= Two-dimensional lattice reconstruction from vector lengths
{title2=$G=\begin{pmatrix}a^2&\sqrt{a^2b^2-A^2}\\\sqrt{a^2b^2-A^2}&b^2\end{pmatrix}$}
The vector-length multiset of a rank-two <Euclidean lattice> determines it up to an orthogonal map. Let $a$ be the shortest nonzero length. Removing two copies of each positive integer multiple of $a$ removes one primitive lattice line; the shortest remaining length $b$ is the shortest independent-vector length. Those two vectors form a lattice <basis>: otherwise a nonzero point in the centred basis parallelogram would be an independent lattice vector of length less than $b$. Lattice-point density determines the <covolume> $A$. The basis <Gram matrix> then has diagonal entries $a^2,b^2$ and absolute off-diagonal entry $\sqrt{a^2b^2-A^2}$. This proves two-dimensional <spectral rigidity> for a <flat torus> through its <dual lattice>.
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