Triangle Relation Planar Algebras: Cohomology, Classification, and New Families

We establish a finite system of consistency equations for triangle relation planar algebras. Within the Yang--Baxter relation class, we characterize triangle relations by $H^2(K;\mathbb C)=0$, where $K$ is the fusion complex recording the support of the fusion coefficients. Together with a reconstruction theorem, this gives an algebraic classification procedure. We classify bi-commutative triangle relation planar algebras with $\dim P_{2,\pm}\leq5$. In ranks two, three, and four, the classification yields respectively $2$, $7$, and $19$ realizable fusion-support pair strata, up to relabelling and shading reversal. These results recover the low-rank examples of Bisch, Jones, Liu, and Lu, and extend the rank-four classification beyond exchange relations. In rank five, we obtain two one-parameter families with different duality types whose generic members do not arise from representations of groups or quantum groups.