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 >= 1 we prove liminf H(nu_g | mu_g)/log g >= (2h-1) Q{genus >= 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 tending to infinity, 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.