r2_exact.py -- verification run 2 (OWR-4531-016)
  [PASS] E0 omega = exp(2 pi i/3): omega^3 = 1, omega^-2 = omega
  [PASS] E1 V(x^3 + y^3) = 0 [Lemma 3.1]
  [PASS] E2 S' = -C^2 identically
  [PASS] E3 C' + S^2 = (1 - x^3 - y^3)/x^2, so C' = -S^2 on the cubic
  [PASS] E4 Q_k = (S, C) = (0, -omega^k) lie on 1 + C^3 = S^3 and the field (-C^2, -S^2) is (-omega^(2k), 0) there
  [PASS] E5 rotation: -omega s^2 = -(omega^-1 s)^2 [proof of (D3)]
  [PASS] E6 sm = eps i => cm^3 = 1 + eps i (eps = -1, 1) [Cor. 3.2(b)]
  [PASS] E7 q + i q_H = 1 and q^3 + q_G^3 - q_H^3 = (s^3 + c^3 - 1)/(s + i)^3 [Cor. 3.2(c), Sec. 5.1]
  [PASS] E8 on the cubic (c/s)^3 = s^-3 - 1 [Cor. 3.2(c)]
  [PASS] E9 for real s, c: conj(s/(s+i)) = s/(s-i), conj(c/(s+i)) = c/(s-i), conj(1/(s+i)) = 1/(s-i) [Cor. 3.2(d)]
  [PASS] E10 b^3 (X^3 - 1) - (aX + c)^3 has coefficients b^3 - a^3, -3a^2 c, -3a c^2, -(b^3 + c^3) [Sec. 5.4]
  [PASS] E11 these four coefficients vanish only for a = b = c = 0   [{a: 0, b: 0, c: 0}]
  [PASS] E12 (1 + s^2)^3 - (1 - s^3)^2 = s^2 (3s^2 + 2s + 3), and 3s^2 + 2s + 3 has no real zero [Rem. 5.4]
  [PASS] E13 (1 + e^{it})/(1 - e^{it}) = i cot(t/2) (identity in v = e^{it/2}) [Lemma 4.1]
  [PASS] E14 tanh(u1 + u2) = (tanh u1 + tanh u2)/(1 + tanh u1 tanh u2) (formula for gamma, identity in e^{2u})
  [PASS] E15 pi p0/h = 2 pi/sqrt 3; omega p0 = -p0/2 + ih; 2ih = p0 (1 + 2 omega); ih = omega p0 + p0/2
  [PASS] E16 1 - |tanh(s/2)|^2 = 4 Re(e^s)/|e^s + 1|^2 = 4 e^sigma cos tau/|e^s + 1|^2 [Lemma 4.2(b)]
  [PASS] E17 tanh(s/2) - 1 = -2/(e^s + 1), tanh(s/2) + 1 = 2/(e^-s + 1)
  [PASS] E18 C0 = 4/(1 - 1/e)^2 = 10.010601 > e; (1 - e^-1)^-2 bounds the cases sigma >= 1, sigma <= -1
  [PASS] E19 1 - |z|^2 <= C0 e^-|sigma| at 2000 random points of the strip |Im s| < pi/2 [(5)]   max ratio 0.399576
  [PASS] E20 1 - |phi(z)|^2 = |phi'(z)| (1 - |z|^2) for phi = e^{i th}(z - b)/(1 - conj(b) z) (200 random cases) [Lemma 2.2]
  [PASS] E21 pseudo-hyperbolic distance invariant under automorphisms (200 random cases) [(H2)]
  [PASS] E22 (1 - r^2)/(1 - r)^2 = (1 + r)/(1 - r) and (1 - r^2)/(1 + r)^2 = (1 - r)/(1 + r) [Lemma 5.3]
  [PASS] E23 for |z| > 1 (off the real axis): psi(1/conj z) - conj psi(z) in {0, 2ih} (3000 random z) [Rem. 6.3]   value 0: 1509 times, value 2ih: 1491 times
  [PASS] E24 (z/2)^n + (1 - z^n/2^n) = 1 and the root has z^n-coefficient -1/(n 2^n) != 0 (n = 3..7): {z/2, (1 - z^n/2^n)^(1/n), 1} is linearly independent [footnote 1, Sec. 1.2]
  [PASS] E25 sum_{j,k >= 1} 2^(-j-k) = 1 [proof of Cor. 1.3]

TOTAL failures: 0
