= Coordinate reflection bijection for tetracode lattices
{title2=$v\mapsto v-2v_ie_i$}
For the <tetracode length-isospectral lattice construction> $L_\pm=(\pm3I+S)\mathbb Z^4$, choose the first coordinate $i$ divisible by three and reverse its sign. Such a coordinate always exists because each nonzero <tetracode> word has one zero coordinate. The reflection preserves the reduction modulo three and switches the two lattice membership conditions modulo four. It is an involution between $L_+$ and $L_-$ preserving every absolute coordinate, so arbitrary positive diagonal weights preserve equality of their vector-length multisets.
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