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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.