Unit Actions on Homometric Five-Point Subsets of Cyclic Groups: Orbit and fixed-point refinement of the seven-family classification

Abstract

Erickson and Jones classified the nontrivial homometry classes of five-point binary bracelets into seven families, Types A-G, and conjectured that each family is invariant under multiplication by units modulo sign. We prove this conjecture and show that the induced actions are diagonal for Type A, multiplicative for Types B-E, affine for Type F, and regular through $U_{20}/\{\pm1\}\cong C_4$ for Type G. We derive closed arithmetic formulas for the numbers of unit orbits and classes fixed by every unit, and determine the orbit-size distributions in all seven types, including explicit stabilizer descriptions for Types A and F.