\mathbbm{1}_m, \quad &
\Gamma_2=\hat{S}_L     \otimes \mathbbm{1}_L \otimes \mathbbm{1}_m,       \\
\Gamma_3=\mathbbm{1}_L \otimes \hat{C}_L     \otimes \mathbbm{1}_m,       &
\Gamma_4=\mathbbm{1}_L \otimes \hat{S}_L     \otimes \mathbbm{1}_m. 
\end{array}
\label{EQ:generic_twist-eater}\end{aligned}$$ While we write the twist using the skew-diagonal form here, we can always rewrite it in the symmetric form by the $SL(4, \mathbb{Z})$ transformation. We call this twist “generic twist” in this article, and the minimal twists (\[EQ:minimal\_sym\_form\]) and (\[EQ:minimal\_skew\_form\]) are particular cases ($m=1$) of the generic twist. As is well known, the TEK model can describe the NCYM theory [@Aoki:1999vr; @Ambjorn:1999ts]. Expanding the matrix model around noncommutative tori background, we can obtain noncommutative $U(m)$ Yang-Mills theory on fuzzy tori. (Note that this interpretation is possible even at finite-$N$.) Because fuzzy torus can be used as a regularization of fuzzy ${\mathbb R}^4$, it is naively possible to give a nonperturbative formulation of the NCYM on fuzzy ${\mathbb R}^4$ by taking a suitable large-$N$ limit in the TEK model. (See appendix \[sec:NCYM\] for details.) However, we will see later it is not the case because of the $\mathbb{Z}_N^4$ symmetry breaking. In the NCYM interpretation the shift and clock matrices can be regarded as matrix realization of a fuzzy torus. From this point of view, twist prescription (\[EQ:generic\_twist\]) provides YM 