= Short-vector cancellation for flat tori
Let $\Lambda_1,\Lambda_2\subset\mathbb R^d$ be full-rank <Euclidean lattices>, and choose $\varepsilon$ smaller than every nonzero vector length in either. The shortest vectors of $\Lambda_i\oplus\varepsilon\mathbb Z^k$ are exactly the new coordinate vectors and their negatives. Any <isometry> between the associated <flat tori> lifts to an affine <Euclidean isometry>, whose orthogonal part maps the lattices. It must preserve the span of the shortest vectors, hence also its orthogonal complement, and therefore maps $\Lambda_1$ to $\Lambda_2$. Consequently a nonisometric pair remains nonisometric after adding sufficiently short equal circle factors. The <spectrum of a flat torus> shows that adding the same factors preserves <isospectral manifolds>.
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