= Balanced spectrum of a nonidentity stabilizer
{title2=$\dim E_{+1}(s)=\dim E_{-1}(s)=2^{n-1}$}
A <stabilizer group> excludes $-I$, so each element squares to $I$ and is Hermitian; an imaginary-phase Pauli element would square to $-I$. A nonidentity stabilizer contains at least one nonidentity local <Pauli matrix> and has zero trace, since the trace of a tensor product is the product of local traces. Its eigenvalues are therefore $\pm1$ with equal multiplicities. This trace fact also gives $\dim\mathcal X_S=2^n/|S|$ from the <stabilizer-projector formula>.
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