Performance Scientific Computing Center (HSCC) of the Beijing Normal University. In this Appendix we list constants A, B, C computed using the computer algorithm as described in Sec. III. The condition (3) holds for for ferromagnetic Potts models with v, m, q > 0. This confirms that qA = C is the exact critical frontier in the ferromagnetic regime. Subnet $^{1}$× 1: Subnet $^{2}$× 2: Subnet $^{3}$× 3: 1. Potts model on $^{n}$× n subnets with pure 2-site coupling K 23 Appendix: Expressions of A,B,C A = q 3 + 9 q $^{2}$v + 33 qv $^{2}$+ (50 + 4 q ) v 3 + 21 v 4 + 3 v 5 B = q $^{2}$v2 + 10 qv 3 + (30 + 2 q ) v 4 + 22 v 5 + 7 v 6 + v 7 C = 9 qv 4 + (54 + 3 q ) v 5 + 63 v 6 + 33 v 7 + 9 v 8 + v $^{9}$. (A.2) A = 29 q $^{3}$v6 + (459 q 2 + 9 q $^{3}$) v 7 + (2592 q + 423 q $^{2}$) v 8 + (5292 + 4185 q + 171 q $^{2}$) v 9 + (12825 + 3258 q + 36 q $^{2}$) v 10 + (15534 + 1539 q + 3 q $^{2}$) v 11 + (12184 + 454 q ) v 12 + (6732 + 78 q ) v 13 + (2688 + 6 q ) v 14 + 768 v 15 + 150 v 16 + 18 v 17 + v $^{18}$, B = q $^{5}$v3 + 21 q $^{4}$v4 + (199 q 3 + 3 q $^{4}$) v 5 + (1040 q 2 + 85 q $^{3}$) v 6 + (2979 q + 844 q 2 + 17 q $^{3}$) v 7 + (3780 + 3834 q + 340 q 2 + 2 q $^{3}$) v 8 + (7182 + 2429 q + 86 q $^{2}$) v 9 + (6858 + 950 q + 13 q $^{2}$) v 10 + (4250 + 233 q + q $^{2}$) v 11 + (1846 + 33 q ) v 12 + (570 + 2 q ) v 13 + 121 v 14 + 16 v 15 + v $^{16}$, C = 29 q $^{3}$v6 + (459 q 2 + 9 q $^{3}$) v 7 + (2592 q + 423 q $^{2}$) v 8 + (5292 + 4185 q + 171 q $^{2}$) v 9 + (12825 + 3258 q + 36 q $^{2}$) v 10 + (15534 + 1539 q + 3 q $^{2}$) v 11 + (12184 + 454 q ) v 12 + (6732 + 78 q ) v 13 + (2688 + 6 q ) v 14 + 768 v 15 + 150 v 16 + 18 v 17 + v $^{18}$. (A.3) v = e K − 1 A = 1 , B = v, C = v 3 + 3 v $^{2}$. (A.1)