Two-State Transfer Invariants for Edge-Weighted Bichromatic Partition Functions on Canonical Recursive 2-Trees

The chromatic polynomial of a 2-tree depends only on its order and therefore does not distinguish many non-isomorphic canonical families. We introduce a multivariate edge-weighted bichromatic partition function PG(q,v,w) in which every triangular face is constrained to use exactly two colors, while v and w record monochromatic and bichromatic edge contributions. Our main result concerns canonical recursive 2-tree families generated by a single active-edge interface. For this class of constructions, the extension behavior depends only on whether the interface vertices received equal or different colors, leading to transfer matrices of a fixed 2×2 size independent of q. This leads to a transfer-word representation for active-edge constructions and explicit partition functions for several canonical 2-tree families. As consequences, we obtain the partition-function identity PSn(q,v,w)=PFn1(q,v,w), equating two families that are non-isomorphic for n≥6, together with the operator factorization Mladder=Msnake2, from which PTLm(q,v,w)=PS2m+2(q,v,w) follows; the latter equality is induced by the graph isomorphism TLm≅S2m+2. For q≥2 and v,w>0, we determine the associated spectral growth rates and show that the Book and Snake transfer systems have equal spectral radii precisely on the locus (q−1)w=2v. Under the same positivity assumptions, this is also the unique parameter locus on which the elementary transfer operators commute.