We study the unsplittable multicommodity fixed-charge network design problem, in which
every commodity follows a single path and each arc is installed at most once. Its linear relaxation
is weak because a commodity may occupy a fraction of an arc. We propose an extended
formulation that indexes the variables of each arc by the number of commodities it carries.
Its relaxation is at least as strong as that of a compact formulation with cardinality bounds
and lifted capacities, and can be strictly stronger. The convex hull of the single-arc relaxation
decomposes by level and projects onto the compact single-arc closure; level counterparts of
cover, configuration, k-split c-strong and packing inequalities are never weaker than the classical
ones, and level-aware cut-set inequalities exploit the extended space beyond that closure. Level
windows and a bound-preserving reduction keep the model small. The root bound improves on
the best published bound for 27 of the 31 Canad instances and equals it on the other four. On 40
random instances the approach proves optimality for 39 against 32 for the compact formulation,
with search trees 2–6 times smaller. An ablation shows that exact per-level separation and the
level-aware cut-set inequalities carry the bound.