Finiteness of spinfoam vertex amplitude with timelike polyhedra and the regularization of full amplitude

This work focuses on Conrady-Hnybida's four-dimensional extended spinfoam model with timelike polyhedra, while we restrict all faces to be spacelike. First, prove the absolute convergence of vertex amplitude when $\mathrm{SU}(1,1)$ boundary states are coherent or canonical basis, their finite linear combinations. Second, based and a proper prescription intertwiner space, construct an arbitrary cellular complex, taking into account sum over intertwiners internal polyhedra. We observe that is infinite for polyhedron has at least two future-pointing past-pointing face normals. In order regularize possible divergence from summing intertwiners, develop quantum cutoff scheme eigenvalue ``shadow operator''. The polyhedra (and spacelike faces) finite, types cutoffs imposed. One imposed $j$ area operator, other shadow operator every