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

Time CrystalsNullModular Z2 Holonomy Unication: From Floquet and Lindblad to Bulk-Integral BF Relative Cohomology Criterion Haobo Ma 1 Wenlin Zhang 2 1 Independent Researcher 2 National University of Singapore Abstract Construct theoretical chain unifying discrete/continuous time crystals with NullModular Z2 holonomy, bulk-integral Z2 BF choice, and relative cohomology invariant. Closed system side, provide rigidity and stability of prethermal discrete time crystals in exponentially long time windows via high-frequency FloquetMagnus and LiebRobinson constraints; under strong disorder provide necessary and sucient structure for π spectral pairing and eigenstate time crystalline order. Open system side, establish spectral criterion for limit cycle time crystals on peripheral spectrum of single-period CPTP channel. Quasiperiodic drive side, construct "temporal quasicrystal" group representation via nite image of Zk time translation group. Above four classes of phenomena interface with unied topologicalalgebraic skeleton: Z2/Zm holonomy and relative cohomology class [K]∈H2(Y, ∂Y ;Z2) of bulk-integral Z2 BF top term; under small causal diamond threshold, if satisfying modularscattering mod-two alignment and parameter two-cycle detectability and generation, then "geometryenergytopology" triplet equivalent, specically [K]=0 ⇐⇒ time crystal "anomaly" vanishes on allowed loops and two-cycles. This paper simultaneously provides Z2 ngerprints for three solvable families ( δ potential, AharonovBohm, topological superconductor endpoint) and engineering schemes with error budgets for superconducting qubits, Rydberg gases, and trapped ions. Core NullModular double cover and BF relative cohomology criterion taken from authors' existing unied principle and restated and proved in time crystal context. Keywords : Discrete time crystal; prethermalization and many-body localization; open system limit cycle; temporal quasicrystal; topological time crystal; π spectral pairing; Z2/Zm holonomy; bulk-integral Z2 BF; relative cohomology; small causal diamond 1 Introduction & Historical Context Spontaneous breaking of time translation symmetry rigorously negated in equilibrium systems, forcing physical carriers of time-ordered phases toward non-equilibrium drive and open dynamics. In periodically driven many-body systems, discrete time crystals characterized by subharmonic response rigidity, long-range temporal correlations, and characteristic spectral ngerprints; subsequent branches of eigenstate ordering, prethermal longevity, dissipative limit cycles, and topological (logical) time crystals constitute cross-platform experimental systems. On other hand, we developed invariants centered on square root determinant branching and mod-two holonomy along geometricinformationscattering unied thread, elevating them to necessary and sucient criterion via bulk-integral Z2 BF relative cohomology language. This paper rigorously bridges these two threads, proving: time crystal " π /unit root" phenomenology at unied criterion level is Z2/Zm 1 holonomy and [K]∈H2(Y, ∂Y ;Z2) non-triviality; conversely, geometryenergy criterion (rst and second order layer on small causal diamond) under alignment threshold implies trivializatio n of such invariants, thereby providing common structure penetrating closed/open/topological/multifrequency. 2 Model & Assumptions 2.1 Closed System (FloquetMBL/Prethermalization) Local many-body system on lattice Λ , periodic drive H(t+T) = H(t) . Floquet unitary F= Texp(−iRT 0Hdt) generates discrete time translation. High-frequency limit ω= 2π/T ≫J admits quasilocal eective Hamiltonian H∗ with exponentially small truncation error; under strong disorder H0 supports l -bit structure. 2.2 Open System (Periodic Lindblad) Density matrix evolution ˙ρ=Lt(ρ) with Lt+T=Lt . Single-period quantum channel E=Texp(RT 0Ltdt) peripheral spectrum determines long-time limit cycle. 2.3 Multi-Frequency Quasiperiodic Mutually irrational frequency family {ωi}k i=1 denes time translation group Zk ; high-frequency prethermalization threshold miniωi≫J ensures quasilocal H⋆ and nite image group representation. 2.4 Topological Time Crystal (Logical Subspace) Stabilizer code (surface code) periodic engineering makes Flogical ≃XLe−iHtop ∗T , non-local logical operators as natural order parameters. 2.5 NullModular and Bulk-Integral BF (Relative Cohomology) Working space Y=M×X◦ , where M small causal diamond domain or more general local spacetime patch, X◦ parameter domain removing discriminant set D . Dene K=π∗ Mw2(TM) + X j π∗ Mµj⌣ π∗ Xwj+π∗ Xρ(c1(LS)) ∈H2(Y;Z2), using [K]∈H2(Y, ∂Y ;Z2) under boundary trivialization and relative cohomology lift. Z2/Zm holonomy computed as mod-two (or unit root) value of square root determinant branch, stabilizing closed paths per "small semicircle/fold-back" rule. 3 Main Results (Theorems and Alignments) Theorem 1 (A: Rigidity and Exponential Lifetime of Prethermal DTC) . Under ω≫J and piecewise nearπ "symmetric kick" conditions, quasilocal unitary U∗ and symmetry element X2=⊮ exist making F=U∗e−iH∗TX U† ∗+ ∆,|∆| ≤ Ce−cω/J . 2 Any local observable O odd under X exhibits 2T subharmonic locking, remaining locked for t≲τ∗∼ecω/J , maintaining Lipschitz stability against small perturbations. Theorem 2 (B: π Spectral Pairing and Eigenstate Order in MBLDTC) . On strongly disordered chain quasilocal unitary U exists making F≃˜ Xe−iHMBLT , spectrum exhibits π pairing, inducing state-independent 2T subharmonic response. Theorem 3 (C: Spectral Criterion for Open System Limit Cycle) . If single-period channel E peripheral spectrum {e2πik/m} with spectral radius <1 elsewhere, then almost all initial states converge to periodmT limit cycle attractor family, constituting m -subharmonic dissipative time crystal. Theorem 4 (D: Multi-Frequency "Temporal Quasicrystal" Group Representation) . Under prethermalization threshold miniωi≫J , nite image of time translation group Zk produces multiple incommensurate subharmonic peaks, forming temporal quasicrystal. Theorem 5 (E: Topological Time Crystal: Non-Local Order and Topological Entanglement) . In logical subspace, non-local loop operators exhibit rigid multiple-period response, consistent with nonzero topological entanglement entropy term. Theorem 6 (F: Unied TopologicalCohomology Criterion) . Under boundary trivialization, relative generation and detectability threshold, following equivalent: [K] = 0 ⇐⇒ for all allowed loops γ:ν√detpS(γ) = +1 and all allowed two-cycles γ2:⟨ρ(c1(LS)),[γ2]⟩= 0. When H2(X◦, ∂X◦) = 0 , above equivalent to mod-two criterion on loops only. Theorem 7 (G: GeometryEnergy ⇒ Topological Triviality) . Under small causal diamond threshold, relative generationdetection, and modularscattering mod-two alignment, if rst order layer gives Gab + Λgab = 8πGTab and second order relative entropy δ2Srel =Ecan ≥0 , then above unied topologicalcohomology criterion holds, i.e., implies [K]=0 and all Z2 holonomies trivial. 4 Proofs 4.1 Prethermalization and π Locking (Theorems A/B) Take FloquetMagnus expansion F= exp{−iTPn≥0Ωn} , truncate at optimal order n∗∼ω/J dening H∗ . Via LiebRobinson and local expansion series renormalization obtain |F−e−iH∗T| ≤ Ce−cω/J . Piecewise nearπ kick UX produces X , obtaining structural decomposition in quasilocal unitary U∗ representation. Under strong disorder UH0U†=f({τz i}) nearly commutes with ˜ X= UXU† , spectrum exhibits π pairing; arbitrary initial state expanded in paired subspace, obtaining state-independent 2T subharmonic response. 4.2 Peripheral Spectrum and Limit Cycle (Theorem C) Perform JordanRiesz decomposition for CPTP E . If peripheral spectrum m unit roots with spectral gap elsewhere, then En exponentially converges on each residue class to periodm cyclic attractor; convergence rate controlled by Liouvillian spectral gap. 3 4.3 Z2/Zm Holonomy and Relative Cohomology (Theorem F) Work with Z2 coecients taking relative cohomology. Bulk-integral Z2 BF action IBF[a, b]=iπZ(Y,∂Y ) b ⌣ δa+ iπZ(Y,∂Y ) b⌣K + iπZ∂Y a ⌣ b, after gauge transformation and boundary term cancellation, summing over [a]∈H1(Y, ∂Y ) , [b]∈Hd−2(Y, ∂Y ) , using nite abelian group character orthogonality obtains partition function projection Ztop ∝δ([K]) , i.e., [K]=0 . By PoincaréLefschetz duality, [K]=0 if and only if Kronecker pairing vanishes for all allowed relative two-cycles [S] ; when H2(X◦, ∂X◦)=0 , reduces to loop mod-two criterion only. 4.4 GeometryEnergy Implies Topological Triviality (Theorem G) Under small causal diamond threshold (Hadamard, no conjugate points, corner prescription, ∇aTab = 0 , xed temperature scale) and invertiblestable hypothesis for weighted null ray transformation, family constraint Rw(λ)(Rkk −8πGTkk) dλ= 0 with Radon-type closure implies Rkk = 8πGTkk ; null cone characterization and Bianchi identity upgrade to tensor equation, obtaining rst order layer Gab + Λgab = 8πGTab . Corner prescription ensures covariant phase space symplectic ux closure, δ2Srel =Ecan ≥0 . If closed path γ exists making ν√detpS(γ) = −1 , then by modularscattering mod-two alignment, construct linear functional corresponding to holonomy in covariant phase space embedding into quadratic form kernel, obtaining Ecan[h, h]<0 contradiction, thus implying all allowed loop holonomies trivial, further by relative generationdetection obtaining [K]=0 . 5 Model Apply 5.1 Z2 Fingerprints for Solvable Families (i) 1D δ potential: Select small loop around complex pole, H1 2iS−1dS=π⇒ν√detpS=−1 . (ii) 2D AharonovBohm: Flux crossing half-ux α=1 2 gives deg(detpS|γ)=1⇒ν=−1 . (iii) Topological superconductor endpoint (Class D/DIII): sgn detpr(0) or sgn Pf r(0) ip synchronizes with ν√detpr . Three families after removing discriminant set have H2(X◦, ∂X◦)=0 , requiring loop mod-two criterion only. 5.2 Platform Mapping and Observables Superconducting qubit 2D array: Logical loop operator spectrum exhibits ω/2 peak only in nonlocal channel, accompanied by non-zero topological entanglement entropy; Rydberg gas: Unit roots in quantum channel peripheral spectrum consistent with uorescence autocorrelation limit cycle; trapped ions: ω enhancement brings exponential lifetime growth with rigid frequency position not drifting; multi-frequency drive: Incommensurate peak positions correspond to nite image of Zk . 6 Engineering Proposals 6.1 Prethermalization Window and Pulse Synthesis Choose ω making e−cω/J ≪ε (gate noise amplitude), ensuring τ∗∼ecω/J covers 102−103 cycles; piecewise sequence implements nearπ ip at τx to amplify 2T locking. 4 6.2 Open System Spectral Gap Engineering Construct Liouvillian spectral gap ∆Liouv via pumpingdecoherence ratio G/κ , suppressing multistable wandering; sample peripheral eigenvalues and limit cycle period around steady-state working point. 6.3 Multi-Frequency Temporal Quasicrystal Two to three mutually irrational frequencies, avoiding accidental integer period recurrence; collect spectrum using incommensurate peak positions to identify nite image; reconstruct unit root values via parameter closed paths. 6.4 Experimental Readout of WilsonLoop Amplitudephase joint scan forming closed path; distinguish ±1 via discrete Fourier peaks in Ramsey/correlation functions; for Zm t unit roots using phase grid. 7 Discussion (risks, boundaries, past work) Boundaries and risks include: absorption-induced locking collapse outside high-frequency window; MBL stability limited in high dimensions and long-range interactions; open system nonMarkovianity causes phase wandering; topological time crystal non-local readout systematic sensitivity to leakage and crosstalk. At unied criterion level, H2 channel detectability requires allowed two-cycle generation of relative two-cohomology; if platform parameter domain two-skeleton insuf- cient, obtains only necessary non-sucient criterion. Compared to existing work, this paper's increment: using bulk-integral Z2 BF relative cohomology class [K] and mod-two holonomy as single invariant , uniformly characterizing closed/open/topological/multi-frequency four classes of time crystals, establishing implication chain between geometryenergy and holonomycohomology under small causal diamond variational threshold. 8 Conclusion Time crystal " π /unit root" phenomenology uniformly characterized by Z2/Zm holonomy and bulkintegral Z2 BF relative cohomology class; when relative generationdetection and modularscattering mod-two alignment hold, geometryenergy criterion on small causal diamond implies topological cohomology trivialization. This structure simultaneously supports prethermal DTC, MBL eigenstate order, open system limit cycle, and topological time crystal, providing group representation and experimental readout for multi-frequency temporal quasicrystals. This framework provides unied theoryengineering channel for cross-platform time-frequency devices, robust storage, and topological logical operations. Acknowledgements, Code Availability Thank publicly available results and platform data establishing experimental background. This paper does not rely on proprietary code; scripts for relative cohomology pairing, Z2/Zm holonomy reconstruction, and peripheral spectrum tting can be reproduced using standard numerical tools following algorithmic steps in appendices. 5 References Select representative theoretical and experimental works: time crystal no-go theorems, Floquet DTC denition, prethermalization upper bounds, eigenstate time crystalline order, dissipative time crystals, topological time crystals and temporal quasicrystals; as well as NullModular double cover and bulk-integral Z2 BF unied principle on geometricinformationscattering thread. A Rigorous Prethermalization Upper Bound and Exponential Lifetime Assume |H(t)| ≤ J , ω≫J . FloquetMagnus Ω0=1 TZT 0 H, Ω1=1 2TiZZ0<t1<t2<T [H(t2), H(t1)] dt1dt2, . . . truncate at n∗∼αω/J , dene H∗=Pn≤n∗Ωn . Via nested commutator tree renormalization prove |F−e−iH∗T| ≤ Ce−cω/J , |d dt⟨H∗⟩| ≤ C′e−cω/J . Piecewise nearπ kick UX makes F≈Xe−iH∗T+O(ϵ) + O(e−cω/J ) , thus for X -odd O⟨O(nT)⟩ ≈ (−1)n⟨O(0)⟩+O(ϵ) + O(e−cω/J ) . Frequency domain exhibits ω/2 locking peak, peak position rigid against parameter perturbations. B MBL π Pairing and Eigenstate Order Unitary U exists making UH0U†=f({τz i}) , nearπ kick under U transformation yields quasilocal ˜ X . If F≃˜ Xe−iHMBLT , then for eigenstate |ψ⟩F|ψ⟩= e−iET ˜ X|ψ⟩, F(˜ X|ψ⟩)=e−i(ET+π)|ψ⟩ , spectrum exhibits π pairing. Arbitrary initial state expanded in paired subspace yields stateindependent 2T subharmonic response. C CPTP Peripheral Spectrum and Limit Cycle Let σ(E) = {λj} . If |λj|<1 for all non-peripheral modes, peripheral modes {e2πik/m} , then mutually disjoint cyclic invariant subspaces exist, making En project arbitrary initial state to periodm limit cycle; convergence rate controlled by ∆Liouv = 1 −max|λj|<1|λj| . D Z2/Zm Holonomy: Mod-Two Robustness of Spectral Flow and Intersection Number Denote discriminant set D⊂X◦ as threshold/embedded eigenvalue or −1 eigenvalue submanifold. For closed path γ stabilized per "small semicircle/fold-back" rule, dene mod-two intersection number I2(γ, D) . Modied determinant detp change only alters quantized phase integer winding number, mod-two projection invariant; partial-wave truncation N→ ∞ and relative trace-class renormalization preserve ν√detpS= (−1)I2(γ,D) . E Bulk-Integral Z2 BF Relative Cohomology Derivation On (Y, ∂Y ) with Z2 coecients construct 6 IBF[a, b]=iπZb ⌣ δa+ iπZb ⌣ K + iπZ∂Y a ⌣ b. Under gauge transformation a7→ a+δλ0 , b7→ b+δλd−3 boundary term cancels, action welldened. Summing over [a] and [b] , using nite abelian group character orthogonality obtains Ztop ∝ δ([K]) ; PoincaréLefschetz duality gives [K] = 0 ⇐⇒ for all [S]∈H2(Y, ∂Y ;Z2) have ⟨K, [S]⟩= 0 . F GeometryEnergy ⇒ Holonomy Triviality: Alignment and Contradiction Method Under corner prescription and covariant phase space framework, closed path modular holonomy de- nes bounded linear functional embedding into quadratic form kernel. If γ exists making ν√detpS(γ) = −1 , functional under mod-two projection provides negative direction, constructing h making Ecan[h, h]< 0 , contradicting second-order non-negativity; thus all allowed closed path holonomies (+1) . Combined with relative generation and detection, implies [K]=0 . G Deformation Retraction and H2= 0 for Three Solvable Families Parameter domains of δ potential, AB, and endpoint scattering after removing discriminant set deformation retract to one-dimensional skeleton, thus H2(X◦, ∂X◦)=0 . Therefore unied criterion reduces to mod-two condition on loops; in this case, constructing reference closed path transverse to D veries [K] = 0 necessary and sucient condition. H ExperimentAlgorithm Checklist (Pseudocode Level) 1. Data acquisition and detrending: Time series O(tn) remove drift via polynomial regression; 2. Peak and locking metric: Discrete Fourier, t peak width Γ and locking ratio R ; 3. Unit root readout: Parameter closed path sample phase, discriminate ±1 or m -th unit root; 4. Relative cohomology test: Select generating family loops/two-cycles, evaluate ν√detpS and ⟨ρ(c1),[γ2]⟩ table, verify all-zero to determine [K]=0 . 7