{"id":"hexagon/2611.00078","pdf":"https://hexagonmath.org/pdf/2611.00078v1","url":"https://hexagonmath.org/abs/2611.00078v1","tags":["math.AG","msc:11H06","msc:11F27","msc:11E20","msc:15A63"],"score":0,"title":"GRAM OBSERVATIONS AND THETA-SERIES RECONSTRUCTION OF EUCLIDEAN LATTICES","domain":"math.NT","source":"hexagon","authors":["Changxia Lin"],"reviews":0,"subject":"math.NT","summary":"A finite collection of lattice-vector lengths probes a positive-definite Gram matrix through rank-one linear functionals. We use this observation map to give a local criterion for non-identifiability and to construct an exact three-dimensional collision at a closed squared-length cutoff of one: two non-isometric lattices have identical complete truncated vector-length data, although their Gram determinants are different. A signed-cycle construction gives a higher\u0002dimensional family whose complete rank-one length multisets agree at every cutoff below an explicit threshold, with an exact…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-11-04","reactions":{},"withdrawn":false,"authorship":"human","formalized":false,"human_understanding":"all"}