Combinatorial and algebraic proofs of Keller's $A_2\square A_2$ $q$-dilogarithm identity

The classical Durfee-square argument gives a combinatorial proof of the pentagon identity for the quantum dilogarithm. Just as the pentagon identity is associated with the $A_2$ quiver, Keller's identity is associated with the square-product quiver $A_2\square A_2$. Previous proofs of Keller's identity use cluster categories or spectral sequences in rapid-decay equivariant cohomology. We give three proofs of Keller's identity: a generating-function proof, an explicit weight-preserving bijection on colored partitions, and a standard-monomial proof using a four-colored arc algebra. Their common mechanism is an iterated Durfee decomposition: two possible pairings give horizontal and vertical decompositions, while a third binary step accounts for the coupling factor. These constructions provide a ``superpotential analogue'' of the Durfee-square argument.