== Remark 5.2 reproduction (exact closedness via orbit-stabiliser/Schreier)
-- positive controls (small characteristic) --
F_4                        |Q|=   4 subgroups(nonzero)=  4 pairs=    16 orbit-reps=    5 | pairs with ABA!<=A or BAB!<=B:     12 | closed pairs:     10 | CLOSED & VIOLATING: 2  (0.0s)
      closed violating rep: A0=['(0, 0)', '(0, 1)'] B0=['(0, 0)', '(0, 1)'] |H|=10
      closed violating rep: A0=['(0, 0)', '(0, 1)'] B0=['(0, 0)', '(1, 0)'] |H|=10
F_9                        |Q|=   9 subgroups(nonzero)=  5 pairs=    25 orbit-reps=    6 | pairs with ABA!<=A or BAB!<=B:     20 | closed pairs:      9 | CLOSED & VIOLATING: 1  (0.0s)
      closed violating rep: A0=['(0, 0)', '(0, 1)', '(0, 2)'] B0=['(0, 0)', '(1, 0)', '(2, 0)'] |H|=120
F_8=F_2[x]/(x^3+x+1)       |Q|=   8 subgroups(nonzero)= 15 pairs=   225 orbit-reps=   24 | pairs with ABA!<=A or BAB!<=B:    217 | closed pairs:     50 | CLOSED & VIOLATING: 6  (0.0s)
      closed violating rep: A0=['(0, 0, 0)', '(0, 0, 1)'] B0=['(0, 0, 0)', '(0, 0, 1)'] |H|=18
      closed violating rep: A0=['(0, 0, 0)', '(0, 0, 1)'] B0=['(0, 0, 0)', '(0, 1, 0)'] |H|=14
      closed violating rep: A0=['(0, 0, 0)', '(0, 0, 1)'] B0=['(0, 0, 0)', '(0, 1, 1)'] |H|=14
      closed violating rep: A0=['(0, 0, 0)', '(0, 0, 1)'] B0=['(0, 0, 0)', '(1, 0, 0)'] |H|=18
      closed violating rep: A0=['(0, 0, 0)', '(0, 0, 1)'] B0=['(0, 0, 0)', '(1, 0, 1)'] |H|=18
      closed violating rep: A0=['(0, 0, 0)', '(0, 0, 1)'] B0=['(0, 0, 0)', '(1, 1, 0)'] |H|=14
F_27=F_3[x]/(x^3+2x+1)     |Q|=  27 subgroups(nonzero)= 27 pairs=   729 orbit-reps=   42 | pairs with ABA!<=A or BAB!<=B:    715 | closed pairs:     14 | CLOSED & VIOLATING: 0  (0.1s)
-- fields listed in Remark 5.2 --
F_25=F_5[x]/(x^2+3)        |Q|=  25 subgroups(nonzero)=  7 pairs=    49 orbit-reps=    8 | pairs with ABA!<=A or BAB!<=B:     42 | closed pairs:      7 | CLOSED & VIOLATING: 0 | closed pairs not of form (cP, c^-1 P): 0  (0.0s)
F_49=F_7[x]/(x^2+1)        |Q|=  49 subgroups(nonzero)=  9 pairs=    81 orbit-reps=   10 | pairs with ABA!<=A or BAB!<=B:     72 | closed pairs:      9 | CLOSED & VIOLATING: 0 | closed pairs not of form (cP, c^-1 P): 0  (0.0s)
F_121=F_11[x]/(x^2+1)      |Q|= 121 subgroups(nonzero)= 13 pairs=   169 orbit-reps=   14 | pairs with ABA!<=A or BAB!<=B:    156 | closed pairs:     13 | CLOSED & VIOLATING: 0 | closed pairs not of form (cP, c^-1 P): 0  (0.4s)
F_125=F_5[x]/(x^3+x+1)     |Q|= 125 subgroups(nonzero)= 63 pairs=  3969 orbit-reps=   96 | pairs with ABA!<=A or BAB!<=B:   3937 | closed pairs:     32 | CLOSED & VIOLATING: 0 | closed pairs not of form (cP, c^-1 P): 0  (3.4s)
-- rings listed in Remark 5.2 --
F_5[e]/(e^2)               |Q|=  25 subgroups(nonzero)=  7 pairs=    49 orbit-reps=   10 | pairs with ABA!<=A or BAB!<=B:     40 | closed pairs:      9 | CLOSED & VIOLATING: 0  (0.0s)
F_7[e]/(e^2)               |Q|=  49 subgroups(nonzero)=  9 pairs=    81 orbit-reps=   12 | pairs with ABA!<=A or BAB!<=B:     70 | closed pairs:     11 | CLOSED & VIOLATING: 0  (0.0s)
F_5 x F_5                  |Q|=  25 subgroups(nonzero)=  7 pairs=    49 orbit-reps=   13 | pairs with ABA!<=A or BAB!<=B:     36 | closed pairs:     13 | CLOSED & VIOLATING: 0  (0.0s)
F_7 x F_7                  |Q|=  49 subgroups(nonzero)=  9 pairs=    81 orbit-reps=   15 | pairs with ABA!<=A or BAB!<=B:     66 | closed pairs:     15 | CLOSED & VIOLATING: 0  (0.0s)
F_5[e]/(e^3)               |Q|= 125 subgroups(nonzero)= 63 pairs=  3969 orbit-reps=  128 | pairs with ABA!<=A or BAB!<=B:   3875 | closed pairs:     94 | CLOSED & VIOLATING: 0  (3.1s)
F_5^3                      |Q|= 125 subgroups(nonzero)= 63 pairs=  3969 orbit-reps=  228 | pairs with ABA!<=A or BAB!<=B:   3808 | closed pairs:    161 | CLOSED & VIOLATING: 0  (3.0s)
F_5[e]/(e^2) x F_5         |Q|= 125 subgroups(nonzero)= 63 pairs=  3969 orbit-reps=  172 | pairs with ABA!<=A or BAB!<=B:   3856 | closed pairs:    113 | CLOSED & VIOLATING: 0  (3.1s)
