In this work, we analyze the infrared divergence of two-loop amplitudes at arbitrary multiplicity in three-dimensional \( \mathcal{N}=6 \) Chern-Simons matter theory. We introduce the Bern-Dixon-Smirnov (BDS) integrand, which captures the full infrared structure while remaining free of unphysical cuts. We show that these local integrands, together with their kinematic prefactors, are naturally organized by the scaffolding triangulations of n = 2k-gon, with distinct triangulations yielding different local representations. Remarkably, this triangulation structure also persists at the level of the integrated functions. This observation provides a graphical proof of both the cancellation of elliptic cuts and the triangulation independence of the integrated result. As a direct consequence, we obtain a simple proof that the integrated BDS integrand coincides with the one-loop maximally-helicity-violating (MHV) amplitude (the BDS ansatz) of \( \mathcal{N}=4 \) super Yang-Mills theory for all n = 2k.
