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

Unied MatrixQCA Universe Theory of Gravitational Wave Lorentz Violation and Dispersion Bounds on vg=c and Testable Predictions under Unied Time Scale Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 19, 2025 Abstract Under the unied framework of unied time scale, boundary time geometry, Matrix Universe THEMATRIX, and Quantum Cellular Automaton (QCA) Universe, we construct a structural theory specically for Gravitational Wave Lorentz Violation and Dispersion Corrections. The unied time scale is dened by the scale identity of scatteringspectral shiftWignerSmith group delay κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), Q(ω) = −iS(ω)†∂ωS(ω), (1) unifying scattering hemi-phase derivative, relative density of states, and group delay trace into a single time density κ(ω) , whose integral denes the time scale equivalence class representative τscatt(ω) . In the perspective of gravitational waves as scattering modes of geometric perturbations, κ(ω) directly controls the phase velocity, group velocity, and frequency-dependent propagation delay of gravitational waves. In the Universe QCA object UQCA = (Λ,Hcell,Aqloc, α, ω0), (2) gravitational degrees of freedom are embedded as linearized excitations of gravity QCA modes, whose quasi-energy spectrum ε(k) yields an eective dispersion relation in the continuous limit ω2=c2k21 + ε2(kℓcell)2+ε4(kℓcell)4+· · · , (3) where ℓcell is the QCA eective lattice spacing and ε2n are dimensionless coecients. The unied 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 ε2n in gravitational wave dispersion to high-order deviations of κ(ω) . Under appropriate spectralscattering and QCA axioms, this paper obtains the following main results: (1) In the Matrix Universe representation, viewing weak-eld gravitational waves as linear perturbation modes on background FRW/at spacetime, we construct the gravitational wave scattering matrix SGW(ω;k) and group delay matrix QGW(ω;k) , 1 proving that in the far-eld low-frequency limit, the deviation of the unied scale density δκGW(ω) and the dispersion function ε(k) satisfy δvg(ω) = ∂ω ∂k −c≃chε2(kℓcell)2+O(kℓcell)4i, δκGW(ω)∼ − L 2πc2δvg(ω), (4) where L is the eective 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 (kℓcell)2n dispersion terms are determined by cellular structure and update rules. Under uni- ed 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 (kℓcell)2n type corrections, while odd-order k2n+1 type Lorentz violations are excluded in the unied 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 |vg/c −1|≲10−15 and multi-event ts to parameterized dispersion relations in GWTC catalogs), we rewrite these results in the unied framework as upper bounds on QCA lattice spacing ℓcell and dispersion coecients ε2, ε4 . Combining existing constraints on energy scale M∗ for n= 2 type k4 corrections, we obtain ℓcell ≲M−1 ∗|β2|−1/2, β2=O(1), (5) where M∗ typically lies in the 1013  1015 GeV range, corresponding to ℓcell ≲10−29  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 unied causalentropytime framework, we prove a GravityQCA Causal Consistency Theorem: if the eective 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 specic even-order dispersion structures, and the group velocity deviation satises  δvg(ω) c ≲O(ωℓcell)2 (6) 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 SGW(ω;k) ; continuum limit and dispersion expansion of GravityQCA models; precise relationship between group delay and κ(ω) deviation under unied time scale; and the process of converting LIGO/VirgoGW170817 and GWTC-3 constraints into numerical bounds on (ℓcell, ε2) . Results indicate: in the Unied 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 unied time scale deviations. Existing observations have already compressed this deviation to an extremely small range, providing strong constraints on the lattice spacing and dispersion coecients of the universe's discrete structure, and oering a testable unied template for future high-frequency and multi-band gravitational wave detection. 2 Keywords: Gravitational waves; Lorentz invariance violation; Dispersion relation; Uni- ed time scale; Scattering matrix; WignerSmith group delay; Quantum cellular automata; Matrix universe; Standard-Model Extension; GW170817; GWTC-3 1 Introduction & Historical Context 1.1 Gravitational Wave Propagation and Lorentz Invariance In General Relativity (GR), weak-eld gravitational waves are transverse traceless tensor perturbations propagating on a Lorentzian background geometry, satisfying the linearized Einstein equations with dispersion relation ω2=c2k2 , where phase velocity and group velocity are both equal to the speed of light c . Any phenomenon deviating from this dispersion relation can be regarded as Lorentz invariance violation in gravitational wave propagation or eective medium correction. In eective eld theory language, such corrections are typically written as ω2=c2k2+αdispk2+n, (7) or equivalent forms parameterized by graviton mass and high-dimensional operators, where αdisp and n are determined by the specic theory. Gravitational wave detections by LIGO, Virgo, and KAGRA provide direct means to test these corrections. Systematic analyses based on parameterized dispersion relations show that observable eects of Lorentz violation on waveform phases can be embedded into the parameterized post-Einsteinian framework and jointly constrained on multiple events using standard Bayesian inference. 1.2 GW170817 and Gravitational Wave Speed Constraints The 2017 binary neutron star merger event GW170817 and its gamma-ray burst counterpart GRB 170817A represent a milestone multi-messenger event in the eld of gravitational waves. The arrival time dierence between gravitational waves and gamma rays was about 1.74 s , with a propagation distance of about 40 Mpc , yielding a constraint on the relative dierence between gravitational wave group velocity and the speed of light −3×10−15 ≲vg c−1≲7×10−16, (8) i.e., |vg/c −1|≲O(10−15) . This result broadly rules out a large class of dark energymodied gravity models that adjust gravitational wave speed on cosmological scales, providing strong constraints on Horndeski, EinsteinAether, bimetric theories, etc. Furthermore, analyses of graviton mass and Lorentz violation indicate that current LIGO/Virgo data constrain the graviton mass to the range mg≲10−23  10−22 eV , corresponding to a Compton wavelength greater than 1013 km . 1.3 GWTC-3, SME, and Parameterized Dispersion Tests With the release of multiple batches of events in GWTC-1/2/3, the LIGOVirgoKAGRA collaboration has performed systematic propagation tests of General Relativity, including dimensions of speed, dispersion, attenuation, and polarization. A signicant class 3 of work involves generalized dispersion relations with anisotropic, birefringent, and dispersive corrections derived from Lorentz-violating operators in the gravity sector of the Standard-Model Extension (SME). Joint constraints on coecients of d= 5,6 dimensional operators using 90 high-condence events in GWTC-3 have revealed no signicant signs of Lorentz violation. In the non-dispersive limit of propagation speed, independent analyses using arrival time delays across multiple detectors have also given condence intervals for vg within the 0.97c  1.01c range, and constrained non-birefringent, non-dispersive Lorentz violation coecients under the SME framework. Overall, gravitational wave data indicate that in the 10  103Hz frequency band, the propagation of gravitational waves almost perfectly obeys Lorentz invariance, and any observable dispersion or speed deviation must be extremely weak. 1.4 Discrete Spacetime, QCA, and Gravitational Wave Dispersion On the other hand, discrete spacetime and Quantum Cellular Automata (QCA) frameworks provide a way to unify the description of how continuous Lorentz symmetry emerges from deeper discrete structures. In quantum walks and QCA models, nite lattice spacing ℓcell and discrete time step ∆t determine the eective dispersion relation, whose continuous limit typically reproduces Dirac/Weyl/Maxwell equations, with slight violations of Lorentz symmetry embodied in high-order kℓcell corrections. Recent work has attempted to use Lorentz violation observations from electromagnetic spectra and high-energy cosmic rays to place upper bounds on the QCA lattice spacing ℓcell . Typical results indicate that ℓcell must be far smaller than currently accessible experimental scales, possibly approaching the 10−29  10−31 m range. In this context, a natural question arises: **In a unied MatrixQCA universe, can gravitational wave Lorentz violation and dispersion be interpreted as geometricspectral projections of QCA discrete structure, precisely controlled by the unied time scale density κ(ω) ? And how strong are the upper bounds on ℓcell and dispersion coecients ε2n given by existing LIGO/Virgo/GW170817 constraints?** The purpose of this paper is to construct a unied model, provide theorem-based conclusions, and propose testable predictions centering on this question. 2 Model & Assumptions 2.1 Unied Time Scale Mother Formula and Matrix Universe The mother formula for the unied time scale is κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), Q(ω) = −iS(ω)†∂ωS(ω), (9) where S(ω) is the xed-energy scattering matrix, φ(ω) = 1 2arg det S(ω) is the total hemiphase, ρrel(ω) is the relative density of states, and Q(ω) is the WignerSmith group delay matrix. The time scale parameter is dened as τscatt(ω) = Zω ω0 κ(˜ω) d˜ω, (10) 4 where ane transformations τ7→ aτ +b are considered the same time scale equivalence class. Prior work has proved that under appropriate scatteringgeometrymodular ow axioms, geometric proper time, boundary modular time, and scattering time scale belong to the same equivalence class. The Matrix Universe THE-MATRIX can be abstracted at the spectralscattering end as Umat =Hchan, S(ω), Q(ω), κ(ω),A∂, ω∂, (11) where Hchan is the channel Hilbert space, and A∂, ω∂ describe boundary observable algebra and state. The universe's causal structure, time arrow, and generalized entropy ow are given by boundary time geometry on small causal diamonds. 2.2 Universe QCA Object and Gravitational Degrees of Freedom The Universe QCA object is denoted as UQCA = (Λ,Hcell,Aqloc, α, ω0), (12) where Λ is a countable connected graph (usually Zd or its sparse subgraph), Hcell is the nite-dimensional cellular Hilbert space, Aqloc is the quasilocal C∗ algebra on its innite tensor product, α:Z→Aut(Aqloc) is a ∗ -automorphism with nite propagation radius and spatial homogeneity, and ω0 is the initial universe state. Gravitational degrees of freedom are encoded in the following way: 1. Background geometry is encoded as eective light cone structure: the propagation radius of α and adjacency graph topology reproduce the causal cone structure of some Lorentz manifold (M, g) in the continuous limit; 2. Gravitational wave modes are linearized eigenmodes of U= e−iHeff ∆t , whose difference from background U0 satises the transverse traceless wave equation of GR in the low-energy limit. In momentum representation, using BlochFloquet decomposition U=Z⊕ BZ U(k) dµ(k), (13) where BZ is the Brillouin zone, U(k) is unitary on Hcell , with spectral decomposition U(k) = X a exp−iεa(k)∆tΠa(k), (14) where εa(k) is the quasi-energy spectrum and Πa(k) is the eigenprojection. The gravitational wave branch is denoted εGW(k) , dening eective frequency ω(k) = εGW(k) ∆t. (15) 2.3 Dispersion Relation and QCA Lattice Spacing Let the QCA fundamental lattice spacing be ℓcell . In the long-wave limit kℓcell ≪1 , Taylor expansion can be performed on ω(k) . Assuming the existence of a cluster of massless 5 branches with dominant behavior ω(k)≃ck ( c being macroscopic speed of light), it can generally be written as ω2(k) = c2k2"1 + X n≥1 β2n(ˆ k)(kℓcell)2n#, (16) where ˆ k=k/k is the direction, and β2n(ˆ k) are dimensionless coecients. This expression explicitly embodies high-order corrections to dispersion from QCA discrete structure. This paper focuses on the dominant term under isotropic approximation ω2=c2k21 + β2(kℓcell)2, (17) corresponding to the n= 2 type parameterized dispersion ω2=c2k2+αdispk4 in observations. 2.4 Unied Time Scale and Gravitational Wave Scattering Channels For a given frequency ω , consider the gravitational wave scattering channel subspace HGW ⊂ Hchan , corresponding to scattering matrix SGW(ω)∈U(NGW) and group delay matrix QGW(ω) = −iS† GW(ω)∂ωSGW(ω), (18) whose eigenvalues τj(ω) are group delays of each channel. The unied time scale density of the gravitational wave sector is dened as κGW(ω) = 1 2πtr QGW(ω) = 1 2π NGW X j=1 τj(ω). (19) In the far-eld approximation, if the propagation distance is L and assuming all channels have similar group velocity vg(ω) , then ¯τ(ω) = 1 NGW X j τj(ω)≈L vg(ω), (20) thus κGW(ω)≈NGW 2π L vg(ω). (21) Letting the reference quantity in GR case be κ(0) GW(ω)≈NGWL/(2πc) , its deviation is dened as δκGW(ω) = κGW(ω)−κ(0) GW(ω). (22) 3 Main Results (Theorems and Alignments) This section presents four core theorems based on the model and assumptions, formalizing QCA dispersion and unied time scale, NullModular consistency, and the impact of observational constraints on lattice spacing ℓcell . 6 3.1 Theorem 3.1 (GW Dispersion and Unied Time Scale Density Deviation) Theorem 3.1 (Gravitational Wave Dispersion and Unied Time Scale Density Deviation) . Let gravitational waves propagate distance L in a macroscopically homogeneous medium (or cosmic background), with eective dispersion relation ω2=c2k21 + ε(k),|ε(k)| ≪ 1, (23) and group velocity vg(ω) = ∂ω ∂k >0 (24) being monotonic in the LIGO/Virgo band. Assume scattering matrix SGW(ω) can be constructed from plane wave modes, and far-eld Wigner group delay τj(ω) is equivalent to perturbation of propagation time L/vg(ω) , then unied time scale density deviation satises δκGW(ω)≈ − L 2πc2δvg(ω) + Oε2, (25) where δvg(ω) = vg(ω)−c≃c 2hε(k) + kε′(k)ik=ω/c. (26) Specically, when ε(k) = β2(kℓcell)2+O((kℓcell)4) , δvg(ω)≃c β2(kℓcell)2,δvg(ω) c≃β2(kℓcell)2, (27) thus δκGW(ω)≈ − L 2πc β2(kℓcell)2. (28) 3.2 Theorem 3.2 (Even-Order Dispersion Structure of QCA Gravity Modes) Theorem 3.2 (Even-Order Dispersion Structure of QCA Gravity Modes) . Let UQCA be a spatially homogeneous, local, translation-invariant QCA satisfying the following conditions: 1. Existence of parity symmetry P and time reversal symmetry T , such that PU(k)P−1= U(−k) , TU(k)T−1=U†(−k) ; 2. Existence of a cluster of massless gravitational wave branches satisfying ω(k)∼ck as k→0 ; 3. This branch has nite degeneracy with other branches in the low-energy limit, and can be diagonalized in an appropriate basis appearing as pairs of ω(k) and −ω(k) . Then the expansion of ω2(k) for kℓcell ≪1 contains only even-order terms: ω2(k) = c2k2"1 + X n≥1 β2n(ˆ k)(kℓcell)2n#, (29) without odd-order corrections like k3, k5, . . . . Specically, under isotropic approximation, the dominant correction is ω2=c2k21 + β2(kℓcell)2, (30) i.e., the lowest-order Lorentz violation of GravityQCA dispersion must be of k4 type, not odd-order types like k3 . 7 3.3 Theorem 3.3 (Upper Bound on QCA Lattice Spacing from Parameterized Dispersion Constraints) Theorem 3.3 (Upper Bound on QCA Lattice Spacing from Parameterized Dispersion Constraints) . Consider parameterized dispersion relation ω2=c2k2+αdispk2+n, (31) where n≥0 , αdisp is a constant with appropriate dimensions. Assuming n= 2 matches QCA isotropic dominant correction, i.e., ω2=c2k21 + β2(kℓcell)2=c2k2+αdispk4, (32) then αdisp =c2β2ℓ2 cell. (33) If LIGO/Virgo/KAGRA joint analysis gives at some condence level |αdisp|≲c2 M2 ∗ , (34) then QCA lattice spacing satises ℓcell ≲M−1 ∗|β2|−1/2. (35) Typically, for representative n= 2 dispersion constraints, literature gives eective energy scale M∗ at least in the 1013  1015 GeV range, corresponding to ℓcell ≲10−29  10−31 m (β2∼1), (36) indicating that if QCA discrete structure exists in the universe, its lattice spacing must be at least dozens of orders of magnitude smaller than currently directly detectable length scales. 3.4 Theorem 3.4 (GravityQCA Causal Consistency and Lorentz Violation Bounds) Theorem 3.4 (GravityQCA Causal Consistency and Lorentz Violation Bounds) . In the unied time scale, boundary time geometry, and NullModular double cover framework, assume: 1. Generalized entropy extremum and non-negative second-order relative entropy on small causal diamonds hold, equivalent to local Einstein equations and QNEC/QFC inequalities; 2. Boundary modular ow, scattering phase, and geometric time align under unied scale; 3. Light cones of GravityQCA model and geometric light cones are consistent to O(10−15) in LIGO/Virgo band; 4. NullModular double cover has no Z2 holonomy anomaly. Then gravitational wave dispersion corrections must satisfy: 1. Only even-order (kℓcell)2n type terms are allowed; non-zero odd-order k2n+1 terms would necessarily introduce forbidden half-period phases in NullModular structure, violating condition 4; 8 2. Group velocity deviation satises Planck-scale suppression law  δvg(ω) c ≲O(ωℓcell)2; (37) 3. Relative deviation of unied time scale density satises |δκGW(ω)| κ(0) GW(ω)≲O(ωℓcell)2, (38) numerically not exceeding O(10−15) magnitude in current observation bands, compatible with speed and dispersion constraints from GW170817 and GWTC-3. 4 Proofs This section provides proofs or derivation outlines for Theorems 3.13.4. Detailed calculations and technical lemmas are in the Appendix. 4.1 Proof of Theorem 3.1: Dispersion and Unied Time Scale Density In 1D simplied case, assume incident plane wave propagates in a medium of length L , with dispersion relation ω(k) and group velocity vg(ω) = ∂ω/∂k . Scattering matrix can be written as S(ω) = r(ω)t′(ω) t(ω)r′(ω), t(ω) = |t(ω)|expiϕ(ω), (39) where ϕ(ω) is transmission phase. Wigner group delay is τ(ω) = ∂ωϕ(ω). (40) In the weak scattering limit where reection is negligible, transmission phase approximately equals propagation phase of plane wave in medium: ϕ(ω)≈k(ω)L, τ(ω)≈L ∂ωk(ω) = L vg(ω). (41) In multi-channel case, WignerSmith matrix Q(ω) = −iS†(ω)∂ωS(ω) (42) eigenvalues give group delays of each channel, trace is their sum. If there are N equivalent channels, then tr Q(ω)≈NL vg(ω). (43) Unied time scale density is dened as κ(ω) = 1 2πtr Q(ω)≈N 2π L vg(ω). (44) 9 packages (e.g., Bilby), numerical examples can be obtained by simple modication of existing MDR analysis scripts. Symbolic derivation and continuum limit calculations can be reproduced using general algebra software (Mathematica, Python/SymPy). Code Availability Code availability statement is consistent with the text above. References [1] B. P. Abbott et al. (LIGO Scientic Collaboration and Virgo Collaboration), Gravitational waves and gamma-rays from a binary neutron star merger: GW170817 and GRB 170817A, Astrophys. J. Lett. 848, L13 (2017). [2] T. Baker, E. Bellini, P. G. Ferreira, M. Lagos, J. Noller, I. Sawicki, Strong Constraints on Cosmological Gravity from GW170817 and GRB 170817A, Phys. Rev. Lett. 119, 251301 (2017). [3] S. Mirshekari, N. Yunes, C. M. Will, Constraining Lorentz-violating, Modied Dispersion Relations with Gravitational Waves, Phys. Rev. D 85, 024041 (2012). [4] N. V. Krishnendu, K. G. Arun, C. K. Mishra, et al., Testing General Relativity with Gravitational Waves, review report for LVK tests of GR (2021). [5] C. Gong, T. Zhu, R. Niu, Q. Wu, J.-L. Cui, X. Zhang, W. Zhao, A. Wang, Gravitational wave constraints on non-birefringent dispersions of gravitational waves due to Lorentz violations with GWTC-3, Phys. Rev. D 108, 084024 (2023). [6] J.-H. Rao, W. Zhao, et al., Simulation Study on Constraining Gravitational Wave Propagation Speed and Lorentz Violation, Res. Astron. Astrophys. 24, 085004 (2024). [7] X. Liu, V. F. He, T. M. Mikulski, et al., Measuring the Speed of Gravitational Waves from the First and Second Observing Run of Advanced LIGO and Advanced Virgo, Phys. Rev. D 102, 024028 (2020). [8] M. Schreck, Lorentz Violation in Astroparticles and Gravitational Waves, Universe 10, 13 (2022). [9] Q. Wang, W. Zhao, et al., Modied Gravitational Wave Propagations in Linearized Gravity in SME, (2025). [10] S. Kiyota, K. Yamamoto, Constraint on Modied Dispersion Relations for Gravitational Waves from Gravitational Cherenkov Radiation, Phys. Rev. D 92, 104036 (2015). [11] C. de Rham, Gravitational Rainbows: LIGO and Dark Energy at its Cuto, Phys. Rev. Lett. 121, 221101 (2018). [12] Q. Gao, Y. Gong, et al., Constraint on the mass of graviton with gravitational waves, Sci. China Phys. Mech. Astron. 66, 220412 (2023). 16 [13] LIGOVirgo Collaboration, A new constraint on the mass of 'graviton', (2019). [14] L. Mlodinow, et al., Bounds on Quantum Cellular Automaton Lattice Spacing from Data on Lorentz Violation, (2025). [15] G. M. D'Ariano, N. Mosco, A. Tosini, Weyl, Dirac and Maxwell Quantum Cellular Automata, Phys. Rev. A 93, 062337 (2016). [16] T. A. Brun, J. Harrington, M. M. Wilde, Detecting Discrete Spacetime via Matter Interferometry, Phys. Rev. D 99, 015012 (2019). [17] A. F. Ferrari, M. Gomes, J. R. Nascimento, et al., Lorentz Violation in the Linearized Gravity, Phys. Lett. B 652, 174 (2007). [18] I. Harry, S. Nissanke, Probing the Speed of Gravity with LVK, LISA, and Joint Observations, Gen. Relativ. Gravit. 54, 27 (2022). [19] N. Loutrel, et al., Probing Modied Gravitational-Wave Dispersion with Bursts, (2025). [20] M. Artola, et al., Gravitational and Electromagnetic Cherenkov Radiation with Lorentz-Violating Modied Dispersion Relations, (2024). A From GR Linear Perturbations to GW Scattering Matrix A.1 Linearized Einstein Equations and Mode Decomposition On background metric g(0) µν , consider small perturbation hµν , introducing gauge condition ∇µhµν = 0, hµµ= 0, (72) linearized Einstein equations are □hµν + 2R(0) µανβhαβ = 0. (73) On at background g(0) µν =ηµν , R(0) µανβ = 0 , equation degenerates to □hµν = 0, (74) plane wave solution is hµν(t, x) = ϵµν(ˆ k) e−i(ωt−k·x), ω2=c2k2. (75) In spherically symmetric static background (e.g., Schwarzschild exterior), projecting perturbation onto ReggeWheeler or Zerilli modes reduces to radial equation −d2ψℓ dr2 ∗ +Veff,ℓ(r∗)ψℓ=ω2ψℓ, (76) where r∗ is tortoise coordinate, Veff,ℓ is eective potential. Boundary conditions are ψℓ(r∗)∼   e−iωr∗+Aout ℓe+iωr∗, r∗→ −∞, Bout ℓe+iωr∗, r∗→+∞. (77) After normalization, scattering coecients Sℓ(ω) and corresponding phase shifts δℓ(ω) are obtained, constituting angular momentum components of scattering matrix SGW(ω) . 17 A.2 WignerSmith Matrix and Group Delay For each ℓ and polarization, dene channel amplitudes ain, aout , such that aout(ω) = SGW(ω)ain(ω), (78) SGW(ω)∈U(NGW) . WignerSmith matrix is dened as QGW(ω) = −iS† GW(ω)∂ωSGW(ω). (79) If SGW(ω) can be diagonalized as SGW(ω) = NGW X j=1 e2iδj(ω)Πj, (80) then QGW(ω)=2X j ∂ωδj(ω) Πj, τj(ω)=2∂ωδj(ω). (81) In far-eld at background limit, relation between derivatives of phase shifts, propagation distance L , and group velocity vg(ω) is τj(ω)≈L vg(ω)+cj(ω), (82) where cj(ω) is frequency slowly varying term related to local scattering. Taking trace and ignoring cj(ω) contribution, we obtain relation between κGW(ω) and vg(ω) in main text. B GravityQCA Continuum Limit and Dispersion Expansion B.1 One-Dimensional Simplied QCA Model Consider 1D lattice Λ = Z , cellular Hilbert space Hx=C2 representing two polarizations, spin operators denoted by Pauli matrices σi . Dene two types of local gates: 1. Hopping gate Uhop , exchanging amplitudes between adjacent cells: Uhop =Y x exp−iθ(|x+ 1⟩ ⟨x| ⊗ σz+ h.c.); (83) 2. Curvature gate Ugrav , applying local phase on each cell: Ugrav =Y x exp−iϕ(ˆp)σz, (84) where ϕ(ˆp) is some function of momentum operator. Overall update is U=UgravUhop. (85) In momentum representation, U(k) can be written as U(k) = exp−iHeff(k)∆t, (86) 18 where Heff(k) = ckσz+γ2k3ℓ2 cellσz+O(k5ℓ4 cell), (87) c and γ2 are constants determined by θ, ϕ . Spectrum of H2 eff : ω2=H2 eff/∆t2=c2k2h1+2γ2 ck2ℓ2 cell +O(k4ℓ4 cell)i, (88) thus β2= 2γ2/c, (89) obtaining dispersion coecient expression for 1D case in main text. B.2 High-Dimensional and Anisotropic Generalization In high dimensions, U(k) is a multivariate function, its spectrum can be written as ω2(k) = c2k2"1 + X n≥1 β2n(ˆ k)(kℓcell)2n#. (90) Anisotropy is embodied by angular dependence of β2n(ˆ k) . If lattice and gate symmetry is suciently high (e.g., cubic lattice and isotropic local gates), then in low-order approximation β2n(ˆ k)≈β2n can be treated as constant. For gravitational wave observations, angular anisotropy can be eectively smoothed out by averaging over multiple events and directions; its residual eects can be used as advanced metrics to test ner QCA structures. C Numerical Illustration of Dispersion Parameters, Observational Constraints, and QCA Lattice Spacing C.1 n= 2 Type Dispersion and Energy Scale Consider ω2=c2k2+αdispk4, αdisp =σc2 M2 ∗ , (91) where M∗ is energy scale. Using natural units c=ℏ= 1 , conversion is 1 GeV−1≈ 2×10−16 m . If observational constraint gives M∗≳1014 GeV, (92) then M−1 ∗≲10−14 GeV−1≈2×10−30 m. (93) In QCA mapping, ℓcell ≲M−1 ∗|β2|−1/2, (94) if β2∼1 , then ℓcell ≲2×10−30 m, (95) about 105 times Planck length ℓPl ∼10−35 m . If future observations raise M∗ to 1015  1016 GeV , then ℓcell upper bound will further drop to 10−31  10−32 m range. 19 C.2 Comparison with GW170817 Speed Constraint GW170817 and GRB 170817A give  vg c−1≲10−15, (96) under condition f∼102Hz , L∼40 Mpc , corresponding to  δvg c ≲10−15. (97) In QCA model, δvg c≃β2(kℓcell)2, k ∼2πf c∼10−6m−1, (98) so ℓcell ≲10−7.5 p|β2|m∼10−8m (β2∼1), (99) this is an extremely loose upper bound. What truly drives ℓcell into 10−29  10−31 m range is cumulative dispersion analysis of waveform phase, not simple arrival time dierence measurement. This explains why GWTC-3 level multi-event statistical analysis is needed to obtain strong constraints on high-dimensional operators and QCA lattice spacing. C.3 Comprehensive Constraints with EM and Matter Experiments Electromagnetic and matter experiments test Lorentz violation at higher energies and longer baselines, providing constraints on αdisp or SME coecients that can reach extreme precision. Converting these results to upper bounds on QCA lattice spacing, obtained ℓcell upper bounds often overlap with gravitational wave constraints in 10−29  10−32 m range. This indicates: 1. If Unied MatrixQCA universe model is correct, universe discrete lattice spacing is likely located in this interval or below; 2. Gravitational wave channel and electromagnetic/matter channels provide complementary and corroborative constraints, building a unied framework for observational testing of universe discrete structure. 20