On foundation of computational universe axiomatic framework U_{comp} = (X,T,C,I) and unified time scale master scale $ \kappa(\omega) = \varphi'(\omega)/\pi = \rho_{rel}(\omega) = (2\pi)^{-1}\tr Q(\omega) this paper constructs fully discrete time crystal theory, unifying it with Null--Modular Z_2 holonomy and time--information--complexity joint geometric structure. We first introduce, on computational universe implementation of reversible quantum cellular automaton (QCA), Floquet--QCA object U_{FQCA} = (X,U_F,C_T,I), where U_F is local Floquet evolution operator with period T, C_T is unified time scale cost of one Floquet step. We give computational universe sense definitions of discrete time translation symmetry and spontaneous breaking, and from complexity geometry and information geometry perspectives, characterize time crystal phase: on any initial state family satisfying local observability and bounded energy density assumptions, exists local observable O whose expectation value exhibits strict period mT rather than T in long-time evolution, where m\ge 2 is integer. Subsequently, on previously constructed causal diamond chain and Null--Modular double cover structure, we introduce cyclic chain of Floquet--QCA time crystals: each Floquet period corresponds to one causal diamond, forming diamond chain \{\Diamond_k\}_{k\inZ}. On this chain, we define for each period modulo-2 time phase label \epsilon_k\inZ_2 induced by scattering phase increment, construct Null--Modular double cover \mathfrak{D}\toD of diamond chain. We prove: existence of period-doubling time crystal (m=2) corresponds precisely to nontrivial Z_2 holonomy of Floquet control loop on Null--Modular double cover, i.e., closed Floquet control loop has no closed lifted path on double cover, thus giving exact correspondence between time crystal parity and Null--Modular holonomy. At engineering level, we consider time crystal readout and robustness under finite complexity budget. By combining unified time scale frequency domain with spectral windowing error control theory (PSWF/DPSS), we construct class of ``finite-order window function observation operators'' for time crystal readout, prove: under conditions that Floquet gap exceeds certain threshold and local noise satisfies finite correlation length assumption, sampling time crystal signal with DPSS type readout window in finite steps can robustly discriminate period-doubling parity with complexity budget N = O(\Delta^{-2}\log(1/\varepsilon)) while error probability not exceeding \varepsilon, where \Delta is Floquet quasienergy gap. Finally, we view time crystals as ``discrete phase lockers'' of unified time scale: on control manifold (M,G), time crystal phase corresponds to class of Floquet control loops with Z_2 holonomy, giving special minimal worldline family in time--information--complexity joint variational principle. We discuss potential experimental role of time crystals as local standards of unified time scale, and complementary relationship with FRB phase metrology and \delta$--ring--AB scattering metrology.
On foundation of computational universe axiomatic framework U_{comp} = (X,T,C,I) and unified time scale master scale κ(ω)=φ′(ω)/π=ρrel(ω)=(2π)−1\trQ(ω)thispaperconstructsfullydiscretetimecrystaltheory,unifyingitwithNull−−ModularZ2holonomyandtime−−information−−complexityjointgeometricstructure.Wefirstintroduce,oncomputationaluniverseimplementationofreversiblequantumcellularautomaton(QCA),Floquet−−QCAobjectUFQCA=(X,UF,CT,I),whereUFislocalFloquetevolutionoperatorwithperiodT,CTisunifiedtimescalecostofoneFloquetstep.Wegivecomputationaluniversesensedefinitionsofdiscretetimetranslationsymmetryandspontaneousbreaking,andfromcomplexitygeometryandinformationgeometryperspectives,characterizetimecrystalphase:onanyinitialstatefamilysatisfyinglocalobservabilityandboundedenergydensityassumptions,existslocalobservableOwhoseexpectationvalueexhibitsstrictperiodmTratherthanTinlong−timeevolution,wherem≥2isinteger.Subsequently,onpreviouslyconstructedcausaldiamondchainandNull−−Modulardoublecoverstructure,weintroducecyclicchainofFloquet−−QCAtimecrystals:eachFloquetperiodcorrespondstoonecausaldiamond,formingdiamondchain{◊k}k\inZ.Onthischain,wedefineforeachperiodmodulo−2timephaselabelϵk\inZ2inducedbyscatteringphaseincrement,constructNull−−ModulardoublecoverD\toDofdiamondchain.Weprove:existenceofperiod−doublingtimecrystal(m=2)correspondspreciselytonontrivialZ2holonomyofFloquetcontrollooponNull−−Modulardoublecover,i.e.,closedFloquetcontrolloophasnoclosedliftedpathondoublecover,thusgivingexactcorrespondencebetweentimecrystalparityandNull−−Modularholonomy.Atengineeringlevel,weconsidertimecrystalreadoutandrobustnessunderfinitecomplexitybudget.Bycombiningunifiedtimescalefrequencydomainwithspectralwindowingerrorcontroltheory(PSWF/DPSS),weconstructclassof‘‘finite−orderwindowfunctionobservationoperators′′fortimecrystalreadout,prove:underconditionsthatFloquetgapexceedscertainthresholdandlocalnoisesatisfiesfinitecorrelationlengthassumption,samplingtimecrystalsignalwithDPSStypereadoutwindowinfinitestepscanrobustlydiscriminateperiod−doublingparitywithcomplexitybudgetN=O(Δ−2log(1/ε))whileerrorprobabilitynotexceedingε,whereΔisFloquetquasienergygap.Finally,weviewtimecrystalsas‘‘discretephaselockers′′ofunifiedtimescale:oncontrolmanifold(M,G),timecrystalphasecorrespondstoclassofFloquetcontrolloopswithZ2holonomy,givingspecialminimalworldlinefamilyintime−−information−−complexityjointvariationalprinciple.Wediscusspotentialexperimentalroleoftimecrystalsaslocalstandardsofunifiedtimescale,andcomplementaryrelationshipwithFRBphasemetrologyandδ--ring--AB scattering metrology.
Abstract We formulate a Floquet drive as parallel transport for a flat, operator-valued connection on the time--phase cylinder $\cM_*=\R_{>0}\times\Sone$, conveniently charted as the punctured complex-time plane $\kappa=t\e^{i\varphi}$. The Hamiltonian is the pullback $H=\hbar\dot\varphi\,\cA_\varphi$, the one-cycle propagator is the holonomy of the connection, and, provided the controller tracks the drive phase in real time, the propagator at fixed winding is exactly invariant under every orientation-preserving reparametrization of the phase. The binary drive of Yao \emph{et al.} embeds as a piecewise-constant connection, for which the identity reduces to pulse-area compensation. The geometric statement adds gauge covariance and isolates the winding class in $\pi_1(\cM_*)\simeq\Z$ as the only surviving discrete datum. In a period-qT discrete time crystal the holonomy cyclically permutes the q symmetry-broken branches, converting that winding class into a Zq observable whose response to integer winding defects is the root-of-unity moment $\E[\e^{2\pi iK/q}]$. We show that phase-locked cycle slips and free-running phase diffusion give distinguishable functional forms, exponential versus wrapped-Gaussian with a 4/π prefactor, and that finite controller bandwidth maps exactly onto residual pulse-area noise. The wrapped-Gaussian law is obtained in closed form for every q, with asymptotic prefactor (q/π)sin(π/q), and a systematic scheduler offset is shown to produce a deterministic, non-decaying square-wave signature that is distinguishable from both stochastic mechanisms. We quantify the statistical identifiability of the two stochastic mechanisms and show that discrimination requires coverage of the early-time crossover rather than precision on an asymptotic prefactor. Exact state-vector simulations up to twelve spins for q=2, and an eight-site Z3 clock chain for q=3, quantify each regime.
In the framework of Unified Time Scale, Matrix Universe THE\text{-MATRIX}, and Quantum Discrete Cellular Automaton Universe U_{qca}, we construct a systematic theory for Quantum Chaos and the Eigenstate Thermalization Hypothesis (ETH). The Unified Time Scale Mother Formula equation \kappa(\omega) =\varphi'(\omega)/\pi =\rho_{rel}(\omega) =(2\pi)^{-1}trQ(\omega) equation unifies 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 U_{mat} independent of specific Hamiltonian choices. The Universe as a QCA object equation U_{qca}=(\Lambda,\mathcal H_{cell},\mathcal A_{qloc},\alpha,\omega_0) equation describes discrete time evolution via local unitary automorphisms \alpha on a countable lattice, and reconstructs relativistic quantum field 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 finite region \Omega\subset\Lambda, the restricted finite-dimensional unitary operator U_\Omega satisfies Discrete Time ETH for its quasi-energy spectrum eigenstates with respect to any local operator O_X (support X\subset\Omega): for the vast majority of eigenstates \psi_n within an energy window, equation \langle\psi_n|O_X|\psi_n\rangle =\langle O_X\rangle_{micro}(\varepsilon_n) +\mathcal O(e^{-c|\Omega|}), equation and the squared average of off-diagonal matrix elements decays exponentially with volume, achieving eigenstate thermalization for local observables. 2. Under the axioms of "Postulated Chaos QCA" (finite propagation radius, translation symmetry, local gates generating high-order unitary designs, no extra extensive conserved quantities), the quasi-energy level statistics of the finite region U_\Omega converge after unfolding to the Wigner--Dyson distribution of the CUE random 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 Unified Time Scale: Under the unified time scale \tau, the "thermalization time scale" and the growth rate of local entropy density are jointly controlled by the energy shell average of the scale density \kappa(\omega) and the QCA light cone structure. We prove a "Unified Time--ETH--Entropy Growth Theorem": for any family of finite-density local initial states, the entropy density increases monotonically with \tau 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 unified 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 unified Matrix--QCA Universe, rather than additionally introduced dynamical postulates. The appendix provides: rigorous definitions 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 specific predictions for numerical verification.
We propose a Floquet time crystal model that responds in arbitrary multiples of the driving period. Such an n-tuple discrete time crystal is theoretically constructed by permutations in a disordered spin chain and is well suited for experimental implementations. Transitions between these time crystals with different periods give rise to a novel phase of matter that we call subspace-thermal discrete time crystals, where states within subspaces of definite charges are fully thermalized at an early time. However, the whole system still robustly responds to the periodic driving subharmonically, with a period being the greatest common divisor of the original two periods. Existing theoretical analysis from many-body localization cannot be used to understand the rigidity of such subspace-thermal time crystal phases. To resolve this, we develop a theoretical framework for the robustness of DTCs from the perspective of the robust $2\ensuremath{\pi}/n$ quasi-energy gap. Its robustness is rigorously proved if the system satisfies a certain condition where the mixing length, defined by the Hamming distance of the symmetry charges, does not exceed a global threshold. Although whether the condition is satisfied in generic disordered systems is unclear, the rigorous proof for DTC properties applies beyond the models considered here and extends to other existing DTCs realized by kicking disordered MBL systems, where the condition is automatically satisfied and conventional MBL-DTCs can be regarded as a special case of the subspace-thermal DTC with the subspace dimension being one, thus offering a systematic way to construct discrete time crystal models. We also introduce the notion of DTC charges that allow us to probe observables that spontaneously break the time-translation symmetry in both regular discrete time crystals and subspace-thermal discrete time crystals. Moreover, our discrete time crystal models can be generalized to systems with higher spin magnitudes or qudits, as well as to higher spatial dimensions.
Construct unified framework with boundary as sole fundamental stage, gluing three mature but usually separate structures as three projections of same object: (i) Based on Birman--Krein--Friedel--Wigner--Smith framework, ``time translation operator'' characterized by scattering phase, spectral shift function, Wigner--Smith time delay matrix; (ii) Based on Tomita--Takesaki modular theory and recent results on causal diamond/null surface modular Hamiltonians, Markov property, QNEC, Null--Modular do
In previous ``computational universe'' framework, universe axiomatized as discrete object U_{comp} = (X,T,C,I), upon which constructed discrete complexity geometry, discrete information geometry, control manifold (M,G) induced by unified time scale, task information manifold (S_Q,g_Q), and time--information--complexity joint variational principle. Unified time scale given by scattering master scale $ \kappa(\omega) = \varphi'(\omega)/\pi = \rho_{rel}(\omega) = (2\pi)^{-1}\tr Q(\omega) unifying phase derivative, spectral shift density, and Wigner--Smith group delay trace as single scale. However, this framework still remains mainly at ``theoretical geometry'' level, has not systematically given how to metrologically measure and calibrate unified time scale and computational universe structure in actual experiments. This paper, on basis of computational universe--unified time scale--spectral windowing readout, constructs cross-platform metrology paradigm using ``phase--frequency'' as sole observable, implementing it on two representative testbeds: cosmological-distance Fast Radio Burst propagation (FRB) and laboratory-scale \delta-ring--Aharonov--Bohm (AB) flux scattering. Core idea: from computational universe perspective, all observables realized through phase--frequency readout under unified time scale; FRB and \delta-ring scattering respectively provide cosmic-scale and laboratory-scale ``homologous readouts'', viewable as implementations of same metrology paradigm at different scales under unified time scale and complexity geometry. Main results of this paper: enumerate \item Under framework of categorical equivalence between computational--physical universes, introduce ``phase--frequency readout functor'' PhFr, sending any physically realizable computational universe object to metrology object containing only phase--frequency data. Prove PhFr compatible with unified time scale master scale: under traceable perturbation and wave operator completeness, PhFr output completely determined by \kappa(\omega) and finite spectral--scattering invariants. \item For FRB, construct ``vacuum polarization windowing upper limit'' model: window FRB frequency-domain phase using PSWF/DPSS type window functions, prove under fixed complexity budget and cosmological distance constraints, any unified time scale variation \delta \kappa(\omega) contribution to FRB phase residual can be bounded by strict upper bound; if observed residual below this bound, obtain unified time scale type upper limit on vacuum polarization or other new physics. \item For \delta-ring--AB flux scattering, restate equivalence between spectral quantization equation f(k,\alpha_\delta,\theta) = \cos(kL) + (\alpha_\delta/k)\sin(kL) - \cos\theta = 0 and ``amplitude-corrected phase closure'' \cos\gamma(k) = |t(k)|\cos\theta and prove under computational universe--control manifold perspective: under spectral observation \{k_n(\theta)\} at fixed (L,\theta), \delta--coupling strength \alpha_\delta and AB flux \theta are identifiable in non-pathological domain (Jacobian full rank), usable as ``laboratory ruler'' for unified time scale--phase metrology. \item Under unified time scale--spectral windowing readout framework, embed FRB and \delta-ring scattering in same ``phase--frequency metrology universe'', prove existence of ``cross-platform scale unification condition'': when FRB phase residual and \delta-ring scattering spectral shift both explained by same \kappa(\omega) model, their windowed readouts belong to same equivalence class on appropriate PSWF/DPSS space, thus can calibrate and consistency-test unified time scale through joint fitting. \item Embed above phase--frequency metrology structure into time--information--complexity variational principle, formalize ``choosing FRB/\delta-ring window functions and control parameters'' as variational problem on joint manifold, give variational conditions for ``simultaneously using cosmic-scale and laboratory-scale phase--frequency readouts to maximize unified time scale identifiability under finite complexity budget''. enumerate This paper thus completes experimental implementation design of ``phase--frequency unified metrology'' within computational universe framework: FRB and \delta$-ring scattering become two-end testbeds of unified time scale and complexity geometry, PSWF/DPSS window functions become natural tools for error control, both jointly constructing cross-scale, cross-platform, yet completely unified phase--frequency metrology system under computational universe perspective.
### Abstract (Part One: Crystalline Time → Quantum Kinematics) We present Part One of a two-part programme that connects a discrete, non-split “crystalline time” framework to the safest core of quantum kinematics, with every nontrivial statement backed by simulation certificates. The aim of Part One is intentionally narrow: we build (i) a geometric lexicon for phases and symplectic structure, (ii) a minimal and assumption-transparent route to the Schrödinger equation via unitary time translation (Stone’s theorem), and (iii) the Heisenberg uncertainty principle via symplectic commutators and the Robertson inequality. Born weights and the Dirac operator are deferred to Part Two, where the structural and interpretive inputs are heavier. Section 1 introduces the geometric dictionary: “phases” are treated as holonomies of a U(1) connection on a complex line bundle, and classical mechanics is formulated on phase space as a symplectic manifold (T*Q, ω = dp ∧ dq). On the framework side, crystalline time is encoded by a non-split extension 0 → Z24 → Z48 → Z2 → 0 with the decomposition K = 2k + p (k ∈ Z24, p ∈ Z2). The fine tick K → K+1 necessarily toggles p, while the coarse tick K → K+2 acts as k → k+1 and preserves p. The leaf observer is p-blind; we model this by a projection/coarse-graining Πp over the Z2 fiber, yielding an effective open-system dynamics on the leaf. Time-kinetic support is regulated by bridge kernels S and J through Kt = Zt(S + εJ), with ε controlling whether the leaf sector is hyperbolic (ε > 0), constrained/elliptic (ε = 0), or ghost-like (ε < 0). All of these statements are verified by discrete operator checks at machine precision. Section 2 derives Schrödinger from one minimal physical principle: strongly continuous unitary time translations U(t). By Stone’s theorem, U(t) = exp(−iHt/ħ), giving iħ ∂t|ψ(t)⟩ = H|ψ(t)⟩. We then show how this “safe tower” is realized inside the framework: define the leaf embedding E+ (pure-even state), the leaf projection P+ (extract ψ+), and the fine-step unitary U on Z48. The cadence-dependent leaf amplitude map A_eff(m) = P+ U^m E+ exhibits an exact even/odd dichotomy (confirmed by closed-form derivation and machine-precision verification). For even cadences m = 2r (i.e., N|48 with m = 48/N even), A_eff(m) is a unitary kernel shift S^r: purity remains 1 and entropy remains 0. For odd cadences m = 2r+1, A_eff(m) = S^r (I+S)/2 is contractive: average retention Tr(A†A)/24 = 0.5, a near-zero singular mode appears, and entropy is produced under the projected density-matrix channel. Numerically, starting from a pure-even state, one fine tick generates substantial odd support (||ψ−|| = 0.6146). A divisor-of-48 scan shows that N=48 (m=1) and N=16 (m=3) produce identical entropy/purity trajectories (up to translation), while all even cadences (e.g., N=24,12,8,6) remain unitary and entropy-free. We further connect the odd-tick projected map to a continuum generator: in the long-wavelength, diffusive scaling limit the comoving odd-tick dynamics converges to ∂t φ = D ∂x^2 φ, with D = Δx^2/(8Δt). This diffusion limit is validated numerically with relative L2 errors ≈ 9.3×10^−6 (smooth profile) and ≈ 1.0×10^−5 (delta profile) at t ≈ 0.01. Finally, the ε-controlled cone speed c^2 = Zx/(2εZt) is confirmed by right-moving packet propagation with v_meas matching c_pred to machine precision for ε ∈ {1, 0.3, 0.1, 0.03}. Section 3 derives uncertainty from symplectic structure. The symplectic backbone motivates the canonical commutator [x, p] = iħ, with discrete phase-space realizations through Weyl pairs. The Robertson inequality ΔAΔB ≥ (1/2)|⟨[A, B]⟩| then yields the Heisenberg bound ΔxΔp ≥ ħ/2. On the leaf kernel Hilbert space, a Gaussian wavepacket saturates the bound numerically: ⟨[Q, P]⟩ ≈ (−9.37×10^−17) + 0.9999999998486 i, ΔQΔP ≈ 0.4999999999337, and (1/2)|⟨[Q, P]⟩| ≈ 0.4999999999243. In parallel, the unitary translation group built from the kernel momentum operator satisfies ||U(t+s) − U(t)U(s)|| ≈ 2.06×10^−14 and ||U(Δx) − S|| ≈ 2.17×10^−14, and a finite-difference Schrödinger check yields a relative residual ≈ 1.09×10^−6 at dt = 10^−6. Part One therefore closes a reproducible bridge: (i) crystalline time plus a p-blind projection yields an exact cadence-dependent split into unitary (even) and contractive/entropic (odd) leaf dynamics, (ii) the unitary kernel sector realizes Stone’s theorem and Schrödinger evolution explicitly, and (iii) the symplectic backbone yields commutators and uncertainty with quantitative saturation. Part Two will address Born weights and the Dirac operator, where the required axioms/structures are heavier and must be handled with the same “Court” discipline.