Planar FMM lists partition all ordered off-diagonal particle pairs

For a cell $C$ let $h(C)$ be its remaining tree height, and for a pair $(C,D)$ use $h(C)+h(D)$. Every nonterminal call split by [F1] selects a nonleaf cell (both coordinates when $C=D$); each child call strictly lowers the height of every split coordinate and leaves any unsplit coordinate fixed. The measure is a nonnegative integer, so every branch terminates. The recursion is finitely branching (at most sixteen child pairs at a simultaneous equal-cell split), hence its finite-depth recursion tree is finite.

If the source cell is split, its child label sets are disjoint and cover the source labels, so the child Cartesian products are a disjoint cover of $I_C\times I_D$. The same statement holds when the target cell is split. When $C=D$ is internal, splitting both ordered coordinates gives all ordered child products, also disjoint. Thus each recursive replacement preserves exactly the parent pair set.

A terminal far cell pair records its entire Cartesian product $I_C\times I_D$. A terminal near pair of leaves records exactly that product with the diagonal labels removed. By step 2.1, induction from the root pair shows these terminal records cover the off-diagonal part of the root product without overlap.

A far cell pair has disjoint circumscribed disks because $R_{CD}>a_C+a_D$; hence its label sets cannot intersect and its product contains no diagonal pair. Removing the diagonal only at terminal leaf pairs therefore leaves exactly all ordered pairs $(j,i)$ with $j\mathrm{\ne }i$, each once.