The Entropy Cost of Infinite-Genus Weil-Petersson Local Limits

Let \(\mu_g\) be normalized Weil–Petersson measure on the moduli space of closed hyperbolic surfaces of genus \(g\). If surface laws \(\nu_g\), with arbitrary measurable rooting kernels, have a pointed smooth weak limit \(Q\), then for every fixed \(h\ge1\) we prove \(\liminf_{g\to\infty} H(\nu_g\mid\mu_g)/\log g\ge(2h-1)Q\{\mathrm{genus}\ge h\}\). Positive mass on infinite-genus surfaces therefore requires superlogarithmic surface relative entropy.

Conversely, a fixed one-ended infinite-genus hyperbolic surface can be obtained as a local limit under every joint entropy budget \(b_g\log g\) with \(b_g\to\infty\), measured relative to area-rooted Weil–Petersson measure. The proof uses classical separating-curve volume estimates, binary entropy, positive-volume core neighborhoods, and a slow diagonal. Two further propositions explain known planar/cylindrical consequences of tangle-freeness and the systolic-cylinder mixture.

This is a complete theorem for the stated model, not a resolution of the broad AIM agenda. The preprint is unrefereed; absolute priority and formal verification are not certified.