Prime-Sum Cycles with Periodic Edge Shifts: A Parity Obstruction and Explicit Witnesses

We study indexed cyclic permutations of the integers 1,...,n for which every adjacent sum, modified by the periodic edge shifts (0,-1,+1), is prime. We prove that no admissible permutation exists when n is congruent to 4 modulo 6, using a parity-balance obstruction and the uniqueness of a possible even prime sum. We provide independently verified examples for n=21, 23, and 26, exhaustive counts for n=3,...,9, and a conditional six-vertex insertion criterion. Necessary parity conditions for general integer shifts are also discussed. The converse existence conjecture remains open. This manuscript is a preprint and has not been peer reviewed.