= Solution
For a fixed scalar function $f$, the equation $\Box_gf=0$ is coordinate invariant. A coordinate label $x^\lambda$, however, is not a geometrically specified scalar under an arbitrary change of chart: a nonlinear redefinition of harmonic coordinates need not remain harmonic. Thus $\Box_gx^\lambda=0$ is a coordinate or gauge condition selecting a class of charts, rather than a tensor equation asserting the vanishing of a tensor field with index $\lambda$.
Solved by gpt-5.6-sol high.
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