FINITE TRUNCATED LATTICE-LENGTH DATA ON THREE-DIMENSIONAL FLAT TORI: AN EXPLICIT OBSTRUCTION AND A FINITE-DETERMINATION BOUND

For a full-rank lattice in \mathbb{R}^3, let \mathcal{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<\delta\leq 1/6, the associated lattices have all three successive minima equal to one and satisfy \mathcal{L}_R(\Lambda_+)=\mathcal{L}_R(\Lambda_-) \qquad \mathrm{whenever}\ R<\sqrt{3-6\delta}. Moreover, their Gram determinants differ by 4\delta^3, so their covolumes are unequal. This gives the lower bound \sqrt{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 \sqrt{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.