A new class of non-Gaussian curvature fluctuations zeta(pr)(x) identical with deltaN(chi(i)) arises from the postinflation preheating behavior of a noninflaton field chi(i). Its billiardlike chaotic dynamics imprints regular log-spaced narrow spikes in the number of preheating e-folds N(chi(i)). We perform highly accurate lattice simulations of supersymmetry-inspired quartic inflaton and coupling potentials in a separate-universe approximation to compute N(chi(i)) as a function of the (nearly homogeneous) initial condition chi(i). The superhorizon modes of chi(i)(x) result in positive spiky excursions in zeta(pr) and hence negative gravitational potential fluctuations added to the usual sign-independent inflaton-induced perturbations, observably manifested in large cosmic structures and as (polarized) temperature cosmic microwave background cold spots.