A new lower bound for the growth rate of Av(1324)

The growth rate of Av(1324) is the last unknown Stanley-Wilf limit of a length-four pattern. Since 2020 the best rigorous lower bound has been the 10.271012 of Bevan, Brignall, Elvey Price and Pantone; we raise it to 10.466425. Their construction lays 1324-avoiders along an infinite staircase of two-cell blocks alternating with single connecting cells, forbidding a 1324 by a local condition on how the points of a block interleave with the components of the cell beside it. Certain points of a block, its leaves, can be exempted from that condition. They exempt them on one axis only, and report that they could not count the possibilities when both are exempted. We do both. The second relaxation is their own argument read in the other axis, and the Harris correlation inequality bounds the joint count below by the product of the two taken separately. Two further ingredients concern the leaves. The non-leaves of a cell cut it into strips, and how many carry each number of leaves is the first open problem that paper lists, unresolved there even in the simplest case. We determine those densities in closed form and show that they concentrate, the number of strips with j leaves having variance O(n^{7/4}) in the size n at every j, where o(n^2) is all the construction needs, so the bound is read at the true distribution and not at the worst case consistent with the known averages. Marking leaves and strips also tilts the ensemble off the densities at which their construction holds it.