  Topological band theory predicts a $\mathbb{Z}$ classification of
three-dimensional (3D) Dirac semimetals (DSMs) at the single-particle level.
Namely, an arbitrary number of identical bulk Dirac nodes will always remain
locally stable and gapless in the single-particle band spectrum, as long as the
protecting symmetry is preserved. In this work, we find that this
single-particle classification for $C_n$-symmetric DSMs will break down to
$\mathbb{Z}_{n/\text{gcd}(2,n)}$ in the presence of symmetry-preserving
electron interactions. Our theory is based on a dimensional reduction strategy
which reduces a 3D Dirac fermions to 1D building blocks, i.e., vortex-line
modes, while respecting all the key symmetries. Using bosonization technique,
we find that there exists a minimal number $N=n/\text{gcd}(2,n)$ such that the
collection of vortex-line modes in $N$ copies of DSMs can be symmetrically
eliminated via four-fermion interactions. While this gapping mechanism does not
have any free-fermion counterpart, it yields an intuitive ``electron-trion
coupling" picture. By developing a topological field theory for DSMs and
further checking the anomaly-free condition, we independently arrive at the
same classification results. Our theory paves the way for understanding
topological crystalline semimetallic phases in the strongly correlated regime.
