The periodic Sylvester tensor equation (PSTE) arises in multidimensional periodic systems. This paper proposes a tensor finite-time convergent gradient neural network (Tensor-FTCGNN) with a global adaptive gain for solving a class of PSTEs. Using the Einstein product, the periodically coupled tensor equations are formulated as a unified linear operator equation on a product tensor space, and the corresponding adjoint operator and global gradient are derived. Tensor linear operator theory and Lyapunov analysis are employed to establish the global finite-time convergence of the ideal continuous-time model and to derive an explicit upper bound for its settling time. For practical numerical implementation, a regularized Tensor-FTCGNN is introduced, which reaches an $ϵ$-dependent high-accuracy region in finite time and subsequently converges exponentially to the exact solution. Numerical experiments compare the proposed regularized model with representative gradient neural network (GNN) models, verify the theoretical finite-time upper bound through the observed Phase-Ⅰ entrance times, and evaluate its convergence behavior over the tested tensor sizes. Finally, the proposed framework is applied to cooperative periodic trajectory reconstruction for a multiarm robotic system, illustrating its potential for multidimensional robotic applications.