Part 1. Proposition 2.4: the points +-1, +-2, +-3 are not free (sharpness of Theorem A)
  ok    A^n = (1 n; 0 1) and B_w^n = B_(nw) for |n| <= 5 and four integer w
  ok    B_1 B_-1 = I, so B_-1 is the inverse of B_1
  ok    k=1: X = A B_(-1) = ((0, 1), (-1, 1)), det 1, trace 1, order exactly 6 (expected 6)
        => (x y^-1)^6 is a relation of <A, B_1> and (x y)^6 of <A, B_(-1)>: 1 and -1 are not free points
  ok    B_2 B_-2 = I, so B_-2 is the inverse of B_2
  ok    k=2: X = A B_(-2) = ((-1, 1), (-2, 1)), det 1, trace 0, order exactly 4 (expected 4)
        => (x y^-1)^4 is a relation of <A, B_2> and (x y)^4 of <A, B_(-2)>: 2 and -2 are not free points
  ok    B_3 B_-3 = I, so B_-3 is the inverse of B_3
  ok    k=3: X = A B_(-3) = ((-2, 1), (-3, 1)), det 1, trace -1, order exactly 3 (expected 3)
        => (x y^-1)^3 is a relation of <A, B_3> and (x y)^3 of <A, B_(-3)>: 3 and -3 are not free points
  m(1) = m(-1) = 4: 1, 2, 3 (resp. -1, -2, -3) are not free, and 4 (resp. -4) is free by Lemma 2.1.
  m(2) = m(3) = 2 and m(4) = 1 likewise (|2*2| = 4, |2*3| = 6, |4| = 4).
Part 2. Proposition 4.2 and Table 1, S3: |G| = 6
  ok    character table of S3: orthonormal, sum of squared degrees = 6
  ok    nontrivial bicyclic units: 6 of type beta, 6 of type gamma; with the unit 1: 7 of one type (expected 7); all a^2 = 0
  conjugation orbits: 1 (beta), 1 (gamma)
  (beta,beta): pairs (orbit rep x all) 6, ab not nilpotent 4, min over these of max|eigenvalue| = 9
      m=1: {'FREE': 4, 'NONFREE': 0, 'UNKNOWN': 0}
      m=2: {'FREE': 4, 'NONFREE': 0, 'UNKNOWN': 0}
      m=3: {'FREE': 4, 'NONFREE': 0, 'UNKNOWN': 0}
      m=4: {'FREE': 4, 'NONFREE': 0, 'UNKNOWN': 0}
  (beta,gamma): pairs (orbit rep x all) 6, ab not nilpotent 6, min over these of max|eigenvalue| = 3
      m=1: {'FREE': 2, 'NONFREE': 4, 'UNKNOWN': 0}
      m=2: {'FREE': 6, 'NONFREE': 0, 'UNKNOWN': 0}
      m=3: {'FREE': 6, 'NONFREE': 0, 'UNKNOWN': 0}
      m=4: {'FREE': 6, 'NONFREE': 0, 'UNKNOWN': 0}
      non-free at m=1, e.g. 1+a = beta_(x=(1,2), h=(2,3)), 1+b = gamma_(x=(2,3), h=(1,2)); nonzero eigenvalues of ab: [-3]
  (gamma,beta): pairs (orbit rep x all) 6, ab not nilpotent 6, min over these of max|eigenvalue| = 3
      m=1: {'FREE': 2, 'NONFREE': 4, 'UNKNOWN': 0}
      m=2: {'FREE': 6, 'NONFREE': 0, 'UNKNOWN': 0}
      m=3: {'FREE': 6, 'NONFREE': 0, 'UNKNOWN': 0}
      m=4: {'FREE': 6, 'NONFREE': 0, 'UNKNOWN': 0}
      non-free at m=1, e.g. 1+a = gamma_(x=(1,2), h=(2,3)), 1+b = beta_(x=(2,3), h=(1,2)); nonzero eigenvalues of ab: [-3]
  (gamma,gamma): pairs (orbit rep x all) 6, ab not nilpotent 4, min over these of max|eigenvalue| = 9
      m=1: {'FREE': 4, 'NONFREE': 0, 'UNKNOWN': 0}
      m=2: {'FREE': 4, 'NONFREE': 0, 'UNKNOWN': 0}
      m=3: {'FREE': 4, 'NONFREE': 0, 'UNKNOWN': 0}
      m=4: {'FREE': 4, 'NONFREE': 0, 'UNKNOWN': 0}
  Table 1 row, S3 same: pairs 12, ab not nilpotent 8, min max|lambda| 9, not free at m=1: 0, free at m=2: 8
  Table 1 row, S3 mixed: pairs 12, ab not nilpotent 12, min max|lambda| 3, not free at m=1: 8, free at m=2: 12
  ok    every pair with ab not nilpotent is decided (FREE or NON-FREE) for m = 1, 2, 3, 4
  ok    the involution g -> g^-1 symmetry: (gamma,gamma) and (gamma,beta) counts equal (beta,beta), (beta,gamma)
  ok    m_beta(S3) = 1, m(S3) = 1, M(S3) = 2 (expected m = 1, M = 2)
  distinct products ab with certified spectrum: 21; time 0.0 s
Part 2. Proposition 4.2 and Table 1, S4: |G| = 24
  ok    character table of S4: orthonormal, sum of squared degrees = 24
  ok    nontrivial bicyclic units: 156 of type beta, 156 of type gamma; with the unit 1: 157 of one type (expected 157); all a^2 = 0
  conjugation orbits: 8 (beta), 8 (gamma)
  (beta,beta): pairs (orbit rep x all) 1248, ab not nilpotent 838, min over these of max|eigenvalue| = 2
      m=1: {'FREE': 826, 'NONFREE': 12, 'UNKNOWN': 0}
      m=2: {'FREE': 838, 'NONFREE': 0, 'UNKNOWN': 0}
      m=3: {'FREE': 838, 'NONFREE': 0, 'UNKNOWN': 0}
      m=4: {'FREE': 838, 'NONFREE': 0, 'UNKNOWN': 0}
      non-free at m=1, e.g. 1+a = beta_(x=(2,3), h=(3,4)), 1+b = beta_(x=(1,2)(3,4), h=(1,3)); nonzero eigenvalues of ab: [-2, 2]
  (beta,gamma): pairs (orbit rep x all) 1248, ab not nilpotent 974, min over these of max|eigenvalue| = 2
      m=1: {'FREE': 926, 'NONFREE': 48, 'UNKNOWN': 0}
      m=2: {'FREE': 974, 'NONFREE': 0, 'UNKNOWN': 0}
      m=3: {'FREE': 974, 'NONFREE': 0, 'UNKNOWN': 0}
      m=4: {'FREE': 974, 'NONFREE': 0, 'UNKNOWN': 0}
      non-free at m=1, e.g. 1+a = beta_(x=(2,3), h=(3,4)), 1+b = gamma_(x=(3,4), h=(2,3)); nonzero eigenvalues of ab: [-3]
  (gamma,beta): pairs (orbit rep x all) 1248, ab not nilpotent 974, min over these of max|eigenvalue| = 2
      m=1: {'FREE': 926, 'NONFREE': 48, 'UNKNOWN': 0}
      m=2: {'FREE': 974, 'NONFREE': 0, 'UNKNOWN': 0}
      m=3: {'FREE': 974, 'NONFREE': 0, 'UNKNOWN': 0}
      m=4: {'FREE': 974, 'NONFREE': 0, 'UNKNOWN': 0}
      non-free at m=1, e.g. 1+a = gamma_(x=(2,3), h=(3,4)), 1+b = beta_(x=(3,4), h=(2,3)); nonzero eigenvalues of ab: [-3]
  (gamma,gamma): pairs (orbit rep x all) 1248, ab not nilpotent 838, min over these of max|eigenvalue| = 2
      m=1: {'FREE': 826, 'NONFREE': 12, 'UNKNOWN': 0}
      m=2: {'FREE': 838, 'NONFREE': 0, 'UNKNOWN': 0}
      m=3: {'FREE': 838, 'NONFREE': 0, 'UNKNOWN': 0}
      m=4: {'FREE': 838, 'NONFREE': 0, 'UNKNOWN': 0}
      non-free at m=1, e.g. 1+a = gamma_(x=(2,3), h=(3,4)), 1+b = gamma_(x=(1,2)(3,4), h=(1,3)); nonzero eigenvalues of ab: [-2, 2]
  Table 1 row, S4 same: pairs 2496, ab not nilpotent 1676, min max|lambda| 2, not free at m=1: 24, free at m=2: 1676
  Table 1 row, S4 mixed: pairs 2496, ab not nilpotent 1948, min max|lambda| 2, not free at m=1: 96, free at m=2: 1948
  ok    every pair with ab not nilpotent is decided (FREE or NON-FREE) for m = 1, 2, 3, 4
  ok    the involution g -> g^-1 symmetry: (gamma,gamma) and (gamma,beta) counts equal (beta,beta), (beta,gamma)
  ok    m_beta(S4) = 2, m(S4) = 2, M(S4) = 2 (expected m = 2, M = 2)
  distinct products ab with certified spectrum: 1857; time 0.2 s
Part 3. Proposition 4.3: the pair of Z S5
  1+a = beta_(x=(1,2), h=(2,3)(4,5)),  1+b = beta_(x=(1,3,4,2,5), h=(1,2)(3,4))  (points 1..5)
  ok    a, b nonzero with a^2 = b^2 = 0
  ok    p(t) = t(t-1)(t+1)(t^2-2t+3)(t^2+2t+3) = t^7 + t^5 + 7t^3 - 9t
  ok    p(ab) = 0 in Z S5
  ok    (p / (t))(ab) != 0, so t divides the minimal polynomial
  ok    (p / (t-1))(ab) != 0, so t-1 divides the minimal polynomial
  ok    (p / (t+1))(ab) != 0, so t+1 divides the minimal polynomial
  ok    (p / (t^2-2t+3))(ab) != 0, so t^2-2t+3 divides the minimal polynomial
  ok    (p / (t^2+2t+3))(ab) != 0, so t^2+2t+3 divides the minimal polynomial
  => the minimal polynomial of ab is p; its nonzero roots are +-1 and +-1 +- i sqrt2,
     with |root|^2 in {1, 3}: every nonzero eigenvalue has modulus < 2, so |2 lambda| < 4 throughout;
     but Im(2(1 + i sqrt2))^2 = 8 > 4, so 2(1 + i sqrt2) is free by Lemma 4.1 (|Im w| > 2).
  ok    exact inequalities |1 + i sqrt2|^2 = 3 < 4 and (2 sqrt2)^2 = 8 > 4
  time 0.0 s
TOTAL failures: 0   (total time 0.2 s)
