The endpoint pencil on a two-vertex subdivision core (light half) #
Every slot of a loopless two-vertex core joins the same two core vertices. The divisor consisting of one chip at each endpoint has rank at least one for arbitrary positive integral slot lengths: a chip removed in the interior of a slot is recovered by the exact segment-reflection potential on that slot.
This packages the uniform hyperelliptic argument for banana graphs. It depends only on the segment-reflection and rank-one layers.
The two endpoint chips on a subdivision whose loopless core has exactly two vertices form a degree-two rank-one divisor. Keeping the vertex-count equality explicit makes this usable for dependent canonical contracted cores.
The endpoint pencil may be padded to any larger exact degree. This is useful when a two-vertex banana occurs as a low-genus structural case inside a theorem whose critical degree is fixed by the ambient genus.