Residue-Class Pair-Capacity Inequalities for Restricted Difference Bases

Abstract

This preprint develops exact residue-class pair-capacity inequalities for sets of integer marks. Reducing marks modulo m partitions unordered mark pairs into undirected residue-difference classes, yielding necessary quadratic inequalities in the residue occupancies for any prescribed set of distinct positive differences. For complete restricted difference bases in [0,N], the paper also proves an exact collision-accounting identity: the slack between pair capacity and the number of required labels in a residue class equals the total excess representation multiplicity in that class. Two fully reproducible examples at length 9999 illustrate a saturated-class collision obstruction modulo 221 and a strict capacity deficit modulo 389. The accompanying standalone verifier reconstructs the 173-mark example, recomputes all 14,878 unordered pair differences, independently cross-checks the capacity formulas against direct pair enumeration for every undirected class of both moduli, and verifies the two obstruction certificates. DOI: 10.5281/zenodo.22160860