The Five-Fold Sumset Cardinalities of Four-Element Sets of Integers

This preprint determines the possible cardinalities of the five-fold sumset 5A when A is a four-element set of integers. After affine normalization to A={0,a,b,c} with 0<a<b<c, the argument studies collisions among the 56 formal five-term sums through a finite rational central hyperplane arrangement. An exact stratum-coverage certificate supplies integer representatives for all 95 feasible generic collision planes and all 663 feasible intersection rays; the no-collision ambient stratum is handled separately by the witness (a,b,c)=(1,16,22). An independent certificate checker reconstructs the arrangement, validates one-to-one coverage, and verifies the recorded representatives. A separate direct brute-force program enumerates every normalized triple with c<=30 and computes the sumsets without using the certificate geometry. The resulting classification is R_Z(5,4)={16,20,21,24,26,27,29,...,39,41,...,48,50,...,56}; equivalently, the excluded values between 16 and 56 are {17,18,19,22,23,25,28,40,49}. The manuscript and computational materials have undergone internal adversarial auditing but have not been peer reviewed. This deposit does not claim definitive priority or independently certified novelty.