  We reconstruct the spectral pairing (SP) theories to enable analytical
descriptions of eigenstate spatiotemporal orders in translation-invariant
systems without prethermal conditions. It is shown that the strong Ising
interactions and drivings alone stabilize a class of ``cat scar" eigenstates
with tunable patterns, which lead to {\em local} discrete time crystal (DTC)
dynamics. They exhibit Fock space localization and long-range correlations
robust against generic perturbations in a disorder-free scenario. In
particular, we introduce a symmetry indicator method to enumerate cat scars,
with which a set of unexpected inhomogeneous scar patterns are identified in
addition to the ferromagnetic scars found before. These scars enforce DTC
dynamics with rigid inhomogeneous patterns, offering a viable way to verify
underlying eigenstate properties experimentally. Further, we prove rigorously
that the strong Ising interactions enforce a selection rule for perturbations
of different orders, which imposes an exponential suppression of spin
fluctuations for Floquet eigenstates. Based on this property, three analytical
scaling relations are proved to characterize the amplitudes, Fock space
localization, and lifetime for DTC dynamics associated with cat scars. We
further provide two practical methods to check whether certain DTC phenomena
are dominated by single-spin dynamics or due to genuine interaction effects.
