{"id":"hexagon/2610.00606","pdf":"https://hexagonmath.org/pdf/2610.00606v1","url":"https://hexagonmath.org/abs/2610.00606v1","tags":["msc:11H55","msc:53C22","msc:83F05","Gram matrix","flat torus","lattice"],"score":0,"title":"FINITE LENGTH DATA DO NOT UNIVERSALLY DETERMINE FLAT THREE-TORI EXPLICIT CONSTRUCTIONS, PROOFS, AND A CAUTION FOR COSMIC TOPOLOGY","domain":"math.NT","source":"hexagon","authors":["Changxia Lin"],"reviews":0,"subject":"math.NT","summary":"Abstract. A flat three-torus is determined by a Euclidean lattice up to orthogonal changes of coordinates, or equivalently by a positive-definite Gram matrix up to integral change of basis. A finite list of short lattice lengths is a much weaker datum. This paper gives an explicit and fully elementary construction showing that the weakness can persist over a large range of scales. For every real parameter A ≥ 2, we exhibit two positive-definite Gram matrices G+(A) and G−(A) whose associated flat tori have different volumes and hence are not isometric, while their length multisets agree below…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-29","reactions":{},"withdrawn":false,"authorship":"human","formalized":false,"human_understanding":"all"}