Discrete Horizon Theory of Black Hole Entropy and Information in Unified Matrix--QCA Universe\\ \large $S_{\mathrm{BH

Discrete Horizon Theory of Black Hole Entropy and Information in Unified Matrix–QCA Universe SBH =A/4 and Information Conservation under Unified Time Scale Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 19, 2025 Abstract Under the axiomatic framework of unified time scale, boundary time geometry, and “Universe as Quantum Cellular Automaton” (QCA), we provide a unified discrete–continuous characterization of black hole entropy and the information paradox. The unified time scale mother formula κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω),(1) where φ(ω) is the total scattering hemi-phase, ρrel(ω) is the relative density of states, and Q(ω) = −iS(ω)†∂ωS(ω) is the Wigner–Smith group delay matrix, has been proven in prior work to be the mother scale of the universe’s unified time scale. On one hand, the boundary time geometry framework indicates that for spacetimes with horizons, the Hawking temperature THand Bekenstein–Hawking entropy SBH =A/(4G) can be restated purely using modular flow of boundary algebras, generalized entropy, and Brown–York quasilocal energy. The “geometry– entropy” structure of black hole thermodynamics can be fully derived from local quantum conditions of small causal diamonds. On the other hand, the Universe QCA object UQCA = (Λ,Hcell,A, α, ω0) (2) uses a countable graph Λ as discrete space, finite-dimensional local Hilbert spaces Hcell and quasilocal C∗-algebras Ato describe local degrees of freedom, and a ∗-automorphism αwith finite propagation radius and its unitary implementation Uto describe discrete time evolution, reconstructing relativistic field theory and geometric structures in the continuum limit. This paper unifies the above two structural lines on the black hole horizon, obtaining the following main results: 1. In the Universe QCA framework, we introduce a “horizon band” sublattice ΓH⊂Λ composed of finite cells on the horizon cross-section ΣH, and provide the inner/outer region Hilbert decomposition H ≃ Hin ⊗ HH⊗ Hout.(3) 1 For families of states satisfying local mixing and stationarity, the cross-horizon entanglement entropy satisfies the area law Sent(ΣH) = ηcell A(ΣH) ℓ2 cell +O(A0),(4) where ℓcell is the QCA effective lattice spacing, and ηcell is the cell entropy density constant. 2. Embedding the above QCA horizon area law into boundary time geometry: by aligning the QCA discrete time step with the horizon modular flow parameter via the unified time scale, we prove that consistency constraints under the small causal diamond limit and generalized entropy extremization enforce ηcell ℓ2 cell =1 4G,(5) thereby yielding Sent(ΣH) = A(ΣH) 4G+O(A0),(6) meaning the QCA horizon model automatically reproduces the coefficient 1/4 of the Bekenstein–Hawking entropy. 3. In the Matrix Universe representation, black hole formation and evaporation are viewed as a class of scattering processes on the channel space. The unitarity of the scattering matrix SBH(ω) ensures information conservation of the entire evolution. The QCA one-step evolution Uis unitary on the full Hilbert space and compatible with the spectral measure of SBH(ω) via the unified scale, thus rewriting the naive paradox of “Hawking radiation turning pure states into mixed states” as an effect of “coarse-graining over massive QCA microscopic degrees of freedom in the effective theory outside the horizon”. 4. On a universe satisfying local mixing, energy constraints, and QCA local scrambling assumptions, we construct a model of “horizon–radiation” partition evolving with discrete time steps, and prove: for the vast majority of initial pure states, the radiation entropy Srad(n) evolves with step number napproximately following Page curve behavior, i.e., first increasing with nto a peak, then decreasing as the horizon area shrinks, and finally returning to zero. This result demonstrates that in the unified Matrix–QCA universe, the black hole information paradox can be reduced at the theorem level to issues of typicality and coarse-graining. Keywords: Black hole entropy; Information paradox; Unified time scale; Boundary time geometry; Quantum cellular automaton; Matrix universe; Page curve; Entanglement entropy area law 1 Introduction & Historical Context Bekenstein first pointed out that to uphold the generalized second law, a black hole itself must carry entropy proportional to its horizon area, obtaining the relation SBH =A 4Gℏ,(7) often written as SBH =A/(4G) in natural units. Hawking subsequently proved in the semiclassical approximation that black holes radiate an approximately thermal particle spectrum at temperature TH=κsurf 2π,(8) 2 where κsurf is the surface gravity, thus establishing the complete form of black hole thermodynamics. However, viewing Hawking radiation as strictly thermal combined with the no-hair theorem leads to the famous black hole information paradox: if the initial state is pure, the Hawking radiation after complete evaporation is approximately thermal, seemingly contradicting the unitarity of quantum theory. On this issue, recent work using the “island” formula and replica wormhole path integral techniques has reproduced the Page curve within the framework of general relativity and holography, providing a class of “information recovery” schemes in the semiclassical sense. On the other hand, since Page’s work on average subsystem entropy, it has been recognized that for typical pure states in high-dimensional Hilbert spaces, the reduced state of any small subsystem is nearly maximally mixed, with entropy approximating the logarithm of that subsystem’s dimension. This “typicality” idea has been systematically developed in studies of random circuits, random matrices, and many-body quantum chaos, playing a central role in many Page curve toy models. Parallel to this, Quantum Cellular Automata (QCA), as discrete time–discrete space models based on local unitary update rules, have been proven to serve as natural discretizations of quantum field theories, possessing complete index theory and GNVW index classification in one-dimensional cases. QCAs can be viewed as discrete universe models and can also be experimentally realized via quantum circuits, trapped ions, and photonic platforms to implement Dirac-type QCA or related random circuit dynamics. In prior work, the unified time scale and boundary time geometry framework unified phase–spectral shift–group delay data from scattering theory with Tomita–Takesaki modular flow, generalized entropy variation, and Gibbons–Hawking–York boundary terms into a “boundary time geometry” system, allowing gravitational field equations, Hawking temperature, and Bekenstein–Hawking entropy to be derived at the level of boundary observable algebras. Meanwhile, the “Matrix Universe” perspective views all observable quantities in the universe as a massive but structurally constrained family of operator matrices, whose spectral structure implements the unified time scale and whose block sparsity pattern encodes causal partial order. The goal of this paper is to endogenous the key questions of black hole entropy and the information paradox within this unified framework, using the QCA universe as the discrete ontology and the Matrix universe as the spectral–scattering representation: 1. What object is the horizon in a discrete universe? Why does its entropy satisfy an area law, and why is the coefficient exactly 1/4? 2. If the universe is strictly unitary at the QCA level, in what manner is information preserved and recovered during black hole formation and evaporation? 3. How does the unified time scale constrain the relationship between the QCA microscopic scale and effective geometric constants (such as G)? The core viewpoint of this paper is: under the constraints of unified time scale and boundary time geometry, as long as the continuous limit of the discrete universe QCA reproduces real-world gravity and field theory, the QCA cross-entanglement entropy on the horizon automatically satisfies the area law, with its coefficient uniquely fixed to 1/4Gthrough matching with the generalized entropy formula; the black hole information paradox is then transformed in the Matrix–QCA representation into a theorem-level problem regarding typicality and coarse-graining, rather than a fundamental crisis of unitarity. 3 2 Model & Assumptions This section provides basic definitions of the unified time scale, boundary time geometry, and universe QCA object, and lists the structural assumptions relied upon by subsequent theorems. 2.1 Unified Time Scale Mother Scale Let (H, H0) be a pair of self-adjoint operators satisfying trace-class perturbation and good scattering conditions, S(ω) be the scattering matrix at energy ω, and spectral shift function ξ(ω) and relative density of states ρrel(ω) = −ξ′(ω) exist. The Birman–Kre˘ın formula gives det S(ω) = exp−2πiξ(ω).(9) Denote total scattering phase Φ(ω) = arg det S(ω) = −2πξ(ω), hemi-phase φ(ω) = 1 2Φ(ω). Define Wigner–Smith group delay matrix Q(ω) = −iS(ω)†∂ωS(ω).(10) The unified time scale mother scale is defined as κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω),(11) and the scattering time scale is τscatt(ω) = Zω ω0 κ(˜ω) d˜ω. (12) Prior work shows: when appropriate local quantum energy conditions and modular flow conditions are met, modular time τmod, geometric time τgeom, and τscatt are affine transformations of each other in the physical domain, thus determining a unique time scale equivalence class [τ]. 2.2 Boundary Time Geometry and Black Hole Thermodynamics In the Boundary Time Geometry (BTG) framework, given a spacetime region (M, g) and its boundary ∂M, considering boundary observable algebra A∂, boundary state ω∂, and appropriate boundary spectral structure, Tomita–Takesaki theory assigns to each pair (A∂, ω∂) a self-adjoint generated one-parameter modular flow σω t, where parameter tis modular time. For static black holes with Killing horizons, BTG yields the following structural conclusions (omitting details): 1. The Hawking temperature THat the horizon equals the KMS temperature of the modular flow, satisfying TH=κsurf /(2π) with surface gravity. 2. Bekenstein–Hawking entropy can be viewed as the von Neumann entropy density of the horizon boundary algebra: for horizon cross-section area A, SBH =σHA, σH=1 4G.(13) 4 3. For small causal diamonds containing the horizon, the extremum and second-order non-negativity of generalized entropy Sgen(Σ) = A(Σ) 4G+Sout(Σ) (14) are equivalent to Einstein equations and their quantum corrections, thus embedding black hole thermodynamics into a unified framework of local quantum gravity conditions. In this paper, the role of BTG is to provide macroscopic geometric boundary conditions and entropy density normalization conditions for the QCA continuous limit. 2.3 Universe QCA Object and Causal Structure The Universe QCA object is defined as a quintuple UQCA = (Λ,Hcell,A, α, ω0),(15) where: 1. Λ is a countable connected graph (typically Zdor its subgraph). 2. Each lattice site x∈Λ carries a finite-dimensional Hilbert space Hx≃ Hcell, the overall Hilbert space formally being the infinite tensor product H=Nx∈ΛHx. 3. Quasilocal C∗-algebra A is the norm closure of the union of bounded operators on finite regions. 4. α:A → A is a ∗-automorphism with a finite propagation radius R, such that the support of any local operator Aafter evolution by αextends only to its R-neighborhood. 5. αis implemented by a unitary operator U, i.e., α(A) = U†AU. 6. Initial state ω0is a state on A, corresponding to the universe state at n= 0. The set of events is E= Λ ×Z, with partial order relation (x, n)⪯(y, m)⇐⇒ m≥n, dist(x, y)≤R(m−n),(16) forming a locally finite causal set, whose continuous limit can be viewed as a causal structure on some Lorentzian manifold. Structure and classification of QCA can be referred to in systematic reviews. 2.4 Matrix Universe Representation and Scattering Mother Scale Consistency The Matrix Universe viewpoint considers all observable structures of the universe as an operator matrix family in some sense THE-MATRIX = {S(ω), Q(ω), . . . },(17) whose block structure encodes channel spaces of different causal regions and observers, and whose spectral data imparts time structure via the unified time scale mother scale κ(ω). Unified time scale requires: the scattering matrix SQCA(ω) generated by the QCA continuous limit should satisfy for its hemi-phase and group delay trace κQCA(ω) = 1 2πtr QQCA(ω) = κ(ω) + O(ε),(18) where εis the error as lattice spacing and time step tend to zero, ensuring QCA microscopic dynamics and macroscopic BTG belong to the same time scale equivalence class. 5 2.5 Structural Assumptions for Horizon QCA Model To realize black hole horizons in the QCA universe, this paper adopts the following assumptions: 1. **Horizon Band Sublattice Assumption**: Exists a subset ΓH⊂Λ and complementary regions Λin,Λout ⊂Λ such that Λ=Λin ∪ΓH∪Λout,Λin ∩Λout =∅,(19) corresponding in the continuous limit to black hole interior, horizon neighborhood, and exterior region respectively; the cell count of ΓHsatisfies NH= #ΓH=A ℓ2 cell +O(A0),(20) where Ais the corresponding geometric horizon cross-section area, and ℓcell is the effective lattice spacing. 2. **Local Mixing and Typicality Assumption**: Within the black hole equilibrium time window, the horizon band and its inner/outer complements form an approximately typical pure state family under energy shell and conservation law constraints, with local reduced states approximating maximal mixing, corresponding to conditions for Page’s average entropy formula. 3. **QCA Local Scrambling Assumption**: The QCA local update rules on horizon– radiation correlated regions are equivalent to local random circuits with sufficient mixing, making the state family after finite steps approximate Haar random under local observation, allowing the use of random circuit theory and ETH results to approximate entropy growth and Page curves. Under these assumptions, the QCA universe provides a mathematizable platform for constructing “discrete horizons” and studying black hole entropy and information recovery. 3 Main Results (Theorems and Alignments) Under the above unified framework and assumptions, the core results of this paper can be organized into the following three main theorems, along with an alignment proposition considering both Matrix Universe and QCA Universe. 3.1 Area Law of QCA Horizon Entanglement Entropy Theorem 3.1 (Horizon QCA Area Law).In the Universe QCA object UQCA, if there exists a horizon band sublattice ΓHsatisfying conditions in 2.5 and Hilbert decomposition H ≃ Hin ⊗ HH⊗ Hout,HH=H⊗NH cell ,(21) and the universe state ρwithin the black hole equilibrium time window belongs to a family of states satisfying local mixing and typicality, then there exists a constant ηcell >0and C=O(1) such that Sent(ΣH) = ηcellNH+O(1) = ηcell A ℓ2 cell +O(A0),(22) 6 where Sent(ΣH)is the cross-horizon entanglement entropy, i.e., Sent(ΣH) = Sρin∪H=Sρout,(23) ρin∪H= trHout ρ,ρout = trHin⊗HHρ. This theorem states: in the horizon band QCA model, the dominant contribution to black hole entropy is given by cross-horizon entanglement entropy, satisfying a strict area law; constant ηcell relates to local Hilbert dimension and effective state space under constraints. 3.2 Coefficient Constraint Matching Generalized Entropy Theorem 3.2 (Coefficient Matching Theorem).Assume Boundary Time Geometry and generalized entropy theory hold, i.e., for each horizon cross-section ΣHthere exists Sgen(ΣH) = A(ΣH) 4G+Sout(ΣH) (24) and local quantum gravity conditions are equivalent to Einstein equations in the small causal diamond limit. If further requiring: macroscopically observed black hole entropy from outside equals cross-horizon entanglement entropy, i.e., SBH(ΣH) = Sent(ΣH),(25) and the first-order area coefficient of SBH(ΣH)is given by geometric part A/(4G), then we must have ηcell ℓ2 cell =1 4G,(26) thereby Sent(ΣH) = A 4G+O(A0).(27) This theorem indicates: once requiring the discrete QCA model to reproduce BTG and generalized entropy structure in the continuous limit, the information density ηcell/ℓ2 cell of horizon cells is no longer an arbitrary microscopic parameter but is uniquely fixed to 1/(4G), interpreting Gas a collective effect of QCA microscopic lattice spacing ℓcell and local state space dimension. 3.3 QCA–Page Curve Type Behavior and Information Recovery Theorem 3.3 (QCA–Page Curve Type Behavior).In Universe QCA object UQCA, consider a black hole formation–evaporation process, giving Hilbert decomposition with discrete time step n H ≃ HBH(n)⊗ Hrad(n)⊗ Hbg,(28) where HBH(n)corresponds to black hole interior and near-horizon region, Hrad(n)to Hawking radiation modes, Hbg to background universe. If satisfied: 1. Overall state |Ψ0⟩is pure, evolving unitarily |Ψn+1⟩=U|Ψn⟩; 2. Horizon cell count decreases monotonically with n, while radiation mode count increases monotonically; 3. 7 QCA local scrambling assumption holds, i.e., under each energy shell and macroscopic constraint, |Ψn⟩approximates a typical pure state family on HBH(n)⊗ Hrad(n), then for almost all initial pure states, Srad(n)≈minlog dBH(n),log drad(n),(29) where dBH(n) = dim HBH(n),drad(n) = dim Hrad(n). Entropy function Srad(n)rises then falls with n, forming Page curve type behavior: early stage dBH ≫drad implies Srad(n)≈log drad(n)monotonically increasing; middle stage dBH ≈drad entropy peaks; late stage dBH ≪drad implies Srad(n)≈log dBH(n)decreasing with black hole depletion, finally tending to zero. This result transplants the Page curve from traditional random matrix and random circuit models into the QCA universe under unified time scale constraints, providing a discrete universe version of information recovery. 3.4 Matrix Universe–QCA Universe Alignment Proposition Proposition 3.4 (Consistency of Scattering Mother Scale and QCA Continuous Limit). Suppose UQCA generates effective Hamiltonian Heff and scattering matrix SQCA(ω)in appropriate limit, satisfying Rindler approximation and BTG modular flow structure near horizon. If unified time scale mother scale κ(ω)comes from macroscopic Matrix Universe scattering data, then in the limit of error ε→0, 1 2πtr QQCA(ω) = κ(ω) + O(ε),(30) thus QCA discrete time step ∆tand unified time scale are affine related. This proposition ensures: ”time” involved in Theorems 3.1–3.3 can be uniformly interpreted as parameters under the same scale class, thus discrete description of black hole entropy and Page curve belongs to the same time geometry structure as continuous geometric description. 4 Proofs This section outlines the main lines of proof for the above theorems and propositions. Detailed technical estimates and boundary case analyses are placed in the appendices. 4.1 Theorem 3.1: Horizon QCA Area Law Consider Hilbert decomposition H=Hin ⊗ HH⊗ Hout,HH=H⊗NH cell ,(31) denote dH= dim HH=dNH cell, din = dim Hin, dout = dim Hout.(32) On the physical subspace Hphys ⊂ H selected by energy shell and conservation law constraints, typicality assumption implies: under natural Fubini–Study measure, reduced states of vast majority of pure states |Ψ⟩ ∈ Hphys ρin∪H= trHout |Ψ⟩⟨Ψ|, ρout = trHin⊗HH|Ψ⟩⟨Ψ|,(33) 8 possess spectra close to maximal mixing. In ideal models ignoring constraints, Page conjectured and proved that when m≤n, the average entropy of the smaller subsystem of a random pure state on Hilbert space Cmn is Sm,n = mn X k=n+1 1 k−m−1 2n≃log m−m 2n,(34) where mis smaller subsystem dimension, nlarger subsystem dimension. Viewing Hin ⊗ HHas one side and Hout as the other, two cases arise: 1. If dindH≤dout, smaller subsystem is Hin ⊗ HH, average entropy close to log(dindH). 2. If dindH> dout, smaller subsystem is Hout, average entropy close to log(dout). In black hole equilibrium case, horizon band cell count NHis large, dH=dNH cell grows exponentially, while din, dout or their logarithms grow only polynomially or constantly relative to NH, so in NH→ ∞ limit, dominant term is ES(ρout)≈NHlog deff +O(1),(35) where deff ≤dcell is effective local dimension corrected by energy constraints and local conservation laws. Let ηcell = log deff ,(36) then Sent(ΣH) = S(ρout) = ηcellNH+O(1) = ηcell A ℓ2 cell +O(A0).(37) Presence of energy shell and local conservation laws reduces dimension of Hphys, but in large system limit, this reduction corresponds to renormalization of ηcell, not changing linear relationship of entropy with NH, see Appendix A. 4.2 Theorem 3.2: Matching Generalized Entropy Coefficient Area law from Theorem 3.1 has form Sent(ΣH) = ηcell A ℓ2 cell +O(A0).(38) On the other hand, in BTG and generalized entropy framework, geometric part of black hole entropy is SBH(ΣH) = A 4G,(39) total generalized entropy is Sgen(ΣH) = A 4G+Sout(ΣH),(40) where Sout corresponds to entropy of visible degrees of freedom outside horizon, related to specific radiation and exterior quantum states. Identifying cross-horizon entanglement entropy as macroscopic black hole entropy, i.e., SBH(ΣH) = Sent(ΣH),(41) then coefficients of first-order area term on both sides must be equal, thus ηcell 1 ℓ2 cell =1 4G.(42) 9 A.2 Energy Shell Constraint and Local Equilibrium In QCA universe, energy is not a priori conserved quantity, but defined by effective Hamiltonian Heff in continuous limit and unified time scale. “Equilibrium state” near horizon can be viewed as set of typical states satisfying fixed energy density and energy flux constraints, its physical subspace dimension satisfying dim Hphys ∼expslocNH,(61) where sloc can be defined via microcanonical or canonical ensemble as local entropy density. Under local ETH assumption, vast majority of eigenstates within energy shell are equivalent to typical states under local observation, thus Page-type average entropy estimate remains valid, only replacing log dcell with sloc, i.e., ηcell =sloc. Local ETH has been widely verified in random circuit and many-body chaos models. A.3 Area Law in Continuous Limit Finally, substituting relation between NHand geometric area A, NH=A ℓ2 cell +O(A0),(62) obtaining Sent(ΣH) = ηcell A ℓ2 cell +O(A0),(63) which is statement of Theorem 3.1. Appendix B: Further Details of QCA–Page Curve Type Behavior This appendix clarifies random circuit and local scrambling assumption used in Theorem 3.3. B.1 Local Random Circuit and t-design In finite size systems, local random circuits consist of layers of two-body or few-body unitary gates, gates in each layer acting on non-overlapping subsets. If gate set generates SU(d) in group theoretic sense, and randomly selected gate sequence constitutes approximate unitary t-design after sufficient depth, its action on initial simple product state produces approximate Haar random state in polynomial time. This conclusion has been widely used in random circuit models of many-body and open systems. In QCA universe, if local update rules can be viewed as periodic action of some fixed gate set, then introducing small random perturbations or multi-step composition can also approximately realize t-design locally, supporting local scrambling assumption. 16 B.2 Hilbert Dimension and Shape of Page Curve During black hole formation–evaporation, horizon cell count NH(n) decreases with time, while radiation mode count (viewed as excitations leaving some causal region) increases with time. If local Hilbert dimension of each cell and mode approximates constant, then log dBH(n)≈sBHNH(n),log drad(n)≈sradNrad(n),(64) where sBH, srad are corresponding local entropy densities. Page time corresponds to position where logarithmic dimensions are equal sBHNH(nPage)≈sradNrad(nPage).(65) Under unified time scale, mapping nto continuous parameter τyields shape of continuous Page curve. Its specific functional form depends on dynamics of NH(τ) and Nrad(τ), determined by specific QCA rules or effective gravity model. Appendix C: Compatibility of Unified Time Scale and QCA Continuous Limit C.1 Effective Hamiltonian and Scattering Matrix In QCA continuous limit, if single-particle sector can be diagonalized in momentum representation, evolution of each momentum kmode approximates U(k) = exp−iω(k)∆t,(66) dispersion relation ω(k) approximates relativistic type in small klimit. Multi-particle sector yields effective Hamiltonian Heff under appropriate approximation, constructing scattering matrix SQCA(ω) via standard scattering theory. Unified time scale mother scale requires κQCA(ω) = 1 2πtr QQCA(ω)≈κ(ω),(67) where κ(ω) comes from continuous macroscopic universe scattering data. This condition constrains parameters of ∆t,ℓcell, and Heff , making QCA time scale compatible with BTG modular time scale. C.2 Rindler Approximation and Modular Flow Near horizon, using Rindler coordinates, metric can be written as ds2=−κ2 surf x2dt2+ dx2+ dy2+ dz2.(68) Modular flow corresponds to translation along t, its KMS temperature is TH=κsurf /(2π). In QCA continuous limit, if local update rules approximate Rindler model near horizon band, unified time scale requires each discrete time step ∆tcorrespond to fixed modular time increment, manifesting in frequency domain as consistency between κQCA(ω) and κ(ω). This compatibility ensures: in unified Matrix–QCA universe, time passage perceived by black hole horizon is completely consistent with time inferred by external observer via scattering mother scale, placing Bekenstein–Hawking entropy, Page curve, and QCA horizon model into unified time geometry structure. 17 References [1] J. D. Bekenstein, “Black holes and entropy”, Phys. Rev. D, 1973. [2] S. W. Hawking, “Particle creation by black holes”, Commun. Math. Phys., 1975. [3] D. N. 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