GRAM OBSERVATIONS AND THETA-SERIES RECONSTRUCTION OF EUCLIDEAN LATTICES

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 higherdimensional family whose complete rank-one length multisets agree at every cutoff below an explicit threshold, with an exact first-shell discrepancy. We then turn to multi-vector data. The full degree-n Gram theta data determines every rank-n Euclidean lattice, by a basis-witness and covolume-index argument. On each compact subset of lattice moduli space, the same argument yields a finite cutoff; a determinant-one cusp family shows that the basis-witness cutoff supplied by this method cannot be uniform on the full moduli space. Finally, a published rank-32 theta-series coincidence, together with a minimum-shell spanning identity, is stabilized by orthogonal direct sums to every rank n ≥ 32. If k(n) denotes the least degree of full Gram theta data that universally determines rank-n Euclidean lattices, we obtain 4 ≤ k(n) ≤ n for n ≥ 32. The paper separates proved statements, cited lattice-theoretic input, and computational checks; it does not claim a solution of the exact minimum-degree problem.