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

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 equation \kappa(\omega) =\varphi'(\omega){\pi} =\rho_{rel}(\omega) =1{2\pi}tr\, Q(\omega), equation where \varphi(\omega) is the total scattering hemi-phase, \rho_{rel}(\omega) is the relative density of states, and Q(\omega)=-i S(\omega)^\dagger\partial_\omega S(\omega) 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 T_H and Bekenstein--Hawking entropy S_{BH}=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 equation \mathfrak U_{QCA} =(\Lambda,\mathcal H_{cell},\mathcal A,\alpha,\omega_0) equation uses a countable graph \Lambda as discrete space, finite-dimensional local Hilbert spaces \mathcal H_{cell} and quasilocal C^\ast-algebras \mathcal A to describe local degrees of freedom, and a \ast-automorphism \alpha with finite propagation radius and its unitary implementation U to 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 \Gamma_{H}\subset\Lambda composed of finite cells on the horizon cross-section \Sigma_{H}, and provide the inner/outer region Hilbert decomposition equation \mathcal H \simeq \mathcal H_{in}\otimes\mathcal H_{H}\otimes\mathcal H_{out}. equation For families of states satisfying local mixing and stationarity, the cross-horizon entanglement entropy satisfies the area law equation S_{ent}(\Sigma_{H}) =\eta_{cell}A(\Sigma_{\mathrm{H})}{\ell_{cell}^2} +O(A^0), equation where \ell_{cell} is the QCA effective lattice spacing, and \eta_{cell} 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 equation \eta_{\mathrm{cell}}{\ell_{cell}^2} =1{4G}, equation thereby yielding equation S_{ent}(\Sigma_{H}) =A(\Sigma_{\mathrm{H})}{4G} +O(A^0), equation 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 S_{BH}(\omega) ensures information conservation of the entire evolution. The QCA one-step evolution U is unitary on the full Hilbert space and compatible with the spectral measure of S_{BH}(\omega) 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 S_{rad}(n) evolves with step number n approximately following Page curve behavior, i.e., first increasing with n to 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.