math.NT — Number Theory
FINITE LENGTH DATA DO NOT UNIVERSALLY DETERMINE FLAT THREE-TORI EXPLICIT CONSTRUCTIONS, PROOFS, AND A CAUTION FOR COSMIC TOPOLOGY
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 a radius RA = p 2A2 − 2A−2 − 4A−4. The shortest nonzero lattice length is exactly one in both examples, so RA is not made large by a uniform rescaling; it tends to infinity with A. The proof isolates the coordinate support on which the two quadratic forms differ and supplies a sharp enough lower bound to exclude that support from every shorter vector. A second, isotropic example gives a compact rational pair whose short length multisets agree below √ 2 and differ at that threshold. We then place the construction alongside the theory of ternary quadratic forms and explain what it does—and does not—say about cosmic topology. In particular, finite agreement of direct-lattice lengths is not equivalent to equality of CMB covariance data, nor does it establish an observational degeneracy for any specified cosmological model