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The 4D Regge action is invariant under 5--1 and 4--2 Pachner moves, whichdefine a subset of (local) changes of the triangulation. Given this fact onemight hope to find a local path integral measure that makes the quantum theoryinvariant under these moves and hence makes the theory partially triangulationinvariant. We show that such a local invariant path integral measure does notexist for the 4D linearized Regge theory.  To this end we uncover an interesting geometric interpretation for theHessian of the 4D Regge action. This geometric interpretation will allow us toprove that the determinant of the Hessian of the 4D Regge action does notfactorize over 4--simplices or subsimplices. It furthermore allows to determineconfigurations where this Hessian vanishes, which only appears to be the casein degenerate backgrounds or if one allows for different orientations of thesimplices.  We suggest a non--local measure factor that absorbs the non--local part ofthe determinant of the Hessian under 5--1 moves as well as a local measurefactor that is preserved for very special configurations.

Discretization independence implies non-locality in 4D discrete quantum gravity