F1 K=1: W L(8T') W^-1 = block diagonal with the blocks theta_x(8T') of rule (3); the rule with x_(j+k) gives the same blocks here (K <= 3: only t^0 and t^3 = 1 occur in G_K, no shift) (4 x 4)
F1 K=2: W L(8T') W^-1 = block diagonal with the blocks theta_x(8T') of rule (3); the rule with x_(j+k) gives the same blocks here (K <= 3: only t^0 and t^3 = 1 occur in G_K, no shift) (16 x 16)
F1 K=3: W L(8T') W^-1 = block diagonal with the blocks theta_x(8T') of rule (3); the rule with x_(j+k) gives the same blocks here (K <= 3: only t^0 and t^3 = 1 occur in G_K, no shift) (48 x 48)
F1 K=4: W L(8T') W^-1 = block diagonal with the blocks theta_x(8T') of rule (3); the rule with x_(j+k) fails (128 x 128)
F1 K=5: W L(8T') W^-1 = block diagonal with the blocks theta_x(8T') of rule (3); the rule with x_(j+k) fails (320 x 320)
F2 group-ring values read from run2_groupring_output.txt: K = [1, 2, 3, 4, 5, 6, 7, 8, 9]
   K=1: sum_x dim ker theta_x = 3, d_K = 3/2 = 1.5000000 (exact over Q) = group ring  (0.0 s)
   K=2: sum_x dim ker theta_x = 4, d_K = 1/2 = 0.5000000 (exact over Q) = group ring  (0.0 s)
   K=3: sum_x dim ker theta_x = 30, d_K = 5/4 = 1.2500000 (exact over Q) = group ring = Proposition 4.2  (0.0 s)
   K=4: sum_x dim ker theta_x = 26, d_K = 13/32 = 0.4062500 (exact over Q) = group ring  (0.0 s)
   K=5: sum_x dim ker theta_x = 62, d_K = 31/80 = 0.3875000 (exact over Q) = group ring  (0.0 s)
   K=6: sum_x dim ker theta_x = 198, d_K = 33/64 = 0.5156250 (exact over Q) = group ring = Proposition 4.2  (0.0 s)
   K=7: sum_x dim ker theta_x = 331, d_K = 331/896 = 0.3694196 (exact over Q) = group ring  (0.1 s)
   K=8: sum_x dim ker theta_x = 746, d_K = 373/1024 = 0.3642578 (exact over Q) = group ring  (0.1 s)
   K=9: sum_x dim ker theta_x = 1818, d_K = 101/256 = 0.3945312 (exact over Q) = group ring = Proposition 4.2  (0.3 s)
   K=10: sum_x dim ker theta_x = 3682, d_K = 1841/5120 = 0.3595703 (exact over Q)  (0.8 s)
   K=11: sum_x dim ker theta_x = 8054, d_K = 4027/11264 = 0.3575107  (0.3 s)
   K=12: sum_x dim ker theta_x = 17958, d_K = 2993/8192 = 0.3653564 = Proposition 4.2  (0.8 s)
   K=13: sum_x dim ker theta_x = 37897, d_K = 37897/106496 = 0.3558537  (1.9 s)
   K=14: sum_x dim ker theta_x = 81440, d_K = 2545/7168 = 0.3550502  (4.4 s)
   K=15: sum_x dim ker theta_x = 175578, d_K = 29263/81920 = 0.3572144 = Proposition 4.2  (10.3 s)
F3 class parts (indices k = -3m - r, m in Z/P) of the generic operators, all x, all three classes:
   no entry between different classes for any x, K = 3, 6, ..., 15
   K=3: sum_x dim ker = 30, d_K = 5/4 = 1.2500000 = Proposition 4.2 = generic  (0.0 s)
   K=6: sum_x dim ker = 198, d_K = 33/64 = 0.5156250 = Proposition 4.2 = generic  (0.0 s)
   K=9: sum_x dim ker = 1818, d_K = 101/256 = 0.3945312 = Proposition 4.2 = generic  (0.0 s)
   K=12: sum_x dim ker = 17958, d_K = 2993/8192 = 0.3653564 = Proposition 4.2 = generic  (0.5 s)
   K=15: sum_x dim ker = 175578, d_K = 29263/81920 = 0.3572144 = Proposition 4.2 = generic  (5.4 s)
   K=18: sum_x dim ker = 1673910, d_K = 92995/262144 = 0.3547478 = Proposition 4.2  (52.4 s)
   K=21: sum_x dim ker = 15588402, d_K = 2598067/7340032 = 0.3539585 = Proposition 4.2  (528.5 s)
F4 rows and segments: 374488 pairs (configuration, class) with P <= 6 (class 0 only for P = 6); 1059276 segment blocks, each = 8 M_s and closed; kernels = balanced segments, or [no U] + [no D]
F5 K=3: 2^-K sum_x sum_i (P_x)_((k,i),(k,i)) = 5/4 at every index k = d_K; segment place checked for 4 configurations  (0.0 s)
F5 K=4: 2^-K sum_x sum_i (P_x)_((k,i),(k,i)) = 13/32 at every index k = d_K  (0.0 s)
F5 K=5: 2^-K sum_x sum_i (P_x)_((k,i),(k,i)) = 31/80 at every index k = d_K  (0.0 s)
F5 K=6: 2^-K sum_x sum_i (P_x)_((k,i),(k,i)) = 33/64 at every index k = d_K; segment place checked for 48 configurations  (0.0 s)
F5 K=7: 2^-K sum_x sum_i (P_x)_((k,i),(k,i)) = 331/896 at every index k = d_K  (0.1 s)
F5 K=8: 2^-K sum_x sum_i (P_x)_((k,i),(k,i)) = 373/1024 at every index k = d_K  (0.2 s)
F5 K=9: 2^-K sum_x sum_i (P_x)_((k,i),(k,i)) = 101/256 at every index k = d_K; segment place checked for 448 configurations  (0.5 s)
checks: 2871128, failures: 0
ALL CHECKS PASSED
time 631.9 s
