= Odd involution pairing bosonic and fermionic states
{title2=$A^2=I,\quad A(-1)^F=-(-1)^FA$}
An odd <involution> $A$ reverses <fermion parity> and is its own inverse, so it identifies the bosonic and fermionic subspaces. They have equal dimension in a finite-dimensional representation. For a <supercharge> $q$ with $q^2=0$ and $\{q,q^\dagger\}=cI$, $c>0$, use $A=(q+q^\dagger)/\sqrt c$. This gives a direct proof of <boson-fermion degeneracy in a supermultiplet> at positive energy. A zero-energy supersymmetric vacuum need not have an accompanying fermionic vacuum.
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