= Rationally independent axis weights obstruct a lattice isometry
{title2=$\sum_i a_i x_i^2=\sum_i a_i y_i^2\Longrightarrow x_i^2=y_i^2$}
Choose positive axis weights linearly independent over the rationals, and suppose both integer <Euclidean lattices> contain $q\mathbb Z^d$ for some positive integer $q$. A length-preserving linear map sends each $qe_i$ to an integer vector with exactly that absolute-coordinate pattern, by the displayed implication. Thus the map must be a diagonal sign change. For the four-coordinate <tetracode>, preserving the code forces all four signs to agree, because each of its four nonzero projective words has support on a different triple of coordinates. A global sign cannot exchange the two odd-vector parity conditions, proving nonisometry for these weights.
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