Tunable intersublattice exchange coupling drives magnetic evolution in Mn$_{3+x}$Ga$_{1-x}$C ($0 \le x \le 0.60$)

We investigate the magnetic and transport evolution in Mn$_{3+x}$Ga$_{1-x}$C ($0 \le x \le 0.60$), where Mn substitution at corner Ga sites induces lattice contraction and suppresses the antiferromagnetic order of Mn$_3$GaC. As $x$ increases, the magnetic ground state of the system undergoes a sequential transition from an antiferromagnetic state, via a canted ferrimagnetic state, to a robust ferrimagnetic state, accompanied by a surge in the magnetic ordering temperature. Saturation magnetic moments reaches a maximum of 3.63~$\mu_{\mathrm{B}}$/f.u. at $x = 0.10$, whereas the topological Hall resistivity peaks at 1.47~$\mu\Omega\cdot$cm for $x = 0.20$ before decreasing with further doping. First-principles calculations demonstrate a $\sim\!40^{\circ}$ canting of face-centered Mn moments at $x = 0.20$, signifying spin frustration, and an eventual antiparallel alignment of face-centered and corner-site Mn moments at higher $x$. These results reveal that intersublattice antiferromagnetic coupling governs the magnetic transformation and emergent transport phenomena, thus providing a microscopic foundation for designing high-ordering-temperature antiperovskites.