Time Crystals and Null--Modular $\mathbb{Z

Time Crystals and Null–Modular Z2Holonomy under Unified Time Scale: Floquet–QCA Time Crystals, Topological Parity, and Engineering Implementation in Computational Universe Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 24, 2025 Abstract On foundation of computational universe axiomatic framework Ucomp = (X, T,C,I) and unified time scale master scale κ(ω) = φ′(ω)/π =ρrel(ω) = (2π)−1tr Q(ω) this paper constructs fully discrete time crystal theory, unifying it with Null– Modular Z2holonomy and time–information–complexity joint geometric structure. We first introduce, on computational universe implementation of reversible quantum cellular automaton (QCA), Floquet–QCA object UFQCA = (X, UF,CT,I), where UFis local Floquet evolution operator with period T,CTis 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 Owhose expectation value exhibits strict period mT rather than Tin long-time evolution, where m≥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 {♢k}k∈Z. On this chain, we define for each period modulo-2 time phase label ϵk∈Z2induced by scattering phase increment, construct Null–Modular double cover e D→Dof diamond chain. We prove: existence of period-doubling time crystal (m= 2) corresponds precisely to nontrivial Z2holonomy 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 1 spectral windowing error control theory (PSWF/DPSS), we construct class of “finiteorder 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(∆−2log(1/ε)) while error probability not exceeding ε, where ∆ 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 Z2holonomy, 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 δ–ring–AB scattering metrology. Keywords: Computational universe; Unified time scale; Quantum cellular automaton; Floquet time crystal; Null-Modular double cover; Z2holonomy; Spectral windowing readout; DPSS 1 Introduction Time crystals initially proposed as phase spontaneously breaking time translation symmetry: system’s ground state or steady state exhibits nontrivial periodic structure in time. Although original “continuous time crystal” idea constrained in strict equilibrium, in periodically driven systems (Floquet systems), Floquet time crystals spontaneously breaking discrete time translation symmetry actually realized. In these systems, time translation group Zsymmetry spontaneously broken to mZ, manifested as observable response to complete Floquet period Thaving superperiod mT, commonly m= 2 period-doubling time crystals. In previous works of this series, we constructed “unified time scale–computational universe” theory at higher level, including: 1. Computational universe axiomatic system Ucomp = (X, T,C,I), viewing universe as reversible evolution on discrete complexity graph; 2. Unified time scale master scale κ(ω) = φ′(ω)/π =ρrel(ω) = (2π)−1tr Q(ω), unifying scattering phase derivative, spectral shift density, and group delay trace as single time scale density; 3. Control manifold (M, G) induced by unified time scale and complexity geometry; 4. Causal diamonds, boundary computation operators, and causal diamond chains {♢k}; 5. Null–Modular double cover and Z2holonomy constructed on diamond chains, selfreference parity and topological complexity; 6. Time–information–complexity joint variational principle and multi-observer consensus geometry. 2 In this framework, time no longer external parameter, but embodiment of unified time scale in scattering–complexity geometry; time direction, time parity, and self-reference structure embodied through Null–Modular double cover and Z2holonomy. Core questions of this paper: 1. How to rigorously define Floquet–QCA time crystals in purely discrete framework of computational universe–unified time scale, giving their geometric–topological characterization? 2. How does period-doubling parity of time crystals relate to Null–Modular Z2holonomy? 3. How to perform stable readout and engineering implementation of time crystals under finite complexity budget? We will see time crystals naturally realized as class of phases in Floquet–QCA in computational universe, Null–Modular double cover provides intrinsic Z2topological invariant, spectral windowing readout provides optimal solution for their observation under finite complexity budget. 2 Preliminaries: Computational Universe, Unified Time Scale, and Floquet–QCA 2.1 Computational Universe and QCA Implementation Recall computational universe object Ucomp = (X, T,C,I), where Xcountable configuration set, T⊂X×Xone-step update relation, Csinglestep cost, Itask information quality function. Standard abstraction of reversible QCA: for lattice site set Λ and finite-dimensional Hilbert space Hxat each site, global Hilbert space H=Nx∈ΛHx, reversible QCA is local unitary operator U:H → H satisfying local causality constraints. In computational universe, view configuration x∈Xas label of some normalized basis vector |x⟩ ∈ H, one-step update relation defined by (x, y)∈T⇐⇒ ⟨y|U|x⟩ = 0 single-step cost C(x, y) given by physical time required to execute Uor its local decomposition once under unified time scale. 2.2 Unified Time Scale and Floquet Evolution On physical side, consider periodically driven system with time-dependent Hamiltonian H(t+T) = H(t), corresponding Floquet evolution operator UF=Texp −iZT 0 H(t) dt, 3 whose eigenvalues e−iεαT,εαare quasienergies. In unified time scale–scattering framework, view UF(ω) as frequency-domain scattering– evolution operator, frequency ωdependence embodied through drive spectrum and system response. For each Floquet period, define local group delay matrix QF(ω) = −iUF(ω)†∂ωUF(ω), whose trace gives local unified time scale density increment κF(ω) = (2π)−1tr QF(ω). In computational universe, we concern “each Floquet period as one causal diamond” discrete version, constructed in Section 3. 2.3 Basic Definition of Floquet Time Crystals In general Floquet system, time translation group Zaction n7→ n+ 1, corresponding to iteration Un F. Time crystal is spontaneous breaking of this symmetry: Definition 2.1 (Floquet Time Crystal, Physical Side).In periodically driven system, if exists local observable Oand initial state family {ρ0}such that for almost all ρ0, expectation sequence ⟨O⟩n= tr(ρ0U†n FOUn F) exhibits strict period m > 1 in long-time limit, i.e., ⟨O⟩n+m=⟨O⟩n, and satisfies no shorter period, system called in Floquet time crystal phase of period mT. Typical case m= 2 time crystal. We reformulate this concept in QCA–computational universe framework. 3 Floquet–QCA Time Crystals in Computational Universe 3.1 Floquet–QCA Object Definition 3.1 (Floquet–QCA Computational Universe).A Floquet–QCA computational universe object is quadruple UFQCA = (X, UF,CT,I), where: 1. Xconfiguration set, as normalized basis vector labels of global Hilbert space H; 2. UF:H → H local Floquet evolution operator corresponding to drive period T; 3. CT:X×X→[0,∞] complexity cost of one Floquet step, satisfying CT(x, y)>0 if ⟨y|UF|x⟩ = 0; 4. I:X→Rtask information quality function. 4 One Floquet evolution step represented on event layer E=X×Zas (x, n)7→ (y, n + 1),⟨y|UF|x⟩ = 0. Complexity cost viewable as integral of unified time scale over single period. 3.2 Discrete Time Translation Symmetry and Breaking In computational universe, view UFas generator of “time translation one step”. For observable O(e.g., local operator Oxacting only on finite region), its discrete time evolution O(n) = U†n FOUn F. For initial state ρ0(viewable as density operator), observation sequence ⟨O⟩n= tr(ρ0O(n)). Definition 3.2 (Floquet Time Crystal in Computational Universe).In Floquet–QCA computational universe, if exists local observable O, integer m≥2, and initial state family R0(satisfying finite density and finite correlation length conditions) such that: 1. For almost all ρ0∈ R0, exists sufficiently large n0such that for all n≥n0 ⟨O⟩n+m=⟨O⟩n, 2. No 1 ≤m′< m exists making same condition hold, then UFQCA called in time crystal phase of period mT. In particular, when m= 2, called period-doubling time crystal. 3.3 Floquet Spectrum and Quasienergy Band Structure Under finite volume or appropriate boundary conditions, UFhas eigendecomposition UF|ψα⟩= e−iεαT|ψα⟩, where εα∈(−π/T, π/T] are quasienergies. Time crystal existence closely related to “symmetry splitting structure” in quasienergy band structure: e.g., in m= 2 case, exists two bands with quasienergies differing by π/T, making coherent superposition in evolution undergo sign flip every two periods. Formally, can adopt structure projecting to subspaces HA,HBsatisfying U2 F|ψ⟩ ≈ e−i2εT |ψ⟩, and UFexchanges HAwith HB. More importantly, in computational universe–complexity geometry, we can translate phase structure of Floquet spectrum to Null–Modular Z2holonomy on causal diamond chains, developed in next section. 5 4 Null–Modular Z2Holonomy and Time Crystal Parity This section constructs Floquet–QCA time crystal implementation on causal diamond chains and Null–Modular double cover, proves correspondence between period parity and Z2holonomy. 4.1 Floquet Period as Causal Diamond Chain View single-period Floquet evolution as one causal diamond ♢F:  Diamond interior vertices are event set from some initial state layer to next layer within complexity budget T;  Diamond boundary are period initial/final events;  Diamond volume evolution given by local decomposition of UF;  Boundary operator K♢Fisomorphic to UFaction on boundary. If system repeatedly driven in time, forms Floquet diamond chain on event layer {♢F,k}k∈Z, where each ♢F,k corresponds to k-th Floquet period. For each ♢F,k, define average unified time scale increment ∆τk=ZΩF wF(ω)κF(ω) dω, in periodically stable case, ∆τk≡∆τproportional to physical period T. 4.2 Modulo-2 Time Phase and Z2Holonomy In third work on diamond chains and Null–Modular double cover, we defined modulo-2 time phase label ϵk∈Z2for each diamond, determined by scattering phase increment modulo 2π. In Floquet case, define effective phase increment per period as ∆φF= arg det UF, ϵF=⌊∆φF/π⌋mod 2. For time crystals, especially period-doubling phase, key structure not single-period phase but two-period closed loop U2 F, and its corresponding scattering phase and group delay. When constructing diamond chain double cover e DF→DF, let edge label of each Floquet period diamond be ϵF. Total parity of Nperiods on closed chain ΣN= N X k=1 ϵFmod 2 = NϵFmod 2. 6 For m= 2 time crystal, natural mechanism makes closed loop of two periods have nontrivial Z2holonomy: e.g., if ϵF= 1, after each period, index on double cover flips once, after two periods flips twice returning to original index, but overall topology of closed path exhibits nontrivial holonomy. More precisely, consider closed loop of Floquet control parameter path ΓF⊂ M (e.g., closed variation of drive protocol parameters in periodic driving), its Null–Modular double cover holonomy holZ2(ΓF)∈Z2 closely related to time crystal period parity. 4.3 Time Crystal Parity and Null–Modular Holonomy Correspondence Theorem 4.1 (Period-Doubling Time Crystal and Z2Holonomy).Let UFQCA be Floquet– QCA computational universe object satisfying: 1. Exists uniform volume limit and finite correlation length initial state family R0; 2. Floquet spectrum has quasienergy gap ∆F>0, exists two bands εα, εβsatisfying εβ≈εα+π/T; 3. On corresponding control manifold closed loop ΓF, Null–Modular double cover holonomy nontrivial, i.e., holZ2(ΓF) = 1. Then UFQCA in time crystal phase of period 2T; conversely, under above regularity conditions, if UFQCA in robust period-2Ttime crystal phase, corresponding Floquet control closed loop’s Null–Modular holonomy is nontrivial element. Proof sketch. “If” direction: Nontrivial holonomy means under two-period closed loop some global Z2quantity flips odd times, in Floquet spectrum corresponds to “parity switching” structure making Floquet subspaces exchange in one period, return to original position in two periods, causing expectation value to exhibit period-2Tflip structure. Using group theory and quasienergy band structure can prove exists local observable O satisfying time crystal condition. “Only if” direction: Period-doubling of time crystal means on Floquet–QCA worldline exists self-reference feedback condition making two periods globally close. Through previous correspondence between self-reference parity and Null–Modular holonomy, can prove corresponding closed loop holonomy nontrivial. Detailed proof in Appendix C. 5 Time Crystal Readout and Engineering Implementation Under Finite Complexity Budget This section discusses stable readout of time crystals under finite complexity budget, gives DPSS-based observation strategy and error upper bound. 7 5.1 Readout Model and Noise Consider local observable Oon local region Λ0⊂Λ, define discrete time sequence an= tr(ρ0U†n FOUn F), n = 0,1, . . . , N −1. In ideal time crystal phase, anexhibits period-mstructure when n≫1, typically m= 2 alternating sequence. With local noise and dissipation, writable as an=sn+ηn, where snideal time crystal signal, ηnnoise, assuming ηnzero-mean, finite correlation length Gaussian process. 5.2 DPSS Window Function Readout To extract period structure within finite complexity steps N, construct windowed Fourier spectrum ba(ω) = N−1 X n=0 wnane−iωn, where {wn}window function sequence. According to previous spectral windowing readout results, DPSS maximizes energy concentration under given length Nand frequency band W, minimizing worst-case error under finite sample number and frequency band constraints. For m= 2 time crystal, ideal signal main frequency at ω=π(normalized angular frequency). Can choose DPSS window function with bandwidth W≪π, focusing on spectral energy near ω≈π. 5.3 Error Upper Bound and Complexity Budget Let DPSS window function be w(0), corresponding eigenvalue λ0≈1, then under finite samples, error variance of main frequency energy estimation satisfies Var ba(π)≤σ2 η|w(0)|2, where σ2 ηnoise variance. To distinguish “with time crystal signal” and “without time crystal signal”, maintain certain signal-to-noise ratio |E[ba(π)]|2 Var(ba(π)) ≥c0, obtaining sample number requirement N=O∆−2log(1/ε), where ∆ Floquet quasienergy gap (controlling time crystal signal amplitude and dissipation time), εerror probability. Theorem 5.1 (Sample Complexity for Finite Complexity Time Crystal Discrimination). Under conditions: 8 1. Floquet–QCA time crystal has quasienergy gap ∆F>0; 2. Noise process {ηn}zero-mean, finite correlation length, bounded variance; 3. Readout window function DPSS basis sequence w(0) under appropriate bandwidth W; to discriminate whether period-2Ttime crystal signal exists with error probability not exceeding ε, required complexity steps Nsatisfies N≥C∆−2 Flog(1/ε), where Cconstant. Proof sketch. Combines DPSS energy concentration, Chebyshev inequality, and large deviation estimation, see Appendix D for details. 6 Unified Perspective: Time Crystals as Discrete Phase Locking of Unified Time Scale From unified time scale–control manifold–causal diamond chain–Null–Modular double cover global perspective, time crystals understandable as special “discrete phase lockers”: 1. Floquet control closed loop ΓFon control manifold (M, G) generates periodic time increment ∆τthrough unified time scale density κ(ω), has Z2holonomy on Null– Modular double cover; 2. Modulo-2 time phase labels ϵkon causal diamond chain {♢F,k}synchronize with Floquet control holonomy, forming “time parity locking”; 3. Time crystal phase existence means in time–information–complexity joint variational principle exists special minimal worldline family, simultaneously stable in “time direction–phase–self-reference parity” three dimensions. At experimental level, time crystals viewable as local standards of unified time scale: compared to FRB and δ–ring–AB scattering “passive measurements”, time crystals provide “actively generated time scale phase structure”. By jointly embedding time crystals, FRB, and δ–ring scattering in phase–frequency metrology universe, can perform consistency testing and joint calibration of unified time scale model across scales (laboratory– interstellar–cosmological) platforms. A Prototypical Existence Theorem for Floquet–QCA Time Crystals This appendix gives typical construction scheme and prototypical existence result for time crystal phases in QCA models. 9