Distance-spectral classification of twelve-point kissing shells in three dimensions

Abstract

Let D={1,2,8/3,3,11/3,4}, the set of squared distances realised between two points of one twelve-neighbour shell of a Barlow (close-packed) structure. Call a twelve-point subset of the unit sphere distance-spectral if all of its pairwise squared distances lie in D. We prove that such twelve-point sets form exactly two congruence classes: every such set is congruent, allowing reflections, to the cuboctahedral or the anticuboctahedral kissing shell. The continuum problem is reduced to a finite exact rational enumeration, and the finite layer is independently reproduced by a second implementation. We then classify two spectral shells joined along an exact unit bond. Besides the ideal Barlow bonded contexts, there is exactly one further congruence class: the stacking chimera. Each constituent first shell is individually admissible, but their union is not embeddable in a Barlow packing; within range 2 the obstruction is a single cross-side pair at squared distance 25/9. All results are zero-tolerance statements; no quantitative stability or crystallization theorem is claimed. The proof-bearing computational evidence and independent replay are archived separately at DOI 10.5281/zenodo.21963603. This publication record also includes an optional offline interactive visualization for exposition; the visualization is not part of the proof-bearing evidence.