Holographic Entropy from Discrete Causal Horizons: A Unitary Solution to the Black Hole Information Paradox in Quantum Cellular Automata

Holographic Entropy from Discrete Causal Horizons: A Unitary Solution to the Black Hole Information Paradox in Quantum Cellular Automata Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract The black hole information paradox originates from the tension between the causal isolation of event horizons in general relativity and the global unitarity of quantum theory. In the semiclassical framework, black holes are treated as thermodynamic systems with temperature TH and entropy SBH =kBA/(4ℓ2 P) , where A is the horizon area. Hawking radiation exhibits a nearly thermal spectrum in the semiclassical limit, implying that a black hole formed from the collapse of a pure state would evolve into a mixed state upon complete evaporation, thus manifesting an apparent violation of unitarity. This paper presents and analyzes a microscopic model of black holes within the discrete ontology of quantum cellular automata (QCA) and the framework of information rate conservation v2 ext +v2 int =c2 . The core idea is that the horizon is not a geometrically absolute causal boundary, but rather a critical layer in the QCA network where the external information transport rate vext is suppressed and the internal phase evolution rate vint saturates, termed the information freezing layer. On this discrete horizon, local connections (links) crossing the surface carry all entanglement and information channels between the interior and exterior regions. After explicitly dening the QCA universe model, optical metric, and discrete horizon, we prove: (1) For any QCA network satisfying local nite degree and Planck-scale lattice spacing a∼ℓP , the number of links crossing the horizon Nlink scales with area as Nlink ∝A/ℓ2 P . (2) If each link is approximately in a maximally entangled state at black hole equilibrium, the von Neumann entropy seen by external observers is SBH = ln 2 ·Nlink ∝A ℓ2 P . (3) Identifying links as punctures in a spin network of SU(2) , adopting the loop quantum gravity area spectrum Aj= 8πγℓ2 Ppj(j+ 1) with j= 1/2 and taking the Immirzi parameter γ= ln 2/(π√3) recovers the BekensteinHawking entropy formula SBH =kBA/(4ℓ2 P) , consistent with existing black hole entropy counting results. Furthermore, treating the black hole and radiation as a pure-state system on a nitedimensional Hilbert space, and assuming QCA evolution is a strictly unitary discrete-time evolution operator U , we utilize the Page theorem and the fast scrambling hypothesis to derive the Page curve for radiation entropy: the radiation entanglement entropy increases monotonically with time in the early stage, reaches a peak when the accessible Hilbert space dimensions of the black hole and radiation become comparable, and subsequently declines to zero as evaporation continues, ensuring the nal radiation state is pure. This resolves the information paradox within this framework. We embed this QCA black hole model into the unied scheme of optical metrics and information volume conservation, demonstrating that the holographic entropy law can be viewed as 1 the continuum extrapolation of an entanglement-counting theorem on discrete causal horizons in QCA. We conclude by discussing connections to loop quantum gravity, AdS/CFT, and analog black hole experiments, and propose several testable observations and simulations, including searches for non-thermal correlations in Hawking radiation and measurements of entanglement structure on analog black hole platforms. Keywords: Black hole information paradox; Quantum cellular automaton; Holographic principle; BekensteinHawking entropy; Entanglement entropy; Page curve; Loop quantum gravity 1 Introduction and Historical Context Black hole thermodynamics establishes a precise connection between geometric area and entropy. Bekenstein rst proposed that black holes must carry entropy proportional to their horizon area to preserve the generalized second law during matter absorption, and estimated the order of magnitude of the entropyarea relation. Subsequently, Hawking calculated in quantum eld theory on curved backgrounds that a static black hole should radiate at temperature TH=ℏκ 2πkBc, where κ is the surface gravity. Combined with the rst and second laws of thermodynamics, this uniquely determines the black hole entropy as SBH =kBA 4ℓ2 P , where ℓP=pGℏ/c3 is the Planck length. In the semiclassical framework, Hawking radiation in scenarios without back-reaction or reecting walls approximately takes the form of blackbody radiation, with a density matrix that appears to external observers as a thermal mixed state. If a black hole originates from the collapse of an initial pure state and evaporates completely, semiclassical calculations suggest that the pure state would irreversibly evolve into a mixed state, violating quantum unitarity. This tension is known as the black hole information paradox. Numerous candidate solutions have been proposed: black hole complementarity emphasizes complementary descriptions of events by dierent observers to prevent inconsistency; the rewall proposal suggests a violent high-energy wall at the horizon to maintain entanglement monotonicity; ER=EPR conjectures that entangled pairs across the horizon are realized by nontrivial Einstein Rosen bridges; and recent island formula and quantum extremal surface approaches recover the Page curve through semiclassical gravitational path integrals. However, these proposals typically rest on an intrinsically continuous spacetime manifold. On the other hand, quantum cellular automata and discrete quantum walks provide discrete frameworks for describing quantum elds and relativistic dynamics. In these frameworks, the universe is modeled as local unitary update rules acting on spatial lattice sites, and equations such as Dirac and Maxwell can emerge in the long-wavelength limit from QCA. This naturally suggests a discrete ontology: at the Planck scale, spacetime is not a smooth manifold but a network of coupled quantum units. In such discrete models, the geometric concepts of event horizons and singularities require reinterpretation. If the underlying dynamics are strictly unitary local updates, there is no mathematical information absorption endpoint; any apparent irreversibility must arise from ignoring partial degrees of freedom. Meanwhile, the black hole entropyarea law and the holographic principle hint 2 that the interior volume degrees of freedom are, in some sense, compressed onto the two-dimensional surface of the horizon. This paper builds upon the previously proposed information rate conservation and optical metric frameworks to introduce a discrete-horizon QCA model providing a unied description of black hole entropy and Hawking radiation. The core steps include: (1) Dening external group velocity vext and internal phase evolution velocity vint in QCA, and adopting the mother-scale identity v2 ext +v2 int =c2 as the axiomatic expression of information rate conservation. (2) Constructing an optical refractive index eld n(x) via local information processing density ρinfo and relating vext to n(x) , thereby dening the discrete horizon as an isosurface where n(x) exceeds a critical value. (3) Proving that the number of discrete connections crossing the horizon is proportional to area, and identifying black hole entropy as the entanglement entropy of this connection set under a maximal entanglement assumption. (4) Viewing connections as spin network punctures, borrowing established loop quantum gravity results for area spectrum and Immirzi parameter to recover the 1/4 coecient. (5) Under the assumption of nite-dimensional Hilbert space and fast scrambling, analyzing radiation entanglement entropy evolution over time using the Page theorem, obtaining a Page curve consistent with unitarity requirements, and showing that the information paradox no longer arises in this discrete model. The greatest dierence from existing continuum frameworks is that this paper fundamentally models the black hole horizon as an information freezing layer in the QCA network, rather than a zero-thickness smooth surface in continuum geometry. Black hole entropy is no longer an ancillary property of geometric quantities, but the statistical outcome of the count of cross-boundary entanglement channels in the discrete network. 2 Model and Assumptions This section provides rigorous denitions of the QCA universe model, optical metric, and discrete horizon used in this paper, and lists key assumptions. 2.1 QCA Universe and Information Rate Conservation Let space consist of a three-dimensional regular lattice Λ∼ =aZ3 with lattice spacing a of order Planck scale a∼ℓP . Each lattice site x∈Λ is associated with a nite-dimensional Hilbert space Hx∼ =Cd , and the global Hilbert space is the tensor product H=O x∈ΛHx. The discrete time step is ∆t , and one step of QCA evolution is realized by a unitary operator U:H → H satisfying strict locality: there exists a nite radius R such that any local operator Ox evolved as U†OxU is supported only within a nite neighborhood of radius R centered at x . This locality denes a nite propagation speed c=Ra ∆t, which can be identied with the speed of light in the continuum limit. 3 Consider a local excitation described as a wavepacket in the long-wavelength limit, whose group velocity in coarse-grained coordinates is denoted vext , while the phase precession rate in internal degrees of freedom (coin or internal spin space) is denoted vint . In previous work, it has been shown that in Dirac-type QCA one can construct an information rate vector (vext, vint) satisfying v2 ext +v2 int =c2, and by using τ as the internal phase evolution parameter, standard special-relativistic proper time and four-velocity normalization can be recovered. We take this relation as the axiomatic statement of information rate conservation: Axiom 1 (Information Rate Conservation) . For any distinguishable local excitation, the external propagation rate vext and internal phase rate vint satisfy v2 ext +v2 int =c2. Intuitively, if vext approaches c , the excitation is approximately massless and propagates nearly at the speed of light between lattice sites; if vext approaches 0 , the excitation hardly propagates in position space but undergoes phase ips and self-referential evolution in the internal Hilbert space at a rate approaching c . 2.2 Local Information Density and Optical Refractive Index Dene the local information processing density ρinfo(x) at each lattice site as the number of eective unitary degrees of freedom that can be implemented per unit time on that cell, normalized to a dimensionless quantity, and assume an upper bound ρmax , corresponding to the case where the cell is lled with local maximal entanglement. We adopt a simple but suciently expressive refractive index model relating gravitational redshift, dening the eective refractive index n(x) as n(x) = 1 1−ρinfo(x)/ρmax . When ρinfo ≪ρmax , n(x)≈1 ; when ρinfo →ρmax , n(x)→ ∞ . Optical path conservation and standard optical metric theory imply that the local eective speed of light satises ceff(x) = c n(x). In the long-wavelength continuum limit of QCA, one can construct an optical metric ds2=−η2(x)c2dt2+η−2(x)γij(x)dxidxj, where η(x) is equivalent to n(x) and γij is the three-dimensional spatial metric. Here we retain only the relation ceff(x) = c/n(x) to characterize the suppression of external group velocity. Combining Axiom ?? , we can write the local external group velocity as |vext(x)|=c n(x)=c1−ρinfo(x) ρmax . When ρinfo(x)→ρmax , |vext(x)| → 0 and vint(x)→c . 4 2.3 Discrete Horizon and Information Freezing Layer Under the above setup, we give the denition of the black hole horizon in this paper. Denition 2 (Discrete Horizon) . Let Ncrit >1 be a given critical refractive index. The discrete horizon H is dened as the approximate discrete set on lattice sites of closed isosurface families satisfying n(x)≥Ncrit in the continuum limit. On the QCA network, the horizon corresponds to a shell region of thickness about one to several lattice spacings: H∆={x∈Λ : Ncrit ≤n(x)< Ncrit +δ}, where δ≪Ncrit . Denition 3 (Information Freezing Layer) . When n(x) is suciently large that |vext(x)| ≪ c and vint(x)≈c , the shell is called the information freezing layer. In this region, spatial migration of excitations is nearly blocked, while phase and entanglement evolution in the internal Hilbert space proceeds at a rate approaching c . Assumption 4 (Volume Freezing and Boundary Storage) . For regions satisfying n(x)≥Ncrit, the eective dynamics of the QCA satisfy: (i) Information propagating from the external region along the radial direction to the information freezing layer nds outward group velocity approaching zero, making it dicult to propagate further into the interior volume region. (ii) Conversely, cells inside the freezing layer can undergo high-frequency entanglement rearrangement and energyinformation redistribution through local unitary coupling. Therefore, from the viewpoint of external observers, the interior volume degrees of freedom are dynamically projected onto a two-dimensional information storage array on the freezing layer. This assumption is qualitatively consistent with the holographic principle. 2.4 Cross-Horizon Connections and Hilbert Space Decomposition Partition the lattice set along the freezing layer surface Σ into the exterior region Out , interior region In , and freezing layer H∆ . The Hilbert spaces of the exterior and freezing layer can be written as Hout =O x∈Out Hx,HH=O x∈H∆Hx. Consider the local adjacency graph of the QCA, and denote by L the set of all nearest-neighbor connections crossing the freezing layer, where each connection ℓ∈ L connects one exterior lattice site to one freezing-layer lattice site. Denition 5 (Horizon Connections) . The horizon connection set L is dened as all edges satisfying: (i) One end belongs to Out , the other to H∆ . (ii) The two endpoints have a direct coupling term in the adjacency graph. 5 The connection count Nlink =|L| will be the key counting object for black hole entropy. In the freezingboundary storage limit, the relevant part of the Hilbert space can be written as HH⊗Hout ∼ =O ℓ∈L H(in) ℓ⊗H(out) ℓ⊗Hrest, where H(in) ℓ and H(out) ℓ are the local subspaces at the two ends of the connection, and Hrest contains all degrees of freedom irrelevant to cross-boundary entanglement. This paper will focus primarily on the entanglement structure on L and prove that black hole entropy can be given by the entanglement von Neumann entropy of these connections. 3 Main Results: Theorems and Alignments Under the above model and assumptions, the core results of this paper can be summarized in the following four theorems. Theorem 6 (Holographic Storage on Discrete Horizon) . In a QCA universe with bounded local degree, nite lattice spacing, and satisfying the information rate conservation axiom, suppose there exists a closed freezing layer H∆ dening the black hole horizon, and propagation modes in the interior region are strongly suppressed dynamically. Then, for the operator algebra accessible to external observers, all information in the black hole interior can equivalently be encoded in the degrees of freedom supported by the connection set L on the freezing layer. That is, there exists an isomorphism HBH ∼ =O ℓ∈L H(in) ℓ⊗Haux, where HBH is the eective black hole Hilbert space visible to external observers, and Haux does not directly entangle with the exterior. Theorem 7 (Connection Count and Area Law) . Suppose the QCA network near the freezing layer has an approximately isotropic connection graph on large scales and has nite average degree. Then, for a suciently large horizon surface Σ , the number of connections crossing the freezing layer satises Nlink =ηA a2+oA a2, where A is the continuum-limit surface area of Σ , and η is a dimensionless geometric factor depending only on the local lattice structure. Theorem 8 (Entanglement Origin of Black Hole Entropy) . Under the conditions of Theorems ?? and ?? , if at equilibrium each connection in the freezing layer is approximately in a maximally entangled state, and correlations among connections can be regarded as disordered on large scales, then the von Neumann entropy of the black hole as seen by external observers is SBH =kBln 2 ·Nlink ≈kBηln 2 A a2. 6 Identifying the lattice spacing with the Planck length a=ℓP and choosing the geometric factor η to match the standard value of the Immirzi parameter in loop quantum gravity, we recover SBH =kBA 4ℓ2 P . Theorem 9 (Unitary Evaporation and Page Curve) . Suppose the black holeradiation joint system at any time t can be described by a pure state |Ψ(t)⟩ on a nite-dimensional Hilbert space Htot =HBH(t)⊗Hrad(t), given by unitary iteration of the global QCA evolution operator. If the internal dynamics on the freezing layer are fast scrambling, and assuming at each stage |Ψ(t)⟩ is close to a Haar-random pure state for the given dimension pair (dBH(t), drad(t)) , then the von Neumann entropy of the radiation approximately satises Srad(t)≈kBmin (ln dBH(t),ln drad(t)) , thus yielding a Page curve that rst rises, then falls, ultimately returning to zero. When the black hole fully evaporates, dBH →1 , radiation entropy Srad →0 , and the joint system remains in a pure state, so the information paradox no longer appears in this framework. Proofs of the above theorems are given in subsequent sections and appendices. 4 Proofs This section provides proof outlines for each theorem, with technical details and complete derivations of related mathematical tools deferred to appendices. 4.1 Proof of Theorem ??: Holographic Storage on Freezing Layer The proof relies on the locality and causal structure of QCA. Step 1. Causal cones and accessibility For any local operator Oout in the exterior region, its Heisenberg picture evolution Oout(t) = U−tOoutUt has support restricted to the causal cone based on the initial support of that operator. The near-zero propagation speed in the information freezing layer means that, on nite time scales, the contribution from the interior volume region to external observational operators can be neglected, while contributions from the freezing layer degrees of freedom dominate. Step 2. Eective operator algebra compression More precisely, consider the operator algebra Aout accessible to external observers. Any A∈ Aout time-evolved in the Heisenberg picture can be written as A(t) = U−tAUt=AH(t)⊗Ibulk + small terms , where AH(t) is supported on the freezing layer and exterior, with the eect of interior volume degrees of freedom suppressed by a small parameter (controlled jointly by vext/c and evolution time). Step 3. Stinespring structure and encoding map In this limit, one can construct a CPTP map encoding the interior volume Hilbert space Hbulk degrees of freedom onto auxiliary degrees of freedom Haux ⊂ HH on the freezing layer, such that expectation values of external operators remain unchanged: trbulk (ρBHAout) = traux (˜ρHAout). 7 By the Stinespring dilation theorem and operator algebra isomorphism, there exists a Hilbert space equivalence HBH ∼ =O ℓ∈L H(in) ℓ⊗Haux, yielding the conclusion of Theorem ?? . Details of the construction are given in Appendix A.1. 4.2 Proof of Theorem ??: Connection Count Proportional to Area This theorem is essentially a geometriccombinatorial statement holding on lattice networks with nite degree and uniformity. Step 1. Local isotropy assumption Assume the connection graph near the freezing layer is approximately uniformly isotropic on suciently large scales, i.e., the average degree deg(x) of each lattice site is statistically constant d0 , and connection directions are uniformly distributed on the sphere. Step 2. Surface area discretization View the approximate discretization of the continuum surface Σ on the lattice as the set of all lattice sites with distance less than a/2 from the surface. The cardinality NΣ of this set satises NΣ=ζA a2+oA a2, where ζ is a constant related to the lattice type (cubic, body-centered cubic, etc.). Step 3. Cross-boundary connection count For each lattice site on Σ , count the connections pointing inward and outward. Because average degree is nite and there are no long-range connections, the total number of cross-boundary connections is proportional to NΣ , i.e., Nlink =ηNΣ=ηζ A a2+oA a2. Merging ηζ into η gives the expression of Theorem ?? . This reasoning shares its origin with the number of surface atoms ∝ surface area calculation in solid-state physics. Detailed derivation is in Appendix A.2. 4.3 Proof of Theorem ??: Entanglement Entropy and Area Law Based on Theorems ?? and ?? , we view cross-boundary connections as entangled two-system subspaces. Step 1. Entropy contribution from a single connection Assume the Hilbert space on each connection ℓ is C2⊗C2 , corresponding to a pair of qubits, naturally in a Bell-type maximally entangled state. Tracing out the exterior, the reduced density matrix of a single exterior qubit is ρ(out) ℓ=1 2I2, with von Neumann entropy Sℓ=−kBtr ρ(out) ℓln ρ(out) ℓ=kBln 2. Step 2. Multi-connection tensor product and independence 8 In the fast-scrambling limit on the freezing layer, entanglement on dierent connections can be approximately treated as statistically independent, and their joint state as seen externally is a tensor product density matrix ρout =O ℓ∈L ρ(out) ℓ, so total entropy is SBH =X ℓ∈L Sℓ=kBln 2 ·Nlink. Step 3. Matching with the area law Substituting the result of Theorem ?? , Nlink ≈ηA a2, we obtain SBH ≈kBηln 2 A a2. Taking the natural lattice spacing a=ℓP , the specic value of η can be xed by a more microscopic spin network model. Step 4. Consistency with loop quantum gravity area spectrum In the loop quantum gravity framework, the horizon is punctured by a spin network, and the area eigenvalue is Aj= 8πγℓ2 Ppj(j+ 1), where γ is the Immirzi parameter. If we identify each qubit connection in QCA as a j= 1/2 puncture, the single-puncture area is A1/2= 4πγ√3ℓ2 P, with spin state space dimension 2j+ 1 = 2 , corresponding to entropy ln 2 . Counting the number of punctures N=A/A1/2 , the total entropy is S=NkBln 2 = A 4ℓ2 P kB if and only if γ=ln 2 π√3. This value is consistent with recent derivations based on loop quantum gravity and Landauer's principle, showing that the present QCA model is compatible with such work regarding entropy area relation. This completes the proof of Theorem ?? . Detailed spin network counting and discussion of the Immirzi parameter are in Appendix B. 4.4 Proof of Theorem ??: Unitary Evaporation and Page Curve This theorem relies on two elements: global unitarity of QCA and average entropy results for Haar-random pure states. Step 1. Finite-dimensional pure state and von Neumann entropy symmetry 9 [16] A. Mallick, C. M. Chandrashekar, Dirac cellular automaton from split-step quantum walk, Sci. Rep. 6 , 25779 (2016). [17] T. A. Brun, M. C. Kimberley, Quantum cellular automata and quantum eld theory in two spatial dimensions, Phys. Rev. A 102 , 062222 (2020). [18] C. Huerta Alderete et al., Quantum walks and Dirac cellular automata on a programmable trapped-ion quantum computer, Commun. Phys. 3 , 89 (2020). [19] P. Arrighi, S. Facchini, M. Forets, Discrete Lorentz covariance for quantum walks and quantum cellular automata, New J. Phys. 16 , 093007 (2014). [20] J. Steinhauer, Observation of quantum Hawking radiation and its entanglement in an analogue black hole, Nat. Phys. 12 , 959 (2016); J. R. Muñoz de Nova et al., Observation of thermal Hawking radiation and its temperature in an analogue black hole, Nature 569 , 688 (2019). [21] W. G. Unruh, Experimental black-hole evaporation?, Phys. Rev. Lett. 46 , 1351 (1981). [22] M. Visser, Dirty black holes: entropy versus area, Phys. Rev. D 48 , 583 (1993). [23] I. Aref'eva et al., Complete evaporation of black holes and Page curves, Symmetry 15 , 170 (2023). [24] X. Wang, R. Li, J. Wang, Page curves for a family of exactly solvable evaporating black holes, Phys. Rev. D 103 , 126026 (2021). [25] M. Cadoni, E. Franzin, S. Mignemi, Unitarity and Page curve for evaporation of 2D AdS black hole, Phys. Rev. D 105 , 024027 (2022). A QCA Formalism and Causal Compression A.1 External Operator Algebra and Equivalent Encoding on Freezing Layer This appendix provides a more detailed construction of the Hilbert space equivalence in Theorem ?? . Let Aout be the bounded operator algebra on the exterior region, generated by local operators supported in the Out region and their limits. Consider the Heisenberg evolution of QCA: Φt:Aout → B(H),Φt(A) = U−tAUt. Due to locality, there exist constants C and velocity vc (of order c ) such that a LiebRobinson-type inequality holds: for any operators A, B with spatial distance dist(suppA, suppB)>0 , we have ∥[Φt(A), B]∥ ≤ C∥A∥∥B∥exp (−µ[dist(suppA, suppB)−vct]+), where [·,·] is the commutator, [·]+ is the positive part, and µ > 0 . Inside the freezing layer, external group velocity |vext| is suppressed, so signals propagating from the interior volume region to the exterior require time scales far longer than the observation time T . Thus, in the time window t≤T , the interior volume region can be viewed as causally silent, and its eect on the external operator algebra can be compressed via a CPTP map onto auxiliary degrees of freedom on the freezing layer. More precisely, x time window [0, T] and consider the evolved operator family Φt(Aout) . Dene Heff =Hout ⊗HH, 16 and projection P:H → Heff , tracing out interior volume degrees of freedom. Since feedback from the interior to the exterior is exponentially suppressed in time T , one can prove there exists a CPTP map E:B(Heff)→ B(Heff) such that for any A∈ Aout ,  PΦt(A)P−Et(PAP) ≤ϵ(T), where ϵ(T) is controlled by T . To an external observer, within accuracy ϵ(T) , the interior volume can be eectively treated as an environment system Haux on the freezing layer. The Stinespring representation theorem guarantees the existence of HBH ∼ =O ℓ∈L H(in) ℓ⊗Haux, corresponding to the statement of Theorem ?? . A.2 Geometric Estimation of Connection Count In the case of cubic lattice structure, lattice sites near the freezing layer can be approximately viewed as a two-dimensional discrete mesh attached to the continuum surface Σ , with lattice spacing a . (1) Partition Σ into area elements ∆A=a2 , each corresponding to one or several lattice sites. (2) For a simple cubic lattice, each lattice site on average has one normal connection crossing the surface (pointing inward or outward) and several tangent connections. Cross-boundary connections mainly come from normal edges. (3) For suciently large A , edge eects can be neglected, and the cross-boundary connection count is approximately Nlink ≈A a2. More general lattice types only change the geometric constant prefactor. This estimate is consistent with the standard result number of surface atoms ∼A/a2  in solidstate physics. B Spin Network Picture and the 1/4 Coecient This appendix connects the QCA connection model to the loop quantum gravity spin network horizon model, explaining why the 1/4 coecient can be recovered under natural assumptions. B.1 Area Spectrum and Immirzi Parameter In loop quantum gravity, the eigenvalue spectrum of the area operator is A= 8πγℓ2 PX ipji(ji+ 1), where the sum runs over all spin network edges puncturing the horizon, and ji is the corresponding spin quantum number. 17 If we assume that in the large-area limit, the main contribution comes from j= 1/2 punctures, then the single-puncture area contribution is A1/2= 4πγ√3ℓ2 P. The internal edge state space dimension corresponding to the puncture is dim H1/2= 2j+ 1 = 2, so each puncture can contribute entropy kBln 2 . Writing the total puncture count as N=A A1/2 =A 4πγ√3ℓ2 P , the total entropy is S=NkBln 2 = kBln 2 4πγ√3 A ℓ2 P . Requiring S=kBA/(4ℓ2 P) gives γ=ln 2 π√3. Recent work shows this value can be independently obtained via Landauer's principle and information erasure cost, strengthening its physical interpretation. B.2 Identication of QCA Connections with Spin Punctures In the present QCA model, the Hilbert space on each connection ℓ is taken as H(in) ℓ⊗H(out) ℓ , where each factor is a two-level system. Identifying H(in) ℓ∼ =H(j=1/2) with the spin1/2 representation, then: (1) Each connection corresponds to one spin network edge puncture on the horizon, contributing area A1/2 . (2) Each connection contributes one bit of maximal entanglement entropy kBln 2 . Thus the QCA connection network on the horizon is isomorphic in state counting to the loop quantum gravity spin network model. The black hole entropyarea relation can be viewed as two expressions of the same counting problem in dierent languages. B.3 Nonj= 1/2 Modes and Correction Terms In more general cases, high-spin j > 1/2 punctures and many-body constraints bring logarithmic corrections to the leading term. Existing research shows −αln A corrections appear in loop quantum gravity black hole entropy, with coecient depending on the microscopic model. In the QCA connection model, these corrections can be understood as: (1) Some connections correspond to higher-dimensional local Hilbert spaces (multi-level systems or multiple line coincidences). (2) Global constraints exist inside the freezing layer (such as total spin, topological number conservation), reducing the allowed entanglement conguration count. These eects do not change the leading area term A/(4ℓ2 P) but yield logarithmic or power corrections at subleading order, corresponding to signatures of dierent quantum gravity schemes. 18 C Page Curve in a Finite-Dimensional QCA Toy Model To concretely demonstrate realization of the Page curve in the QCA context, consider the following simple model: (1) Take the initial black hole Hilbert space dimension dBH(0) = 2N , corresponding to N connections or N freezing-layer bits. (2) At each discrete time step, release one qubit from the black hole to radiation, suciently entangling it with already radiated qubits and remaining black hole degrees of freedom via a random unitary gate. C.1 Iterative Process After the k -th step, the black hole has N−k remaining qubits, and radiation has k qubits. Under the fast scrambling assumption, the joint state |Ψk⟩ ∈ C2⊗(N−k)⊗C2⊗k can be approximately viewed as a Haar-random pure state. By the Page theorem, if k≤N/2 , then radiation entropy is approximately Srad(k)≈kBkln 2 −1 2, while when k≥N/2 , the smaller subsystem is the black hole, and radiation entropy becomes Srad(k)≈kB(N−k) ln 2 −1 2. After normalization, a symmetric Page curve is obtained, reaching a peak ≈kB(Nln 2 −1/2) at k=N/2 , then linearly declining as k increases, with nal state k=N having Srad(N)≈0 . C.2 Correspondence with Continuous-Time Evaporation In continuous-time models, black hole mass M(t) and horizon area A(t) slowly decrease with Hawking radiation. The Page curve of radiation entropy over time can be obtained by viewing k as the number of released eective degrees of freedom and interpolating with continuous parameter t . Recent work's numerical simulations show that, under the assumption of unitary evaporation and fast scrambling, such simplied models yield Page curves qualitatively consistent with more sophisticated gravityquantum eld theory models. C.3 Realization in QCA Context In QCA, the above toy model can be realized as follows: (1) Initialize N cells in the freezing layer to a highly entangled state, isolated from the external environment. (2) At each time step, map one freezing-layer cell's degrees of freedom to an external radiation link via a local unitary gate, while applying suciently deep local random circuits inside the freezing layer to achieve fast scrambling. (3) After each step, perform complete tomography on the external radiation subsystem, computing von Neumann entropy to obtain the discrete Page curve. This process can be realized on any programmable quantum computing platform, providing a feasible path for testing the quantitative relationship among freezing layerconnectionPage curve. 19 D Remarks on Extensions and Open Problems (1) Rotating and charged black holes For Kerr and ReissnerNordström black holes, horizon structure and temperatureangular momentum charge relations are more complex. The QCA model can simulate these eects by introducing anisotropy in angular momentum and charge ow density on the freezing layer; the connection count and entropyarea relation are expected to retain the leading form. (2) Multiple horizons and inner horizon stability In cases with inner horizons (e.g., ReissnerNordström, KerrNewman), multiple freezing layers can be constructed, and the topological structure and information transport of connection networks on them can be analyzed. The discrete model may provide a new perspective on microscopic mechanisms of mass ination and inner horizon instability. (3) Relation to the island formula Recent island formulas reconstruct radiation entropy calculations by introducing quantum extremal surfaces. The freezing layer in QCA can be viewed as a kind of discrete extremal surface, whose position is jointly determined by information rate and entanglement structure. How to unify the variational problems of the two into a discretecontinuum hybrid extremal problem is a direction worth deep investigation in the future. (4) Observational constraints and complexity bounds Although information is strictly preserved in QCA, decoding information in black hole radiation may require exponentially large circuit complexity. How to introduce complexity geometry and computability boundaries in this framework will be an important bridge connecting the information paradox and computational complexity theory. 20