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

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.