Open toric strings have endpoint syndromes

In the mod-two support of a primal trail, each internal visit to a vertex contributes one incoming and one outgoing edge, hence even incidence. After pairing arrivals and departures at every visit, the distinct endpoints have odd incidence and every other vertex has even incidence. By [F3], a $Z$ string anticommutes with a star exactly when the number of shared edges is odd. Thus it anticommutes with exactly $A_{q_0}$ and $A_{q_1}$. It commutes with every plaquette because both use only $Z$ factors.

In the dual graph, the same incidence count gives odd overlap exactly at the two endpoint faces. A dual edge crosses the primal edge carrying the same qubit, so [F3] makes the $X$ string anticommute with exactly the two endpoint plaquettes. It commutes with all stars because both use only $X$ factors. By [F2], these signs are precisely the stated e and m endpoint syndromes.

For a closed edge-simple primal trail, every vertex has even degree in its mod-two edge support: arrivals and departures pair at every visit, including the common initial and final vertex. Thus every star overlap is even, and every plaquette overlap is even because both strings use only $Z$. For a closed dual trail, the same incidence count gives even overlap with every plaquette, and its X string commutes with all stars. Hence both closed strings have zero syndrome.