  We establish an unexpected phenomenon of strong regularization along normal
convergence on Wiener chaoses. For every sequence of chaotic random variables,
convergence in law to the Gaussian distribution is upgraded to
superconvergence: the regularity of the densities increases along the
convergence, and all the derivatives converges uniformly. Our findings
strengthen known results regarding modes of convergence for normal
approximation on Wiener chaoses. Without additional assumptions, convergence in
total variation is established by Nourdin & Peccati, and later on amplified to
convergence in relative entropy by Nourdin, Peccati & Swan. Our result is then
extended to the multivariate setting, and for polynomial mappings of a Gaussian
field provided the projection on the Wiener chaos of maximal degree admits a
non-degenerate Gaussian limit. While our findings apply to any context
involving polynomials of a Gaussian field, we emphasize applications regarding:
improved Carbery-Wright estimates near Gaussianity; normal convergence in
entropy and in Fisher information; superconvergence for the spectral moments of
GOE; moments bounds for the inverse of strongly correlated Wishart-type
matrices; superconvergence in the Breuer-Major Theorem. Our proofs leverage
Malliavin's historical idea to establish smoothness of the density via the
existence of negative moments of the Malliavin gradient, and we develop a new
paradigm to study this problem. We relate the existence of negative moments to
spectral quantities associated with the Malliavin Hessian. This link relies on
an adequate choice of the Malliavin gradient, which provides a novel decoupling
procedure of independent interest. Previous attempts to establish convergence
beyond entropy have imposed restrictive assumptions ensuring finiteness of
negative moments for the Malliavin derivatives. Our analysis renders these
assumptions superfluous.
