency. This “unified constraint” can in principle be tested by observational data (cosmology, black hole physics, gravitational wave dispersion, high-energy cosmic rays, etc.). The paper structure is as follows: Section 2 introduces QCA universe and finite entropy density postulate; Section 3 derives global constraint of cosmological horizon entropy on total entropy capacity; Section 4 analyzes local area law at black hole horizon and surface element degrees of freedom counting; Section 5 gives volume–surface matching and unified constraint equations with rigorous derivation; Section 6 discusses how to embed this framework in larger theory of unified time scale κ(ω) and parameter vector Θ; Sections 7 and 8 discuss observational constraints and future prospects; appendices provide derivation details and simplified model examples. 2 Quantum Cellular Automaton Universe and Finite Entropy Density Postulate 2.1 QCA Universe Object We use QCA language to characterize a discrete universe object. Take three-dimensional lattice 2 Λ≃aZ3, where ais lattice spacing; in this paper we denote ℓcell ≡a as QCA’s “cell linear scale”. Each lattice site x∈Λ is equipped with finite-dimensional Hilbert space Hcell, entire universe Hilbert space is H=O x∈ΛHcell(x). Time evolution given by a family of unitary operators in discrete time steps UΘ:H → H where Θ is finite-dimensional real parameter vector encoding local update rule structure (such as neighborhood range, internal degree of freedom coupling, local gauge structure, etc.). We assume UΘhas strict locality: there exists finite radius rsuch that each update step only propagates influence within rneighborhood, thus defining finite “light cone” structure with maximum signal speed denotable as c. 2.2 Finite Entropy Density Postulate: Maximum von Neumann Entropy per Cell In this discrete universe, we introduce core postulate: Postulate 2.1 (Finite entropy density postulate) There exists constant ηcell >0 such that for any region R⊂Λ, its corresponding local Hilbert subspace HR=O x∈RHx any physically realizable state’s von Neumann entropy S(ρR) satisfies S(ρR)≤ηcell NR, NR≡ |R|, where NRis number of cells in region. In other words, maximum entropy each cell can carry is controlled by a unified upper bound ηcell. From information theory perspective, ηcell can be understood as “maximum effective bits (or nats) each cell can encode”. If each cell Hilbert space dimension is d, then roughly have ηcell ≲ln d. In following derivation, we do not depend on specific value of d, only depend on ηcell as abstract entropy budget parameter. In three-dimensional isotropic lattice, number of cells in region with volume Vis approximately Ncell(V)≈V ℓ3 cell , therefore total entropy capacity upper bound of this region is Scap(V) = ηcell Ncell(V)≈ηcell V ℓ3 cell . 3 This simple volume–entropy relation has tension with “area–entropy relation” universally appearing in gravitational systems: if QCA is true microscopic ontology of universe, then must exist additional constraints or projection mechanisms such that in specific limits, effectively realizable entropy exhibits area scaling rather than volume scaling. This paper precisely attempts to clarify this volume–surface transformation mechanism in “cosmological horizon” and “black hole horizon” two extreme cases. 3 Cosmological Horizon Entropy and Global Entropy Capacity Constraint 3.1 de Sitter Cosmological Horizon and Entropy Consider de Sitter universe dominated by cosmological constant Λ >0, its Hubble parameter is H=rΛ 3, de Sitter horizon radius is RdS =r3 Λ. Horizon area is AdS = 4πR2 dS = 4π3 Λ=12π Λ. Gravitational theory gives de Sitter horizon entropy SdS =AdS 4ℓ2 P =12π 4 Λ ℓ2 P =3π Λℓ2 P . Volume within horizon radius is VdS =4π 3R3 dS =4π 33 Λ3/2 . 3.2 Comparison of Global Entropy Capacity and Horizon Entropy In QCA universe, number of cells within horizon radius is approximately Ncell(VdS)≈VdS ℓ3 cell = 4π 3R3 dS ℓ3 cell . According to finite entropy density postulate, total entropy capacity upper bound of this volume is Scap(VdS) = ηcell Ncell(VdS)≈ηcell 4π 3R3 dS ℓ3 cell . Consider two possible physical requirements: 1. Weak form requirement: Maximum entropy accessible to entire universe not less than cosmological horizon entropy, i.e., Scap(VdS)≳SdS; 4 2. Strong form requirement: Cosmological horizon entropy is “saturation entropy” of accessible degrees of freedom within this volume, i.e., Scap(VdS)≈SdS. To obtain clear constraint relations, this paper adopts strong form requirement; weak form can be viewed as multiplying strong form by O(1) coefficient correction. Proposition 3.1 (Global entropy constraint of cosmological horizon) Under strong form requirement, parameters (ηcell, ℓcell) of finite entropy density QCA universe and cosmological constant Λ satisfy ηcell 4π 3R3 dS ℓ3 cell ≈πR2 dS ℓ2 P . Proof Dividing both sides by πand simplifying gives ηcell 4 3 R3 dS ℓ3 cell ≈R2 dS ℓ2 P . Canceling first order RdS: ηcell 4 3 RdS ℓ3 cell ≈1 ℓ2 P . Substituting RdS =p3/Λ gives ηcell 4 3p3/Λ ℓ3 cell ≈1 ℓ2 P . Rearranging gives √Λ≈4 3ηcell √3ℓ2 P ℓ3 cell . Combining constant factors, writing more concise form: √Λ≈3ηcell ℓ2 P 2ℓ3 cell , thus Λ≈3ηcell ℓ2 P 2ℓ3 cell 2 =3 22η2 cell ℓ4 P ℓ6 cell . Proof complete. The relation we obtain can be summarized as: Λ≈3 22η2 cell ℓ4 P ℓ6 cell This is global constraint of cosmological horizon on QCA parameters: given maximum entropy density per cell ηcell and discrete scale ℓcell, natural order of cosmological constant is determined by above formula. Conversely, observed extremely small Λ imposes strong constraints: either ℓcell far larger than Planck length ℓP, or ηcell extremely small, or some combination of both. 5 4 Black Hole Horizon, Local Area Law and Surface Element Degrees of Freedom Counting 4.1 Black Hole Horizon and Bekenstein–Hawking Area Law For non-rotating black hole with radius R, its event horizon area is A= 4πR2, Bekenstein–Hawking entropy is SBH =A 4ℓ2 P . This formula shows: black hole horizon entropy only proportional to area, independent of volume. In QCA universe framework, when effective gravitational potential of local region is sufficiently strong, signals cannot escape and its boundary can be understood as some “information frozen layer”, corresponding to event horizon in continuous theory. We will assume: on this discrete “horizon screen”, exists a set of surface element degrees of freedom that can effectively encode information, whose maximum encodable entropy should reproduce Bekenstein–Hawking formula. 4.2 Cell Arrangement and Surface Element Entropy Density on Horizon Screen On isotropic lattice, horizon cross-section can be approximately discretized as area elements ∆A∼ℓ2 cell. If each such area unit on horizon can carry maximum entropy ηface, then total entropy upper bound is Smax face ≈ηface A ℓ2 cell . Requiring this counting to reproduce Bekenstein–Hawking area law in black hole limit, i.e., ηface A ℓ2 cell ! =A 4ℓ2 P . This immediately gives: Proposition 4.1 (Relation between surface element entropy density and Planck scale) If black hole horizon entropy completely given by maximum entropy budget of discrete surface elements on QCA horizon screen, then surface element entropy density ηface must satisfy ηface =ℓ2 cell 4ℓ2 P This relation directly expresses surface element entropy density as “square of cell linear scale relative to Planck length”. Physically understandable as: if ℓcell approaches ℓP, then each horizon surface element carries O(1) entropy; if ℓcell far larger than ℓP, then each surface element corresponds to many Planck unit areas, thus carrying larger entropy. 6 5 Volume–Surface Matching and Unified Constraint Equations So far, we have separately obtained two constraint relations on (ℓcell, ηcell) from “cosmological horizon volume–entropy capacity” and “black hole horizon area–entropy budget”. However, not yet explained is: how volume element entropy upper bound ηcell relates to horizon surface element entropy upper bound ηface. Intuitively, degrees of freedom on black hole horizon screen should originate from entanglement structure of one or several layers of QCA volume elements near horizon. When effectively “projecting” internal degrees of freedom of these volume elements onto horizon surface, there may exist redundancy, constraints or selectivity, causing available degrees of freedom number on horizon to be lower than simple volume counting. This “projection efficiency” can be described by dimensionless function ξ(Θ). 5.1 Definition of Projection Efficiency Function ξ(Θ) Assume in shell near horizon with thickness approximately χ∼ℓcell, volume element number density is n(shell) cell ∼1 ℓ3 cell . Each volume element in this shell has maximum entropy ηcell, with effective number of volume elements participating in horizon projection along normal direction (along normal) ∼ χ/ℓcell ∼ O(1). Under isotropic and simple geometry assumptions, this complex detail can be encapsulated as dimensionless efficiency factor ξ(Θ) ∈(0,1], defined as: Definition 5.1 (Volume–surface projection efficiency) Let ξ(Θ) represent ratio of “effective degrees of freedom accessible on horizon surface elements” to “maximum usable degrees of freedom of volume elements in shell near horizon”, then in simplest approximation have ηface =ξ(Θ) ηcell. Here we have absorbed geometric thickness factor into ξ(Θ): ξ(Θ) determined by QCA update rule UΘ, its value reflects microscopic entanglement structure, local gauge constraints and information flow projection manner onto horizon. Combining Proposition 4.1 with this definition, immediately get ξ(Θ) ηcell =ℓ2 cell 4ℓ2 P , thus obtaining: Proposition 5.1 (Surface mapping expression of volume element entropy upper bound) Under black hole area law and volume–surface projection assumption, maximum entropy upper bound ηcell of QCA volume element satisfies ηcell =ℓ2 cell 4ξ(Θ) ℓ2 P This is purely relation determined by discrete scale and projection efficiency: given Θ, can calculate or fit ξ(Θ), thus determining ηcell. 7 5.2 Unified Constraint of Cosmological Constant–Discrete Scale–Encoding Efficiency Now, jointly solving Proposition 3.1 and Proposition 5.1, obtain unified constraint containing only (Λ, ℓcell, ξ(Θ)). From Proposition 3.1 get Λ≈3 22η2 cell ℓ4 P ℓ6 cell . Substituting ηcell from Proposition 5.1: ηcell =ℓ2 cell 4ξ(Θ) ℓ2 P , obtaining Λ≈3 22ℓ2 cell 4ξ(Θ) ℓ2 P2ℓ4 P ℓ6 cell . Simplifying numerator: ℓ2 cell 4ξ(Θ) ℓ2 P2 ℓ4 P=ℓ4 cell 16 ξ(Θ)2ℓ4 P ℓ4 P=ℓ4 cell 16 ξ(Θ)2. Thus Λ≈3 22ℓ4 cell 16 ξ(Θ)2ℓ6 cell =3 221 16 ξ(Θ)2 1 ℓ2 cell . Note (3/2)2= 9/4, thus Λ≈9 4 1 16 ξ(Θ)2 1 ℓ2 cell =9 64 1 ξ(Θ)2 1 ℓ2 cell . i.e., Λℓ2 cell ≈9 64 1 ξ(Θ)2 This gives main result of this paper: Theorem 5.2 (Unified constraint equation) In finite entropy density QCA universe, if 1. Total entropy capacity Scap(VdS) within horizon radius saturates cosmological horizon entropy SdS; 2. Black hole horizon entropy completely given by maximum entropy budget of discrete surface elements on QCA horizon screen; 3. Effective degrees of freedom of horizon surface elements come from projection of volume element degrees of freedom in shell near horizon, with projection efficiency ξ(Θ); then cosmological constant Λ, QCA cell spacing ℓcell, single-cell entropy upper bound ηcell and encoding efficiency ξ(Θ) must simultaneously satisfy ηcell =ℓ2 cell 4ξ(Θ) ℓ2 P ,Λℓ2 cell ≈9 64 1 ξ(Θ)2. In other words, in this framework: 8  Given ℓcell and ξ(Θ), ηcell and Λ are jointly fixed;  Given observed Λ and some physical prior on ηcell, can invert allowed region of (ℓcell, ξ(Θ));  For specification of any three, fourth quantity no longer free. This constitutes strongly constrained “parameter coupling”: cosmological constant, black hole entropy and QCA microscopic structure are no longer mutually independent input constants, but locked by unified framework. 6 Embedding with Unified Time Scale κ(ω)and Parameter Vector Θ In larger theoretical plan, QCA update rule UΘtypically through scattering theory and boundary time geometry relates to unified time scale function κ(ω) = φ′(ω) π= ∆ρrel(ω) = 1 2πtr Q(ω). where Q(ω) is Wigner–Smith time delay operator, ∆ρrel is relative state density relative to some reference background. This formula shows: local definition of time flow rate equivalent to metric of state density in frequency domain. In this unified time identity framework, “volume–surface projection efficiency” ξ(Θ) can be more specifically expressed as some spectral functional. For example, can set ξ(Θ) = Zhor W(ω, Θ) κ(ω) d ln ω Zbulk κ(ω) d ln ω , where:  Denominator represents “total time flow density of volume elements across all frequency bands”;  Numerator represents time flow density contribution from “those frequency bands that can be projected and encoded to surface element degrees of freedom through horizon”;  W(ω, Θ) is bandpass window function determined by update rule UΘ, characterizing which frequency modes can effectively couple to degrees of freedom on horizon. In main derivation of this paper, we only view ξ(Θ) as dimensionless parameter depending on Θ, not concerning its specific spectral expression. However, once κ(ω) and W(ω, Θ) are given in some specific QCA model, can transform cosmological constant constraint Λℓ2 cell ≈9 64 1 ξ(Θ)2 into “spectral selection” constraint on update rule UΘ, thus establishing computable chain between scattering phase, time delay, state density and cosmological constant. 9