  We improve the previously best known upper bounds on the sizes of
$\theta$-spherical codes for every $\theta<\theta^*\approx 62.997^{\circ}$ at
least by a factor of $0.4325$, in sufficiently high dimensions. Furthermore,
for sphere packing densities in dimensions $n\geq 2000$ we have an improvement
at least by a factor of $0.4325+\frac{51}{n}$. Our method also breaks many
non-numerical sphere packing density bounds in smaller dimensions. This is the
first such improvement for each dimension since the work of Kabatyanskii and
Levenshtein~\cite{KL} and its later improvement by Levenshtein~\cite{Leven79}.
Novelties of this paper include the analysis of triple correlations, usage of
the concentration of mass in high dimensions, and the study of the spacings
between the roots of Jacobi polynomials.
