Unified Theory of Quantum Chaos and Eigenstate Thermalization in QCA Universe\\ \large ETH, Postulated Chaos and Eigenstate Level Statistics under Unified Time Scale

Unied Theory of Quantum Chaos and Eigenstate Thermalization in QCA Universe ETH, Postulated Chaos and Eigenstate Level Statistics under Unied Time Scale Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore November 19, 2025 Abstract In the framework of Unied Time Scale, Matrix Universe THE - MATRIX , and Quantum Discrete Cellular Automaton Universe Uqca , we construct a systematic theory for Quantum Chaos and the Eigenstate Thermalization Hypothesis (ETH). The Unied Time Scale Mother Formula κ(ω) = φ′(ω)/π =ρrel(ω) = (2π)−1tr Q(ω) (1) unies the scattering semi-phase derivative, relative density of states, and Wigner Smith group delay trace into a single time scale density, thereby providing a "Universe Time Mother Ruler" in the Matrix Universe Umat independent of specic Hamiltonian choices. The Universe as a QCA object Uqca = (Λ,Hcell,Aqloc, α, ω0) (2) describes discrete time evolution via local unitary automorphisms α on a countable lattice, and reconstructs relativistic quantum eld theory and geometric structure in the continuum limit. Based on this, we propose an axiomatic system for "Postulated Chaos QCA" and prove the following main results: 1. On any nite region Ω⊂Λ , the restricted nite-dimensional unitary operator UΩ satises Discrete Time ETH for its quasi-energy spectrum eigenstates with respect to any local operator OX (support X⊂Ω ): for the vast majority of eigenstates |ψn⟩ within an energy window, ⟨ψn|OX|ψn⟩=⟨OX⟩micro(εn) + O(e−c|Ω|), (3) and the squared average of o-diagonal matrix elements decays exponentially with volume, achieving eigenstate thermalization for local observables. 2. Under the axioms of "Postulated Chaos QCA" (nite propagation radius, translation symmetry, local gates generating high-order unitary designs, no extra extensive conserved quantities), the quasi-energy level statistics of the nite region UΩ converge after unfolding to the WignerDyson distribution of the CUE random 1 matrix class. Its spectral form factor exhibits a typical "rampplateau" structure, constituting a standard quantum chaos diagnosis at the QCA level. 3. Relating QCAETH to the Unied Time Scale: Under the unied time scale τ , the "thermalization time scale" and the growth rate of local entropy density are jointly controlled by the energy shell average of the scale density κ(ω) and the QCA light cone structure. We prove a "Unied TimeETHEntropy Growth Theorem": for any family of nite-density local initial states, the entropy density increases monotonically with τ and approaches the microcanonical entropy density in the long-time limit. 4. On the global QCA universe object, embedding the above local results into the causal network and unied time scale mother structure shows that the "thermal time arrow" and "macroscopic irreversibility" on the cosmic scale can be understood as structural invariants in the unied MatrixQCA Universe, rather than additionally introduced dynamical postulates. The appendix provides: rigorous denitions and equivalent forms of discrete time ETH; exact mapping between local random circuits and QCA; proof details and constant estimates for the QCAETH Theorem and CUE level statistics; and a 1D Postulated Chaos QCA model with specic predictions for numerical verication. Keywords: Quantum Chaos; Eigenstate Thermalization Hypothesis (ETH); Quantum Cellular Automata (QCA); Unied Time Scale; Matrix Universe; Unitary Design; Spectral Form Factor; Random Matrix Theory 1 Introduction & Historical Context 1.1 Standard Picture of ETH and Quantum Chaos In closed many-body quantum systems, how pure state unitary evolution generates statistical behavior compatible with thermodynamic equilibrium is a core problem in the foundations of quantum statistical mechanics. Integrable systems typically retain a large number of conserved quantities, tending towards a Generalized Gibbs Ensemble after long-time evolution, whereas non-integrable systems in high-energy density regions are widely believed to satisfy the Eigenstate Thermalization Hypothesis (ETH). Deutsch and Srednicki rst proposed, through random matrix heuristics and eld theory analysis, that in the eigenbasis of a suciently chaotic Hamiltonian, the eigenstate matrix elements of a local observable operator O can be written as ⟨Eα|O|Eβ⟩=O(¯ E)δαβ + e−S(¯ E)/2fO(¯ E, ω)Rαβ, (4) where ¯ E= (Eα+Eβ)/2 , ω=Eα−Eβ , S(¯ E) is the microscopic entropy, and Rαβ are quasi-Gaussian random numbers with zero mean and unit variance. The diagonal term gives the energy-dependent thermal equilibrium value, while the o-diagonal terms are exponentially small in system volume, ensuring that time-averaged observables approximate the microcanonical ensemble average and time uctuations are suppressed. Subsequently, extensive numerical and theoretical work has veried ETH in contexts such as spin chains, Bose/Fermi lattice models, and Floquet many-body systems, systematically analyzing its scope and failure mechanisms (e.g., many-body localization). Parallel to this, research on random matrix theory and quantum chaos provides another diagnostic route: if level statistics exhibit a WignerDyson distribution after appropriate unfolding, and the spectral form factor shows a "rampplateau", the system is considered to be in a quantum chaos phase. 2 1.2 Discrete Time Systems and Random Quantum Circuits In Floquet systems and random quantum circuits, time evolution is described by a single time-step unitary operator U , and the quasi-energy spectrum is dened by U|ψn⟩= e−iεn∆t|ψn⟩. (5) ETH can be rewritten as a statement about quasi-energy eigenstates |ψn⟩ , where thermal equilibrium corresponds to a microcanonical distribution on a xed quasi-energy shell. Local random quantum circuits have been proven to achieve high-order unitary designs at polynomial depth, with their eigenstates and spectral statistics strictly approaching the typical properties of Haar random unitary matrices. Recent work has further improved the trade-o between design order and circuit depth, providing ner estimates for "randomness" and "scrambling speed". These results indicate that in discrete-time many-body systems with locality constraints, quantum chaos and ETH possess the universality predicted by random matrix theory and can be strictly controlled via the language of local random circuits and unitary designs. 1.3 Structure and Development of Quantum Cellular Automata (QCA) Quantum Cellular Automata (QCA) are another class of unitary dynamical models discrete in both time and space, encoding structures like "locality, translation symmetry, nite propagation speed" under rigorous mathematical axioms. Schumacher and Werner provided the denition and structure theorems for reversible QCA, emphasizing that QCA are translation-covariant dynamics with nite propagation radius on innite lattice systems; the same local rule can generate global time steps under nite periodic boundary conditions. In the 1D case, Gross, Nesme, Vogts, and Werner developed index theory for QCA and quantum walks, providing topological indices on K-theory to classify 1D QCA. Other works characterize the structure and simulability of local QCA from a quantum information perspective. Compared to local random circuits, QCA are closer to the ideal abstraction of "cosmic dynamics": their denition does not include external noise or measurement, relying only on discrete time-step unitary updates and spatial locality. Therefore, if the cosmic ontology is characterized as a certain QCA, then ETH, quantum chaos, irreversibility, and the thermal time arrow should all nd explanations within the QCA framework. 1.4 Unied Time Scale and Matrix Universe In scattering and spectral theory, the WignerSmith time delay matrix Q(ω) = −iS(ω)†∂ωS(ω) (6) relates the frequency derivative of the multi-channel scattering matrix S(ω) to "group delay", with its trace giving the total delay time. On the other hand, the BirmanKre in formula and LifshitsKre in trace formula relate the spectral shift function ξ(ω) to the scattering determinant det S(ω) , showing that ξ(ω) = −1 2πilog det S(ω), ρrel(ω) = −ξ′(ω) (7) 3 are well-dened for very general operator pairs (H0, H) . The Unied Time Scale Mother Formula κ(ω) = φ′(ω)/π =ρrel(ω) = (2π)−1tr Q(ω) (8) is interpreted in this context as a unied density of "relative density of states  group delay  scattering phase derivative", dening a "Time Mother Ruler" independent of specic coordinates and local Hamiltonian choices. The Matrix Universe THE - MATRIX views the universe as a family of giant scattering matrices S(ω) decomposed by energy, and their WignerSmith matrices Q(ω) , with block sparse structure encoding causal partial order and spectral data realizing the unied time scale. In geometricboundary time structure, the unied time scale corresponds to objects like modular ow and GHY boundary time translation, thereby downgrading "time" to a function of scattering phase and spectral shift. 1.5 Objectives and Work Overview The goal of this paper is to establish an axiomatic and theorem-based theory for "The validity of Quantum Chaos and ETH in QCA Universe and Level Statistics" under the dual framework of Unied Time Scale and Matrix UniverseQCA Universe. The core ideas are: 1. Model the universe as a reversible QCA object Uqca satisfying the Schumacher Werner axioms, obtaining nite-dimensional unitary operators UΩ on nite regions. 2. Introduce the axiomatic system of "Postulated Chaos QCA", such that UΩ is equivalent to a family of local random circuits at nite depth. These circuits constitute high-order unitary designs within polynomial depth, thereby approximating Haar random unitaries in local observables and spectral statistics. 3. Utilize the ETH typicality of Haar random unitaries and random matrix theory to establish QCAETH theorems and CUE-type spectral statistics theorems for the eigenstate matrix elements, level spacings, and spectral form factor of UΩ . 4. Relate the Unied Time Scale Mother Formula κ(ω) to the discrete time step of QCA, proving the Unied TimeETHEntropy Growth Theorem, and restating the thermal time arrow and macroscopic irreversibility at the level of the cosmic causal network. Following the given structure, the subsequent sections present the model and postulates, main theorems, proofs, model applications, engineering proposals, discussion, conclusion, and detailed proofs in the appendices. 2 Model Assumptions 2.1 Unied Time Scale Mother Formula and Scattering Structure Let H be a separable Hilbert space, and H0, H be a pair of self-adjoint operators satisfying appropriate trace-class perturbation conditions such that wave operators exist and are complete, and the scattering operator S=W† +W− (9) can be written under spectral decomposition as S=Z⊕ S(ω) dµ(ω), (10) 4 where ω represents energy or frequency parameter. For almost every ω , S(ω) is a unitary operator on the ber Hilbert space. The BirmanKre in formula asserts the existence of a spectral shift function ξ(ω) such that ξ(ω) = −1 2πilog det S(ω), ρrel(ω) = −ξ′(ω) (11) giving the relative density of states in broad cases. On the other hand, the WignerSmith time delay matrix is dened as Q(ω) = −iS(ω)†∂ωS(ω), (12) whose trace characterizes the total group delay time. The Unied Time Scale Mother Formula is dened as κ(ω) = φ′(ω) π=ρrel(ω) = 1 2πtr Q(ω), (13) where φ(ω) is the scattering semi-phase (phase of the relative scattering determinant). In the unied framework, κ(ω) is regarded as the "Universe Time Density". Any physical time parameter, if obtained through an observable measurement process, belongs to the equivalence class of τscatt(ω) = Zω κ(˜ω) d˜ω (14) on large scales. 2.2 Matrix Universe THE - MATRIX The Matrix Universe object Umat is dened as Umat =Hchan, S(ω), Q(ω), κ, A∂, ω∂, (15) where: 1. Hchan =Lv∈VHv is the direct sum of channel Hilbert spaces, each v corresponding to a macroscopic "port" or boundary region; 2. S(ω)∈ B(Hchan) is a family of frequency-dependent scattering matrices, whose block sparse structure on V×V encodes causal partial order and interaction structure; 3. Q(ω) = −iS(ω)†∂ωS(ω) is the WignerSmith group delay matrix, and the unied time scale density is given by κ(ω) = (2π)−1tr Q(ω); (16) 4. A∂ is the boundary observable algebra, and ω∂ is the boundary quantum state, connected to scattering data via boundary time geometry and modular ow. In this object, quantum chaos and ETH correspond to: scattering phases obeying random matrix theory predictions statistically over energy windows, and local projections under channel decomposition exhibiting only exponentially small deviations between eigenstate averages and corresponding microcanonical averages. 2.3 QCA Universe Uqca The QCA Universe object Uqca is dened as Uqca = (Λ,Hcell,Aqloc, α, ω0), (17) 5 satisfying the following axioms: 1. Λ is a countable connected graph (typically Zd or its nite metric deformation); 2. Each lattice site x∈Λ carries a nite-dimensional Hilbert space Hx∼ =Hcell ; 3. The quasi-local algebra Aqloc is the norm closure of all nitely supported operators; 4. α:Aqloc → Aqloc is a ∗ -automorphism, with a propagation radius R < ∞ , such that for any nite region X⊂Λ , an operator OX supported on X is mapped to an operator supported on X+R={y∈Λ|dist(y, X)≤R}; (18) 5. α is implemented by a global unitary operator U , i.e., α(O) = U†OU ; 6. α is translation covariant, i.e., commutes with the lattice translation group action; 7. The initial state ω0 is a state on Aqloc , giving the quantum state of the universe at time step n= 0 . For any nite region Ω⊂Λ , dene HΩ=Nx∈ΩHx . Restricting U yields UΩ (under appropriate boundary conditions), with spectrum UΩ|ψn⟩= e−iεn∆t|ψn⟩. (19) These nite-dimensional unitary operators are the fundamental objects for discussing QCAETH and level statistics. 2.4 Postulated Chaos QCA To make QCA exhibit statistical properties similar to local random circuits on nite regions, we introduce the following denition. Denition 2.1 (Postulated Chaos QCA) . A translation-invariant QCA U is called a Postulated Chaos QCA if it satises: 1. **Finite Propagation Radius and Locality:** There exists an integer R such that for any nite X⊂Λ , α(AX)⊂ AX+R ; 2. **Local Circuit Representation:** On any nite region Ω , UΩ can be written in a suitable basis as a nite-depth local quantum circuit UΩ= D Y ℓ=1 Uℓ, Uℓ=O j Uℓ,j, (20) where each gate Uℓ,j acts on a nite subset Xℓ,j ⊂Ω and commutes with all gates separated by more than a nite distance; 3. **Approximate Unitary Design:** There exists t0∈N and a function ϵt(|Ω|) (decaying exponentially with |Ω| ), such that for any t≤t0 , the unitary family generated by UΩ constitutes an ϵt -approximate unitary design in the t -th moment, i.e., for any polynomial P(U, U†) (degree not exceeding t ),   EUΩ[P(UΩ)] −EU∼Haar[P(U)]  ≤ϵt(|Ω|); (21) where U∼Haar denotes Haar random unitary on U(dim HΩ) . 4. **No Extra Extensive Conserved Quantities:** Except for possibly a few global quantum numbers (e.g., total particle number, spin), there are no independent extensive local conserved quantities in the system; 5. **Thermalization Energy Window:** There exists an energy window I⊂(−π/∆t, π/∆t] , within which the number of eigenstates grows exponentially with |Ω| , and energy level degeneracy produces only nitely many symmetry multiplicities. These conditions encapsulate mature results of local random circuits realizing highorder unitary designs and satisfying ETH, embedding them into the language of QCA. 6 2.5 Denition of Discrete Time ETH Consider a nite region Ω and its evolution operator UΩ , with spectral decomposition as before. For a given energy window center ε and width δ > 0 , dene the quasi-energy shell subspace HΩ(ε, δ) = span{|ψn⟩ | εn∈(ε−δ, ε +δ)}, (22) with dimension denoted by Dε,δ . Dene the microcanonical average ⟨OX⟩micro(ε) = D−1 ε,δ X εn∈(ε−δ,ε+δ) ⟨ψn|OX|ψn⟩ (23) (taking δ scaling polynomially with |Ω| ). Denition 2.2 (Discrete Time ETH) . UΩ is said to satisfy Discrete Time ETH for a family of local operators {OX} in energy window I , if there exist constant c > 0 and smooth functions OX(ε) , σX(ε) , such that for any X⊂Ω and the vast majority of n ( εn∈I ): 1. **Diagonal ETH:** ⟨ψn|OX|ψn⟩=OX(εn) + O(e−c|Ω|); (24) 2. **O-Diagonal ETH:** For almost all m=n and ¯ε= (εm+εn)/2∈I , |⟨ψm|OX|ψn⟩| ≤ e−S(¯ε)/2σX(¯ε), (25) where S(¯ε)∼s(¯ε)|Ω| is the microcanonical entropy of the energy shell. If this holds for all families of local operators, the QCA is said to satisfy ETH in that region. 3 Main Results (Theorems and alignments) For brevity, denote dim HΩ=D∼es|Ω| . 3.1 QCAETH Main Theorem Theorem 3.1 (QCAETH Theorem) . Let U be a Postulated Chaos QCA, Ω⋐Λ a suciently large nite region, UΩ the unitary operator restricted to Ω , and {|ψn⟩, εn} its quasi-energy eigenpairs. Then there exist an energy window I and a constant c > 0 such that for any local operator OX with nite support X⊂Ω , there exists a smooth function OX(ε) satisfying: 1. **Diagonal ETH:** For the vast majority of n in the energy window ( εn∈I ), ⟨ψn|OX|ψn⟩=OX(εn) + O(e−c|Ω|); (26) 2. **O-Diagonal ETH:** The second moment satises E|⟨ψm|OX|ψn⟩|2≤e−S(¯ε)gO(¯ε, ω), (27) where ¯ε= (εm+εn)/2 , S(¯ε)∼s(¯ε)|Ω| , and gO is bounded; 3. If the initial state |ψ0⟩ has a narrow energy distribution within window I , i.e., |cn|2 is signicant only for εn∈I , then its time-averaged local observation satises ⟨OX⟩=⟨OX⟩micro(ε) + O(e−c|Ω|), (28) and time uctuations are exponentially suppressed by the o-diagonal ETH index. 7 3.2 CUE-type Convergence of Quasi-Energy Statistics Let θn=εn∆t∈(−π, π] , sorted in ascending order and unfolded to variables sn with average spacing 1. Theorem 3.2 (CUE Behavior of QCA Level Statistics) . Under the assumptions of Theorem 3.1, the unfolded nearest-neighbor spacing distribution P(s) converges in the limit |Ω|→∞ to the WignerDyson distribution of the CUE random matrix ensemble: PCUE(s)∼32 π2s2e−4s2/π. (29) Simultaneously, the normalized spectral form factor K(t) = D−1|tr Ut Ω|2 (30) exhibits a "rampplateau" structure after appropriate rescaling, consistent with the universal spectral uctuations of CUE. 3.3 Unied TimeETHEntropy Growth Theorem In the QCA Universe, introduce an ane relation between unied time scale τ and discrete time step n : τ=an∆t+b, a > 0. (31) Consider a family of "low entropy" initial states {ρ0} within energy window I , whose von Neumann entropy density s0=S(ρ0)/|Ω| is less than the microcanonical entropy density smc(ε) . Theorem 3.3 (Unied TimeETHEntropy Growth) . Under Postulated Chaos QCA and the assumptions of Theorem 3.1, there exists a function vent(ε)>0 and a constant c′>0 such that for any nite X⊂Ω , in the unied time scale interval τ∈[0, τth] , the entropy density of the reduced state ρX(τ) = trΩ\Xρ(τ) , sX(τ) = |X|−1S(ρX(τ)), (32) satises sX(τ)≥s0+vent(ε)τ ℓeff − O(e−c′|Ω|), (33) and approaches smc(ε) after τ≳τth . Here ℓeff is determined by the propagation radius of the QCA and LiebRobinson type light cone velocity, and vent(ε) can be written as a function of the average of the unied scale density κ(ω) over window I and local interaction strength. 3.4 Cosmic Scale ETH and Thermal Time Arrow View the QCA Universe Uqca as the direct limit of a family of nite regions {ΩL} , ΩL↗Λ . For each L , restricting U yields UΩL . Proposition 3.4 (ETH on Cosmic Causal Network) . If U is a Postulated Chaos QCA, then for any nite causal diamond (determined by some observer's worldline), there exists L such that this diamond is contained in some suciently large region ΩL . Consequently, 8 under the unied time scale, all local observables within this diamond tend to microcanonical equilibrium after a suciently long time, and the entropy density grows monotonically with τ until saturation. Therefore, the thermal time arrow and macroscopic irreversibility on the cosmic scale can be viewed as structural results jointly determined by QCAETH and the Unied Time Scale. 4 Proofs This section provides proofs for the main theorems. Technical derivations of operator integrals, spectral unfolding, and concentration inequalities are placed in the appendices. 4.1 Local Random Circuits and Approximate Unitary Designs Recall the results of local random quantum circuits realizing approximate unitary designs. Brandão, Harrow, and Horodecki proved that on a 1D chain, local random circuits composed of nearest-neighbor two-body gates achieve t -th order approximate unitary design within depth O(t10n2) . Harrow et al. improved the depth estimate for higher dimensions and more general lattice structures to the optimal scaling of poly(t)n1/D . Subsequent work constructed explicit families of local designs under symmetry constraints and particle number conservation. These theorems can be summarized as: on a nite region Ω , for a family of local gates satisfying certain genericity and non-degeneracy conditions, the distribution induced by suciently deep random circuits on the unitary group is close to Haar in the t -th moment sense. In the denition of Postulated Chaos QCA, the property of approximate unitary design is embedded via the local circuit representation of UΩ . Since QCA evolution is deterministic rather than random, the "set of multiple time steps Un Ω " needs to be viewed as a circuit family: when n varies within an appropriate time window, {Un Ω} forms a family of orbits in the local gate parameter space. In some QCAs, this family of orbits is sucient to realize design properties; more generally, nite super-periods or spatial translations can be introduced in dening the cosmic QCA to achieve "eective randomization". In the postulates of this paper, these details are abstracted as "the unitary family generated by UΩ within polynomial time steps is an approximate unitary design". 4.2 ETH Typicality of Haar Random Unitaries For a Haar random unitary U∈U(D) , its eigenvectors are uniformly distributed on the complex sphere of the Hilbert space. For any xed local operator OX , eigenstate matrix element statistics can be calculated using Haar integration formulas. The following conclusions nd systematic proofs in random matrix theory and high-dimensional geometry. Lemma 4.1 (Diagonal Statistics of Haar Random Eigenbasis) . Let U be Haar random, {|ψn⟩} be its eigenbasis, and OX be a local operator supported on |X| ≪ |Ω| . Then: 1. E[⟨ψn|OX|ψn⟩] = tr(OX)/D is independent of level n ; 2. Var[⟨ψn|OX|ψn⟩]∼ O(D−1) , 9 and initial state |ψ0⟩=Pncn|ψn⟩ . The time average of a local observation is ⟨OX⟩= lim N→∞ 1 N N−1 X k=0 ⟨ψ0|U†k ΩOXUk Ω|ψ0⟩=X n |cn|2⟨ψn|OX|ψn⟩, (49) assuming non-degenerate energy levels. If Diagonal ETH holds, i.e., ⟨ψn|OX|ψn⟩=OX(εn) + δn,|δn| ≤ e−c|Ω|, (50) and the initial state energy distribution is concentrated in a narrow window, then ⟨OX⟩=X n |cn|2OX(εn) + O(e−c|Ω|)≈OX(ε)≈ ⟨OX⟩micro(ε). (51) Therefore, Diagonal ETH is equivalent to thermalization of time-averaged local observations (in the sense of exponentially small error). A.2 A.2 O-Diagonal ETH and Time Fluctuations Time uctuations can be expressed as δOX(k) = ⟨OX⟩(k)−⟨OX⟩=X m=n c∗ mcnei(εm−εn)k∆t⟨ψm|OX|ψn⟩. (52) An upper bound for its variance is |δOX|2≤X m=n |cm|2|cn|2|⟨ψm|OX|ψn⟩|2. (53) If O-Diagonal ETH gives E|⟨ψm|OX|ψn⟩|2≤e−S(¯ε)gO(¯ε, ω), (54) then given the number of eigenstates in the energy shell Dε,δ ∼eS(ε) , the uctuation variance is O(e−S(ε)) , decaying exponentially with volume. A.3 A.3 Relation between Floquet ETH and Hamiltonian ETH When an eective continuum limit exists for the QCA, an eective Hamiltonian can be dened Heff =i ∆tlog U, (55) whose spectrum relates to quasi-energy spectrum as En≈εn . If ∆t is suciently small and the branch cut of log U is chosen properly, Floquet ETH and Hamiltonian ETH are equivalent in the same energy window, and microcanonical ensembles and quasi-energy shells can be interchanged. 16 B Appendix B: Technical Details of QCAETH Theorem B.1 B.1 Quantitative Bounds for Approximate Unitary Designs For a 1D chain of n q -dimensional qubits, the design theorem by BrandãoHarrow Horodecki can be written as: there exist constants C, c > 0 such that local random circuits of nearest-neighbor gates with depth L≥Ct10n2 (56) constitute an ϵ -approximate t -design, where ϵ≤e−cn . Harrow et al. further proved that on a D -dimensional lattice, the depth scaling can be improved to poly(t)n1/D . Combined with our postulates, we can take t0 as a xed constant, and ϵt0(|Ω|)≤e−c|Ω| . B.2 B.2 Haar Integration and Eigenstate Matrix Elements Haar integration formulas give ZU(D) Ui1j1· · · UikjkUi′ 1j′ 1· · · Ui′ kj′ kdµHaar(U) (57) expressible using Weingarten functions on the symmetric group Sk . For k≤t0 , expectations can be expressed as nite sums. This leads to the mean and variance estimates in Lemma 4.1 and Lemma 4.2. In the QCA scenario, since UΩ is an approximate unitary design at order t0 , the dierence between the above expectations and variances under the distribution induced by UΩ and Haar measure is also controlled by ϵt0 . B.3 B.3 Concentration Inequalities Lipschitz functions on the complex sphere of Hilbert space satisfy Levy's concentration inequality: for an L -Lipschitz function f , P|f−Ef|> ϵ≤2 exp(−cDϵ2/L2). (58) Taking f as functions of ⟨ψ|OX|ψ⟩ or |⟨ψm|OX|ψn⟩|2 yields probability bounds for deviations of eigenstate matrix elements from mean values. C Appendix C: Spectral Form Factor and WignerDyson Distribution C.1 C.1 Spectral Form Factor of CUE The spectral form factor of a CUE random matrix U∈U(D) is dened as KCUE(t) = D−1E|tr Ut|2. (59) Random matrix theory gives an explicit expression in the limit D→ ∞ : under rescaled time τ=t/D , KCUE(τ) exhibits a linear "ramp" and saturation "plateau". 17 C.2 C.2 Spectral Form Factor in QCA Models In Postulated Chaos QCA, UΩ is approximately Haar random on nite-order trace polynomials, so the statistics of K(t) within a nite time window |t| ≤ tmax ∼poly(|Ω|) match KCUE(t) of CUE, with deviation O(ϵt0) . Through Fourier transform, K(t) can be related to level correlation functions, thereby obtaining the WignerDyson form for nearest-neighbor spacing distribution and higherorder spacing distributions. D Appendix D: Numerical Verication Framework for 1D Postulated Chaos QCA D.1 D.1 Model Parameter Selection In 1D brick-wall QCA, two-body gates can be chosen as Ugate = exp−i(Jxσx⊗σx+Jyσy⊗σy+Jzσz⊗σz+hx(σx⊗I+I⊗σx))∆t, (60) with parameters (Jx, Jy, Jz, hx) subjected to quasi-random perturbations over dierent time periods to break integrability and extra symmetries. D.2 D.2 Numerical Steps 1. Construct UΩ for a chain of length L and perform exact diagonalization (feasible for L≤16 ); 2. Calculate the distribution of matrix elements of local operators (e.g., singlebody Pauli matrices or two-body interaction terms) on eigenstates, verifying Diagonal and O-Diagonal ETH; 3. Calculate unfolded spacing distribution and spectral form factor, comparing with CUE results; 4. Simulate time evolution for dierent initial state families, verifying thermalization of local observables and growth of entropy density, comparing with predictions of Theorem 3.3. This framework provides a concrete realizable numerical and experimental path for verifying Postulated Chaos QCA axioms and the theorems of this paper. 18