The double-dimer CLE4 scaling-limit conjecture in the half-plane Temperleyan setting

Theorem 1.1: for two independent dimer coverings of the Temperleyan square lattice in the upper half-plane with the standard infinite-volume law, the loop collection $F(\mathcal L_\delta)$, mapped to the disk, is tight and converges in law as $\delta\to0$ to $F(\mathcal L)$ with $\mathcal L$ nested $CLE_4$, matching every macroscopic loop as an unparametrized curve with all nesting generations. New inputs: a bound on disjoint traversals of a slab and control of signed crossing counts, from a column transfer matrix. Not shown: bounded or general simply connected domains, non-Temperleyan boundary conditions, or other lattices.

Theorem 1.1: for two independent dimer coverings of the Temperleyan square lattice in the upper half-plane with the standard infinite-volume law, the loop collection $F(\mathcal L_\delta)$, mapped to the disk, is tight and converges in law as $\delta\to0$ to $F(\mathcal L)$ with $\mathcal L$ nested $CLE_4$, matching every macroscopic loop as an unparametrized curve with all nesting generations. New inputs: a bound on disjoint traversals of a slab and control of signed crossing counts, from a column transfer matrix. Not shown: bounded or general simply connected domains, non-Temperleyan boundary conditions, or other lattices.