Exact spectral gap of normalized velocity relaxation ($⟨Lf,f⟩=-\Vert (I-\Pi )f{\Vert }^2,\Vert f_t-\Pi f_0\Vert =e^{-t}\Vert f_0-\Pi f_0\Vert$)

For the normalized velocity-reset collision operator on the weighted Hilbert space, $L=\Pi -I$ vanishes on the equilibrium direction and equals $-I$ on its orthogonal complement. The spectral gap is exactly one. Conservation of the projected component and the displayed energy identity give exact exponential decay of the remaining component; the full solution need not decay to zero.