A combinatorial proof of Jacobi's elliptic identity via alternating permutations

We provide a unified combinatorial framework connecting Entringer numbers, Dumont-Viennot snakes, and elliptically weighted continued fractions, which gives a structural interpretation of the Jacobi elliptic identity \begin{equation} $\mathrm{s}\mathrm{n}$'(u)=$\mathrm{c}\mathrm{n}$(u),$\mathrm{d}\mathrm{n}$(u), \end{equation} where $\mathrm{s}\mathrm{n}$, $\mathrm{c}\mathrm{n}$ and $\mathrm{d}\mathrm{n}$ are the Jacobi elliptic functions. This framework allows the decomposition of weighted snakes corresponding to the derivative of $\mathrm{s}\mathrm{n}$ into canonical $\mathrm{c}\mathrm{n}$- and $\mathrm{d}\mathrm{n}$-components, bridging classical combinatorics and elliptic function theory.