== A. The seeds
ok    M_2: the two lists are tau_3 and tau_5 on Z/8
ok    M_2: fixed-point-free involutions
      M_2: cycles of i -> tau(i)+1: [[0, 4], [1, 7, 5, 3], [2, 6]]; of i -> tau'(i)+1: [[0, 6, 4, 2], [1, 5], [3, 7]]
      M_2: genus 2, cone angles (multiples of pi) [2, 2, 2, 2, 4, 4]
ok    M_2: genus 2, cone angles 4pi, 4pi and four regular classes
ok    M_2: the cycles are those printed
ok    M_2: tau(i) and tau'(i) have the parity opposite to i (b separates)
ok    M_2: no bigon
ok    M_2: b is one curve; squares along b: [0, 3, 6, 1, 4, 7, 2, 5]
ok    M_2: every intersection point is good
      M_2: singular corners: top [1, 3, 5, 7], bottom [0, 2, 4, 6]
ok    M_2: singular corners = odd top corners and even bottom corners
ok    M_2: singular corners and translation parts of the transition maps have x + y even: Lambda in {x + y even}
ok    M_2: the diagonal of Q_i (i even) and the anti-diagonal (i odd) are saddle connections: (1,1), (-1,1) in Lambda
ok    M_3: the two lists are tau_3 and tau_5 on Z/12
ok    M_3: fixed-point-free involutions
      M_3: cycles of i -> tau(i)+1: [[0, 4, 8], [1, 11, 9, 7, 5, 3], [2, 6, 10]]; of i -> tau'(i)+1: [[0, 6], [1, 9, 5], [2, 8], [3, 11, 7], [4, 10]]
      M_3: genus 3, cone angles (multiples of pi) [2, 2, 2, 3, 3, 3, 3, 6]
ok    M_3: genus 3, cone angles 6pi, 3pi (four times) and three regular classes
ok    M_3: the cycles are those printed
ok    M_3: tau(i) and tau'(i) have the parity opposite to i (b separates)
ok    M_3: no bigon
ok    M_3: b is one curve; squares along b: [0, 3, 10, 1, 8, 11, 6, 9, 4, 7, 2, 5]
ok    M_3: every intersection point is good
      M_3: singular corners: top [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], bottom [1, 3, 5, 7, 9, 11]
ok    M_3: all top corners and the odd bottom corners are singular: (1,0) and (0,1) in Lambda, Lambda = Z^2
== B. Matrices and traces of Section 8
ok    M_2, d = 2: u = (4161 -512; 512 -63), v = (8257 -512; 67072 -4159), traces 4098
ok    n = 8: tr(A B^-1) = -62
ok    M_2, d = 2, m = 1: 2 elements, one |trace| with 16 digits, log(lambda) ~ 36.045  (Table 1: 16 digits, 36.045)
ok    M_2, d = 2, m = 2: 4 elements, one |trace| with 58 digits, log(lambda) ~ 133.089  (Table 1: 58 digits, 133.089)
ok    M_2, d = 2, m = 3: 8 elements, one |trace| with 185 digits, log(lambda) ~ 424.222
ok    M_2, d = 2, m = 4: 16 elements, one |trace| with 564 digits, log(lambda) ~ 1297.620
ok    M_2, d = 2, m = 5: 32 elements, one |trace| with 1702 digits, log(lambda) ~ 3917.815
ok    M_2, d = 2, m = 6: 64 elements, one |trace| with 5116 digits, log(lambda) ~ 11778.401  (Table 1: 5116 digits, 11778.401)
ok    M_2, d = 3, m = 1: 2 elements, one |trace| with 52 digits, log(lambda) ~ 119.316
ok    M_2, d = 3, m = 2: 4 elements, one |trace| with 178 digits, log(lambda) ~ 407.854
ok    M_2, d = 3, m = 3: 8 elements, one |trace| with 554 digits, log(lambda) ~ 1273.468
ok    M_2, d = 3, m = 4: 16 elements, one |trace| with 1681 digits, log(lambda) ~ 3870.312
ok    M_2, d = 3, m = 5: 32 elements, one |trace| with 5065 digits, log(lambda) ~ 11660.843  (Table 1: 5065 digits, 11660.843)
ok    M_2, d = 4, m = 1: 2 elements, one |trace| with 98 digits, log(lambda) ~ 223.482
ok    M_2, d = 4, m = 2: 4 elements, one |trace| with 335 digits, log(lambda) ~ 770.258
ok    M_2, d = 4, m = 3: 8 elements, one |trace| with 1047 digits, log(lambda) ~ 2410.588
ok    M_2, d = 4, m = 4: 16 elements, one |trace| with 3185 digits, log(lambda) ~ 7331.576  (Table 1: 3185 digits, 7331.576)
ok    M_3, d = 2, m = 1: 2 elements, one |trace| with 19 digits, log(lambda) ~ 42.531  (Table 1: 19 digits, 42.531)
ok    M_3, d = 2, m = 2: 4 elements, one |trace| with 69 digits, log(lambda) ~ 157.413
ok    M_3, d = 2, m = 3: 8 elements, one |trace| with 219 digits, log(lambda) ~ 502.058
ok    M_3, d = 2, m = 4: 16 elements, one |trace| with 668 digits, log(lambda) ~ 1535.995
ok    M_3, d = 2, m = 5: 32 elements, one |trace| with 2015 digits, log(lambda) ~ 4637.803  (Table 1: 2015 digits, 4637.803)
ok    M_3, d = 3, m = 1: 2 elements, one |trace| with 62 digits, log(lambda) ~ 141.969
ok    M_3, d = 3, m = 2: 4 elements, one |trace| with 211 digits, log(lambda) ~ 485.545
ok    M_3, d = 3, m = 3: 8 elements, one |trace| with 659 digits, log(lambda) ~ 1516.274
ok    M_3, d = 3, m = 4: 16 elements, one |trace| with 2002 digits, log(lambda) ~ 4608.459  (Table 1: 2002 digits, 4608.459)
ok    M_2, d = 2, m = 1: t = 4508549040455682
ok    M_2, d = 2, m = 2: t = 6307048613500535725718847010810217734170642900311114223618
ok    M_3, d = 2, m = 1: t = 2958790142831846402
== C. The words w_eps
ok    m = 0: 1 words of length 1 with 1 letters x; R sigma_0 = sigma_1 R; R(W_m) = W_m; 1 cyclic words
ok    m = 1: 2 words of length 6 with 3 letters x; R sigma_0 = sigma_1 R; R(W_m) = W_m; 2 cyclic words
ok    m = 2: 4 words of length 21 with 9 letters x; R sigma_0 = sigma_1 R; R(W_m) = W_m; 4 cyclic words
ok    m = 3: 8 words of length 66 with 27 letters x; R sigma_0 = sigma_1 R; R(W_m) = W_m; 8 cyclic words
ok    m = 4: 16 words of length 201 with 81 letters x; R sigma_0 = sigma_1 R; R(W_m) = W_m; 16 cyclic words
ok    m = 5: 32 words of length 606 with 243 letters x; R sigma_0 = sigma_1 R; R(W_m) = W_m; 32 cyclic words
ok    m = 6: 64 words of length 1821 with 729 letters x; R sigma_0 = sigma_1 R; R(W_m) = W_m; 64 cyclic words
ok    m = 1: the two words are x y x^2 y^2 and y^2 x^2 y x
== D. The coincidence for (d, m) = (2, 1)
ok    M_2: i -> -1-i commutes with tau and tau' (the reflection x -> -x is an isometry)
ok    M_3: i -> -1-i commutes with tau and tau' (the reflection x -> -x is an isometry)
ok    theta followed by the conjugation with beta alpha beta exchanges u_0 and v_0
ok    theta(w_0(u_0,v_0)) is conjugate to w_1(u_0,v_0) in F_2; w_0(u_0,v_0) is conjugate neither to w_1(u_0,v_0) nor to its inverse
ok    exchanging x and y maps x y x^2 y^2 to a rotation of y^2 x^2 y x
ok    M_2: N = B A B diag(-1, 1) has determinant -1 and exchanges u and v
ok    M_2: N g_0 N^-1 = (v^2 u^2)^-1 g_1 (v^2 u^2)
ok    M_2: N preserves Lambda = {x + y even}
ok    M_3: N = B A B diag(-1, 1) has determinant -1 and exchanges u and v
ok    M_3: N g_0 N^-1 = (v^2 u^2)^-1 g_1 (v^2 u^2)
== E. The second certificate: twisted homology of M_2 for the 15 characters (d = 2, 3, 4; m <= 6)
ok    cell structure: 8 vertices, 16 edges, 6 faces with [4, 4, 4, 4, 8, 8] sides; every edge occurs once in each direction on the face boundaries; Euler characteristic -2
ok    16 characters of H_1(S; Z/2), among them the holonomy character of q
ok    trivial character: H_1(S; Z) has rank 4 and the two twists act as the identity (Torelli)
ok    holonomy character: the periods of the twisted cycles form the lattice with basis ((2, 2), (0, 4)) = 2 Lambda, and P vanishes on the face boundaries
ok    holonomy character: P tau_a = A P and P tau_b = B P on the twisted cycles, with A = (1 8; 0 1), B = (1 0; 8 1) (so f_eps acts on H_1(S; Z_chi_0)/torsion = 2 Lambda by g_eps)
ok    15 non-trivial characters: rank of H_1(S; Z_chi)/torsion is 2, chain maps preserve cycles and face boundaries, matrices have determinant 1
      character (values on the 9 edges outside the tree) | tau_a | tau_b on L' | distinct |traces| of the f_eps, d = 2, m = 1, ..., 6
      +++++++--       | (1, 0, 0, 1) | (5, 4, -4, -3) | m=1: 2 | m=2: 2 | m=3: 2 | m=4: 2 | m=5: 2 | m=6: 2
      +++++--++       | (1, 0, 0, 1) | (-3, 4, -4, 5) | m=1: 2 | m=2: 2 | m=3: 2 | m=4: 2 | m=5: 2 | m=6: 2
      +++++----       | (1, 0, 0, 1) | (1, 0, 0, 1) | m=1: 2 | m=2: 2 | m=3: 2 | m=4: 2 | m=5: 2 | m=6: 2
      ++--+++++       | (1, 0, 0, 1) | (-3, -4, 4, 5) | m=1: 2 | m=2: 2 | m=3: 2 | m=4: 2 | m=5: 2 | m=6: 2
      ++--+++--       | (1, 4, 0, 1) | (5, -4, 4, -3) | m=1: 69933736962 | m=2: (40 digits) | m=3: (127 digits) | m=4: (387 digits) | m=5: (1168 digits) | m=6: (3511 digits)
      ++--+--++       | (1, 0, 4, 1) | (5, 4, -4, -3) | m=1: 69933736962 | m=2: (40 digits) | m=3: (127 digits) | m=4: (387 digits) | m=5: (1168 digits) | m=6: (3511 digits)
      ++--+----       | (1, 4, 0, 1) | (1, 0, 0, 1) | m=1: 2 | m=2: 2 | m=3: 2 | m=4: 2 | m=5: 2 | m=6: 2
      +-++-++++       | (1, 0, 0, 1) | (5, -4, 4, -3) | m=1: 2 | m=2: 2 | m=3: 2 | m=4: 2 | m=5: 2 | m=6: 2
      +-++-++--       | (1, 4, 0, 1) | (5, -4, 4, -3) | m=1: 69933736962 | m=2: (40 digits) | m=3: (127 digits) | m=4: (387 digits) | m=5: (1168 digits) | m=6: (3511 digits)
      +-++---++       | (1, 0, 4, 1) | (-3, 4, -4, 5) | m=1: 69933736962 | m=2: (40 digits) | m=3: (127 digits) | m=4: (387 digits) | m=5: (1168 digits) | m=6: (3511 digits)
      +-++-----       | (1, 4, 0, 1) | (1, 0, 0, 1) | m=1: 2 | m=2: 2 | m=3: 2 | m=4: 2 | m=5: 2 | m=6: 2
      +----++++       | (1, 0, 0, 1) | (1, 0, 0, 1) | m=1: 2 | m=2: 2 | m=3: 2 | m=4: 2 | m=5: 2 | m=6: 2
      +----++--       | (1, -4, 0, 1) | (1, 0, 0, 1) | m=1: 2 | m=2: 2 | m=3: 2 | m=4: 2 | m=5: 2 | m=6: 2
      +------++       | (1, 0, 4, 1) | (1, 0, 0, 1) | m=1: 2 | m=2: 2 | m=3: 2 | m=4: 2 | m=5: 2 | m=6: 2
      +-------- (hol) | (1, -4, 0, 1) | (25, 36, -16, -23) | m=1: 4508549040455682 | m=2: (58 digits) | m=3: (185 digits) | m=4: (564 digits) | m=5: (1702 digits) | m=6: (5116 digits)
ok    d = 2, m = 1: |trace| = t (16 digits) on L' of the holonomy character, and for no other character and no word (other values: [1, 11] digits)
ok    d = 2, m = 2: |trace| = t (58 digits) on L' of the holonomy character, and for no other character and no word (other values: [1, 40] digits)
ok    d = 2, m = 3: |trace| = t (185 digits) on L' of the holonomy character, and for no other character and no word (other values: [1, 127] digits)
ok    d = 2, m = 4: |trace| = t (564 digits) on L' of the holonomy character, and for no other character and no word (other values: [1, 387] digits)
ok    d = 2, m = 5: |trace| = t (1702 digits) on L' of the holonomy character, and for no other character and no word (other values: [1, 1168] digits)
ok    d = 2, m = 6: |trace| = t (5116 digits) on L' of the holonomy character, and for no other character and no word (other values: [1, 3511] digits)
ok    d = 3, m = 1: |trace| = t (52 digits) on L' of the holonomy character, and for no other character and no word (other values: [1, 36] digits)
ok    d = 3, m = 2: |trace| = t (178 digits) on L' of the holonomy character, and for no other character and no word (other values: [1, 120] digits)
ok    d = 3, m = 3: |trace| = t (554 digits) on L' of the holonomy character, and for no other character and no word (other values: [1, 374] digits)
ok    d = 3, m = 4: |trace| = t (1681 digits) on L' of the holonomy character, and for no other character and no word (other values: [1, 1136] digits)
ok    d = 3, m = 5: |trace| = t (5065 digits) on L' of the holonomy character, and for no other character and no word (other values: [1, 3421] digits)
ok    d = 4, m = 1: |trace| = t (98 digits) on L' of the holonomy character, and for no other character and no word (other values: [1, 66] digits)
ok    d = 4, m = 2: |trace| = t (335 digits) on L' of the holonomy character, and for no other character and no word (other values: [1, 227] digits)
ok    d = 4, m = 3: |trace| = t (1047 digits) on L' of the holonomy character, and for no other character and no word (other values: [1, 708] digits)
ok    d = 4, m = 4: |trace| = t (3185 digits) on L' of the holonomy character, and for no other character and no word (other values: [1, 2153] digits)
ok    d = 2, m = 1: the other characters give |trace| 69933736962 (four characters) and 2 (ten characters)
running time: 0.5 s
TOTAL failures: 0
