= 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 $\operatorname{erfc}(\delta_c/\sqrt{2S_{\max}})<1$. 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.
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