Odd branching obstructs multiplicity-one integral current structures

Giuliano Basso asked whether every $n$-dimensional integral current space admits an integral current structure with the same characteristic set and multiplicity one almost everywhere. We give a negative answer already in dimension one. The obstruction is a parity phenomenon at odd-valence branch points. More precisely, if a one-dimensional integral current has unit multiplicity on the arms incident to an isolated vertex $v$, then the coefficient of its boundary at $v$ has the same parity as $°(v)$. Hence infinitely many odd-valence vertices force infinite boundary mass for every full-support unit-multiplicity current. We construct a compact geodesic, doubling, $1$-Ahlfors regular graph-like continuum $G$ of finite length and a boundaryless current $T\in\I_1(G)$ with $\set(T)=G$, while no $S\in\I_1(G)$ with $\set(S)=G$ can have multiplicity one $\Hh^1$-almost everywhere. We classify all full-support integral cycles on $G$. In particular, the least possible essential maximal multiplicity is exactly two, and the minimum mass of such a cycle is $3/2$. For finite metric graphs we identify a complementary sharp defect: the minimum boundary mass among unit-multiplicity currents equals the number of odd-degree vertices. Finally, by taking products with flat tori and using the precise Ambrosio--Kirchheim slice representation, we obtain boundaryless compact geodesic Ahlfors-regular counterexamples in every dimension.