Unified Matrix--QCA Universe Theory of Gravitational Wave Lorentz Violation and Dispersion\\ \large Bounds on $v_{\mathrm g

Under the unified framework of unified time scale, boundary time geometry, Matrix Universe THE--MATRIX, and Quantum Cellular Automaton (QCA) Universe, we construct a structural theory specifically for ``Gravitational Wave Lorentz Violation and Dispersion Corrections''. The unified time scale is defined by the scale identity of scattering--spectral shift--Wigner--Smith group delay equation \kappa(\omega) =\varphi'(\omega){\pi} =\rho_{rel}(\omega) =1{2\pi}tr Q(\omega), \qquad Q(\omega)=-i S(\omega)^\dagger\partial_\omega S(\omega), equation unifying scattering hemi-phase derivative, relative density of states, and group delay trace into a single time density \kappa(\omega), whose integral defines the time scale equivalence class representative \tau_{scatt}(\omega). In the perspective of ``gravitational waves as scattering modes of geometric perturbations'', \kappa(\omega) directly controls the phase velocity, group velocity, and frequency-dependent propagation delay of gravitational waves. In the Universe QCA object equation U_{QCA} =(\Lambda,\mathcal H_{cell}, \mathcal A_{qloc},\alpha,\omega_0), equation gravitational degrees of freedom are embedded as linearized excitations of ``gravity--QCA modes'', whose quasi-energy spectrum \varepsilon(k) yields an effective dispersion relation in the continuous limit equation \omega^2 =c^2k^2\bigl[1+\varepsilon_2(k\ell_{cell})^2 +\varepsilon_4(k\ell_{cell})^4+\cdots\bigr], equation where \ell_{cell} is the QCA effective lattice spacing and \varepsilon_{2n} are dimensionless coefficients. The unified time scale requires the QCA discrete time step to be in the same equivalence class as geometric proper time, boundary modular time, and scattering time scale, thereby directly linking \varepsilon_{2n} in gravitational wave dispersion to high-order deviations of \kappa(\omega). Under appropriate spectral--scattering and QCA axioms, this paper obtains the following main results: (1) In the Matrix Universe representation, viewing weak-field gravitational waves as linear perturbation modes on background FRW/flat spacetime, we construct the gravitational wave scattering matrix S_{GW}(\omega;k) and group delay matrix Q_{GW}(\omega;k), proving that in the far-field low-frequency limit, the deviation of the unified scale density \delta\kappa_{GW}(\omega) and the dispersion function \varepsilon(k) satisfy equation \delta v_{\mathrm g}(\omega) =\partial\omega{\partial k}-c \simeq c\Bigl[\varepsilon_2(k\ell_{cell})^2 +\mathcal O\bigl((k\ell_{cell})^4\bigr)\Bigr], \qquad \delta\kappa_{GW}(\omega) \sim -L{2\pi c^2}\delta v_{\mathrm g}(\omega), equation where L is the effective propagation distance. (2) We construct a class of ``Gravity--QCA Models'' in the QCA Universe, whose linearized degrees of freedom reproduce the transverse traceless gravitational wave equation of General Relativity (GR) in the long-wave limit, while high-order (k\ell_{cell})^{2n} dispersion terms are determined by cellular structure and update rules. Under unified time scale and boundary time geometry constraints, combining discrete symmetries and Null--Modular double cover consistency, we prove that in the absence of chiral anomalies and with time reversal conservation, gravitational wave dispersion only allows even-order (k\ell_{cell})^{2n} type corrections, while odd-order k^{2n+1} type Lorentz violations are excluded in the unified framework. (3) Utilizing constraints on gravitational wave propagation speed and dispersion from LIGO--Virgo--KAGRA and the multi-messenger event GW170817/GRB 170817A (e.g., speed constraint \lvert v_{\mathrm g}/c-1\rvert\lesssim10^{-15} and multi-event fits to parameterized dispersion relations in GWTC catalogs), we rewrite these results in the unified framework as upper bounds on QCA lattice spacing \ell_{cell} and dispersion coefficients \varepsilon_2,\varepsilon_4. Combining existing constraints on energy scale M_\ast for n=2 type k^4 corrections, we obtain equation \ell_{cell} \lesssim M_\ast^{-1}\lvert\beta_2\rvert^{-1/2}, \qquad \beta_2=\mathcal O(1), equation where M_\ast typically lies in the 10^{13}--10^{15}\,GeV range, corresponding to \ell_{cell}\lesssim10^{-29}--10^{-31}\,m. This result is of comparable magnitude to independent constraints on discrete spacetime and QCA lattice spacing based on electromagnetic and matter interferometry experiments. (4) In the unified causal--entropy--time framework, we prove a ``Gravity--QCA Causal Consistency Theorem'': if the effective light cone of Gravity--QCA remains consistent with the causal light cone of boundary time geometry in the LIGO/Virgo frequency band, then allowed Lorentz violations must exhibit specific even-order dispersion structures, and the group velocity deviation satisfies equation \biggl\lvert\delta v_{\mathrm g(\omega)}{c}\biggr\rvert \lesssim \mathcal O\bigl((\omega\ell_{cell})^2\bigr) equation Planck-scale suppression law, with high-order contributions to group delay being exponentially suppressed under current observational precision. (5) The appendix provides: construction from GR linear perturbations to Matrix Universe scattering matrix S_{GW}(\omega;k); continuum limit and dispersion expansion of Gravity--QCA models; precise relationship between group delay and \kappa(\omega) deviation under unified time scale; and the process of converting LIGO/Virgo--GW170817 and GWTC-3 constraints into numerical bounds on (\ell_{cell},\varepsilon_2). Results indicate: in the Unified Matrix--QCA Universe Theory, gravitational wave Lorentz violation and dispersion are not arbitrary high-dimensional operator perturbations, but geometric--spectral projections of QCA discrete structure and unified time scale deviations. Existing observations have already compressed this deviation to an extremely small range, providing strong constraints on the lattice spacing and dispersion coefficients of the universe's discrete structure, and offering a testable unified template for future high-frequency and multi-band gravitational wave detection.