Six-point cyclic homometry: classification and enumeration
K. Fathi · Zenodo (CERN European Organization for Nuclear Research) · 2026
This preprint presents a computer-assisted classification and exact enumeration of nontrivial homometry classes of six-element subsets of the finite cyclic group Z/nZ, for every positive integer n. Homometric configurations have identical multisets of pairwise Lee distances. Equivalence is taken under translations and reflections, with repeated and antipodal distances allowed. Main results A complete cover by thirteen explicit modular families and twenty-four primitive finite seeds, whose moduli are at most 52. Explicit inverse tests give disjoint parameter domains and canonical codes for complete nontrivial homometry classes. Compact exact formulas, valid at every modulus, for the number of homometry fibres containing two, three, four, or five dihedral classes. A sharp upper bound of five classes per fibre. There is exactly one five-class fibre when 31 divides n, obtained by dilating the classical Singer fibre; there are none otherwise. A complete description of fibres with at least three classes using five geometric models and five finite primitive cores. Simpler classifications for odd moduli and prime moduli, together with a classification of six-point homometry on the real line without a distinct-distance assumption. Exact counts when every prime divisor of n exceeds 31, and asymptotic density formulas that distinguish even moduli from the two odd divisibility sectors. The proof combines anchored reconstruction, integer-lattice containments, Smith normal forms, exact intersection arrangements, and positive-cone generating functions. A complete finite census through modulus 135 covers 626,805,157 dihedral classes. Independently written exact implementations replay the decisive certificates. The results address the six-point case of the cyclic homometry classification and enumeration problem posed by Erickson and Jones (arXiv:2412.08997v4). Homometry is relevant to reconstruction from unlabelled distance data, with connections to crystallography and phase retrieval. The classification concerns six-point configurations; higher cardinalities are outside its scope. The accompanying archive contains the LaTeX manuscript, proof and enumeration programs, machine-readable certificates, reproduction instructions, and a file checksum manifest.