Existence, stability, and synchronization of fractional spatiotemporal chaotic systems with purely integral boundary conditions

We establish a rigorous operator framework for drive–response time-fractional reaction–diffusion equations under two integral moment constraints. The constrained state and energy spaces are explicitly defined, and the diffusion operator is constructed by a coercive closed form. We prove density of its domain, compactness of its resolvent, identification of its square-root space, and characterization of its strong domain; an explicit trace-surjectivity argument distinguishes the essential moment constraints from the natural zero-flux relations selected by the form. Nonhomogeneous moments are treated through an explicit lifting. For nonlinearities depending on the state and its spatial derivative, fractional resolvent estimates yield unique global mild solutions in the energy space without treating the unbounded elliptic operator as bounded or restarting the Caputo problem. A spectral Galerkin approximation lemma justifies the passage from the finite-dimensional energy inequality to the mild solution. Then, the synchronization error satisfies a computable Mittag–Leffler estimate in which the gradient mismatch is explicitly absorbed by diffusion. A moment-preserving spectral–L1 discretization verifies the analytical gain threshold, moment conservation, fractional orders below $ 0.9 $, the theoretical decay bound, and temporal self-convergence approaching the expected order $ 2-\alpha $. An auxiliary three-mode benchmark embeds a globally Lipschitz saturated Lorenz vector field in the constrained partial differential equation (PDE) state space. Its double-scroll geometry, median $ 0 $–$ 1 $ statistic $ K_{01} = 0.993 $, finite-time sensitivity, and synchronization demonstrate compatibility with chaotic invariant-modal dynamics; no claim of high-dimensional diffusion-generated spatiotemporal chaos is made.