Finite Truncated Lattice-Length Data on Three-Dimensional Flat Tori: An Explicit Obstruction and a Finite-Determination Bound

For a full-rank lattice in R^3, let L_R be the multiset of lengths of all nonzero lattice vectors of length at most R, counted with vector multiplicity. We give an explicit pair of positive-definite Gram matrices with identical diagonal entries and matching length data on every coordinate subspace of dimension at most two, but with different determinants. For 0<δ≤ 1/6, the associated lattices have all three successive minima equal to one and satisfy L_R(Λ_+)=L_R(Λ_-) whenever\ R<√3-6δ. Moreover, their Gram determinants differ by 4δ^3, so their covolumes are unequal. This gives the lower bound √3 for the normalized worst-case identification-radius constant. We also state precisely how Schiemann's finite-determination bound for Seeber–Eisenstein reduced ternary positive-definite real quadratic forms gives the complementary upper bound √7/2. The observation in this paper is the primal-lattice length multiset; it is not the Laplace spectrum of the flat torus. We make no priority claim for the explicit construction pending a systematic comparison with prior work. A separate Lean 4 script is supplied for one elementary cubic identity only; no other proof in the paper is claimed to be formally verified.