= Multiplicity-free complete dyadic spectrum
A <Hermitian operator> on a $2^n$-dimensional <Hilbert space>, with $2^n$ distinct <eigenvalues> all belonging to $\{c/2^n:0\leq c<2^n\}$, must occupy this entire grid. This follows from the <pigeonhole principle>: the set of possible values and the set of distinct <eigenvalues> have the same <cardinality>. Its minimum <eigenvalue> is zero, so the operator is necessarily singular. Distinctness therefore does not imply invertibility; it forces a one-dimensional <kernel> in this specific complete-grid setting.
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