 Halo first crossing with a finite variance cutoff

A small-scale spectral cutoff bounds the smoothing variance by $S_{\max }$. A sharp-k excursion remains Brownian up to this terminal variance, with total crossed fraction $\mathrm{erfc}({\delta }_c/\sqrt{2S_{\max }})<1$. The ordinary first-crossing density remains valid before the cutoff. Below the sharp-filter cutoff radius the variance is constant, so there are no new crossings and the idealized differential halo abundance vanishes. The missing fraction need not be artificially renormalized to one.