A. Facts (F1), (F2) of Theorems 3.4 and 3.5, exactly in the trace algebra over Q[t]/(m)
   A0 multiplication table against matrices: 720 comparisons (u_1, u_2, u_3; K and O; p = 10007, 10009)
   A1 L1 (line (0,t,t), modulo g = 2t^4-4t^2+1): u_3 + u_2 = 0, tr u_2 = 0 (K); u'_4 - u'_3 = 0, tr u'_2 = tr u'_3 = 0 (O); controls u_3 != u_2, u'_3 != +-u'_2; tr u_1 = 2t^2 - 2
   A2 L2 (line (1,t,t), modulo h(t)): u_4 - u_2 = 0, tr u_2 = 1, tr u_1 = t^2 - 1 (K); control u_4 != -u_2; in convention O no relation u'_i = +-u'_j (i < j <= 4)
   A2 L2 (line (1,t,t), modulo h(-t)): u_4 - u_2 = 0, tr u_2 = 1, tr u_1 = t^2 - 1 (K); control u_4 != -u_2; in convention O no relation u'_i = +-u'_j (i < j <= 4)
   (Gauss's lemma: g and h are primitive in Z[t], so these congruences hold in Z[t].)
B. Lemma 3.2 and Theorems 3.4, 3.5 with explicit matrices, all odd primes p <= 2000
   302 odd primes: root criteria of Lemma 3.2 hold for g, h and h(-t);
   instances (prime, root): L1/K 288 (u_3 = -u_2 != +-1, tr u_2 = 0), L1/O 288 (u'_4 = u'_3 != +-1, tr u'_3 = 0), L2/K 584 (u_4 = u_2 != +-1, tr u_2 = 1); kappa != 2 in all
C. Constants
   Thompson's PSL(2,q), q <= 10^4: 626 groups (6 with q = 2^r, 3 with q = 3^r, 617 primes 7..9973); the certificate files contain exactly these q
   first prime > 10^4: 10007 (= 2 mod 5, 7 mod 16, 4 mod 7); |PSL(2,10007)| = 501050730168; |PSL2_2_17| = 2.252e+15, |PSL2_3_11| = 2.780e+15, |Sz_2_11| = 3.601e+16
   minimal simple groups of order < |PSL(2,10007)|: 629, all covered by Section 4
   densities among Thompson residues mod 560: K 48/96, O 24/96; covered below 130 (K): [7, 13, 17, 43, 47, 83, 97, 113, 127]; not covered: [23, 37, 53, 67, 73, 103, 107]; covered for O below 260: [17, 47, 97, 113, 127, 193, 223, 257]
D. Exhaustive counts (pairs whose sequence never reaches 1)
   PSL(2,5) = A_5: K 480, O 720 of 3600 pairs; PSL(2,2): 0, 0 of 36; PSL(2,3): 0, 0 of 144
E. PSL(2,q) certificates (u_M != +-1 and u_{M+L} = +-u_M, recomputed)
   convention K: 51 of 51 groups OK (17304899 steps, 45 s); corrupted periods rejected for q <= 49
   convention O: 51 of 51 groups OK (20193793 steps, 50 s); corrupted periods rejected for q <= 49
   groups: q <= 49: [4, 7, 8, 13, 17, 23, 27, 32, 37, 43, 47]; extension fields: [128, 243, 2048, 2187, 8192]; primes: [907, 997, 1033, 1097, 2017, 2393, 2687, 2833, 2837, 2887, 3187, 3413, 3533, 3607, 3613, 3727, 3847, 4003, 4657, 4703, 5023, 5333, 6113, 6197, 6553, 7477, 7523, 7537, 7673, 7687, 8017, 9133, 9743, 9767, 9887]
F. PSL(3,3) certificates
   6 certificates OK (det 1, u_M != 1, u_(M+L) = u_M; three for each convention)
G. Lemma 4.2: Suzuki model (generators of Bandman et al. 2006, Section 4.1) and the Sz(8) certificates
   q = 8: U product formula (4096 products), M(c) action (448), T M(c) T = M(c^-1), form preserved; T U(a,b) T in B T U for all 63 pairs (a,b) != 0: |<U,M,T>| = q^2(q^2+1)(q-1) = 29120
   q = 8: closure of five generators has 29120 elements and contains every U(a,b), M(c), T
   q = 32: U product formula (3000 products), M(c) action (3000), T M(c) T = M(c^-1), form preserved; T U(a,b) T in B T U for all 1023 pairs (a,b) != 0: |<U,M,T>| = q^2(q^2+1)(q-1) = 32537600
   q = 128: U product formula (3000 products), M(c) action (3000), T M(c) T = M(c^-1), form preserved; T U(a,b) T in B T U for all 16383 pairs (a,b) != 0: |<U,M,T>| = q^2(q^2+1)(q-1) = 34093383680
   sz8_kourovka_sample300.txt: 218 of 218 certificates OK (words reproduce x, y; x, y in the group of order 29120; form preserved; u_M != 1, u_(M+L) = u_M)
   sz8_other_sample300.txt: 207 of 207 certificates OK (words reproduce x, y; x, y in the group of order 29120; form preserved; u_M != 1, u_(M+L) = u_M)
TOTAL failures: 0  (99 s)
