check_note.py -- mpmath 1.3.0 at 50 digits; sympy 1.14.0

(i) reflection-law orbits: the five rows, controls of the wrong parity, side rows
  [PASS] a=1 b=0.5 N=3 tau=1 lam=0.245678061214 : closure 3.53e-49, definitions 2.83e-49; constant k808,k810,k103,k109,k112,k113 max spread 3.09e-49; varying k110,k807,k809,k106,k111 min spread 0.2
  [PASS] a=1 b=0.5 N=4 tau=1 lam=0.2 : closure 6.04e-50, definitions 6.41e-50; constant k110,k807,k809,k106,k113 max spread 2.05e-50; varying k808,k810,k103,k109,k112,k111 min spread 0.199
  [PASS] a=1 b=0.5 N=5 tau=1 lam=0.148624447437 : closure 7.79e-50, definitions 2.94e-50; constant k808,k810,k103,k109,k112,k113 max spread 1.24e-50; varying k110,k807,k809,k106,k111 min spread 0.000292
  [PASS] a=1 b=0.5 N=6 tau=1 lam=0.111111111111 : closure 4.8e-49, definitions 4.13e-49; constant k110,k807,k809,k106,k113 max spread 3.41e-50; varying k808,k810,k103,k109,k112,k111 min spread 0.0128
  [PASS] a=1 b=0.5 N=7 tau=1 lam=0.0851455314456 : closure 3.82e-49, definitions 2.22e-49; constant k808,k810,k103,k109,k112,k113 max spread 6.15e-50; varying k110,k807,k809,k106,k111 min spread 1.43e-6
  [PASS] a=1 b=0.5 N=8 tau=1 lam=0.0669380892315 : closure 7.31e-49, definitions 3.82e-49; constant k110,k807,k809,k106,k113 max spread 3.58e-50; varying k808,k810,k103,k109,k112,k111 min spread 0.00086
  [PASS] a=1 b=0.5 N=9 tau=1 lam=0.0538375047585 : closure 5.47e-49, definitions 3.17e-49; constant k808,k810,k103,k109,k112,k113 max spread 6.68e-50; varying k110,k807,k809,k106,k111 min spread 6.21e-9
  [PASS] a=1 b=0.5 N=10 tau=1 lam=0.0441581104773 : closure 1.06e-48, definitions 4.12e-49; constant k110,k807,k809,k106,k113 max spread 3.03e-50; varying k808,k810,k103,k109,k112,k111 min spread 6.65e-5
  [PASS] a=1 b=0.5 N=5 tau=2 lam=0.249977112539 : closure 4.9e-47, definitions 7.29e-47; constant k808,k810,k103,k109,k112,k113 max spread 7.73e-47; varying k110,k807,k809,k106,k111 min spread 0.198
  [PASS] a=1 b=0.5 N=7 tau=3 lam=0.249999881991 : closure 2.44e-44, definitions 1.92e-44; constant k808,k810,k103,k109,k112,k113 max spread 2.37e-44; varying k110,k807,k809,k106,k111 min spread 0.198
  [PASS] a=1 b=0.5 N=8 tau=3 lam=0.249682007598 : closure 8.2e-48, definitions 5.24e-48; constant k110,k807,k809,k106,k113 max spread 1.24e-48; varying k808,k810,k103,k109,k112,k111 min spread 0.171
  [PASS] a=1 b=0.5 N=9 tau=4 lam=0.249999999392 : closure 2.98e-42, definitions 1.69e-42; constant k808,k810,k103,k109,k112,k113 max spread 3.12e-42; varying k110,k807,k809,k106,k111 min spread 0.198
  [PASS] a=1 b=0.5 N=10 tau=3 lam=0.234696063681 : closure 1.85e-48, definitions 2.18e-48; constant k110,k807,k809,k106,k113 max spread 1.03e-49; varying k808,k810,k103,k109,k112,k111 min spread 0.00473
  [PASS] a=1 b=0.5 N=12 tau=5 lam=0.249998356381 : closure 1.22e-45, definitions 1.23e-45; constant k110,k807,k809,k106,k113 max spread 6.26e-46; varying k808,k810,k103,k109,k112,k111 min spread 0.171
  [PASS] a=1 b=0.8 N=3 tau=1 lam=0.565614696083 : closure 4.26e-49, definitions 5.69e-49; constant k808,k810,k103,k109,k112,k113 max spread 2.18e-49; varying k110,k807,k809,k106,k111 min spread 0.011
  [PASS] a=1 b=0.8 N=4 tau=1 lam=0.390243902439 : closure 2.33e-49, definitions 2.16e-49; constant k110,k807,k809,k106,k113 max spread 1.02e-50; varying k808,k810,k103,k109,k112,k111 min spread 0.0289
  [PASS] a=1 b=0.8 N=5 tau=1 lam=0.271998690002 : closure 3.85e-49, definitions 2.66e-49; constant k808,k810,k103,k109,k112,k113 max spread 3.88e-50; varying k110,k807,k809,k106,k111 min spread 1.21e-6
  [PASS] a=1 b=0.8 N=6 tau=1 lam=0.197530864198 : closure 1.74e-49, definitions 1.12e-49; constant k110,k807,k809,k106,k113 max spread 5.93e-51; varying k808,k810,k103,k109,k112,k111 min spread 0.000364
  [PASS] a=1 b=0.8 N=7 tau=1 lam=0.149025430483 : closure 6.5e-49, definitions 3.47e-49; constant k808,k810,k103,k109,k112,k113 max spread 3.74e-50; varying k110,k807,k809,k106,k111 min spread 4.48e-10
  [PASS] a=1 b=0.8 N=8 tau=1 lam=0.116059503984 : closure 8.98e-49, definitions 3.8e-49; constant k110,k807,k809,k106,k113 max spread 1.45e-50; varying k808,k810,k103,k109,k112,k111 min spread 9.43e-6
  [PASS] a=1 b=0.8 N=9 tau=1 lam=0.092772878308 : closure 8.59e-49, definitions 3.38e-49; constant k808,k810,k103,k109,k112,k113 max spread 2.81e-50; varying k110,k807,k809,k106,k111 min spread 2.95e-13
  [PASS] a=1 b=0.8 N=10 tau=1 lam=0.0757704897974 : closure 5.07e-49, definitions 1.96e-49; constant k110,k807,k809,k106,k113 max spread 7.75e-51; varying k808,k810,k103,k109,k112,k111 min spread 2.35e-7
  [PASS] a=1 b=0.8 N=5 tau=2 lam=0.634530346522 : closure 1.56e-48, definitions 2.59e-48; constant k808,k810,k103,k109,k112,k113 max spread 1.29e-48; varying k110,k807,k809,k106,k111 min spread 0.00671
  [PASS] a=1 b=0.8 N=7 tau=3 lam=0.639649388452 : closure 1.17e-48, definitions 1.43e-47; constant k808,k810,k103,k109,k112,k113 max spread 8.75e-48; varying k110,k807,k809,k106,k111 min spread 0.00671
  [PASS] a=1 b=0.8 N=8 tau=3 lam=0.619218354684 : closure 2.56e-48, definitions 4.13e-48; constant k110,k807,k809,k106,k113 max spread 4.88e-49; varying k808,k810,k103,k109,k112,k111 min spread 0.00801
  [PASS] a=1 b=0.8 N=9 tau=4 lam=0.639978032329 : closure 2.61e-46, definitions 3.48e-46; constant k808,k810,k103,k109,k112,k113 max spread 3.43e-46; varying k110,k807,k809,k106,k111 min spread 0.00687
  [PASS] a=1 b=0.8 N=10 tau=3 lam=0.502766113943 : closure 1.79e-49, definitions 1.68e-49; constant k110,k807,k809,k106,k113 max spread 3.43e-50; varying k808,k810,k103,k109,k112,k111 min spread 2.86e-5
  [PASS] a=1 b=0.8 N=12 tau=5 lam=0.63860628185 : closure 1.24e-48, definitions 4.83e-48; constant k110,k807,k809,k106,k113 max spread 2.21e-48; varying k808,k810,k103,k109,k112,k111 min spread 0.00683
  [PASS] a=2 b=1.2 N=3 tau=1 lam=1.3870440561 : closure 2.11e-48, definitions 1.27e-48; constant k808,k810,k103,k109,k112,k113 max spread 8.37e-49; varying k110,k807,k809,k106,k111 min spread 0.0993
  [PASS] a=2 b=1.2 N=4 tau=1 lam=1.05882352941 : closure 4.35e-49, definitions 1.71e-49; constant k110,k807,k809,k106,k113 max spread 1.25e-50; varying k808,k810,k103,k109,k112,k111 min spread 0.112
  [PASS] a=2 b=1.2 N=5 tau=1 lam=0.763261630453 : closure 4.96e-49, definitions 2.26e-49; constant k808,k810,k103,k109,k112,k113 max spread 7.62e-50; varying k110,k807,k809,k106,k111 min spread 7.27e-5
  [PASS] a=2 b=1.2 N=6 tau=1 lam=0.5625 : closure 1.02e-48, definitions 2.46e-49; constant k110,k807,k809,k106,k113 max spread 1.69e-50; varying k808,k810,k103,k109,k112,k111 min spread 0.00483
  [PASS] a=2 b=1.2 N=7 tau=1 lam=0.427698589537 : closure 1.16e-48, definitions 2.97e-49; constant k808,k810,k103,k109,k112,k113 max spread 5.61e-50; varying k110,k807,k809,k106,k111 min spread 1.78e-7
  [PASS] a=2 b=1.2 N=8 tau=1 lam=0.334650960999 : closure 1.79e-48, definitions 3.63e-49; constant k110,k807,k809,k106,k113 max spread 2.66e-50; varying k808,k810,k103,k109,k112,k111 min spread 0.000246
  [PASS] a=2 b=1.2 N=9 tau=1 lam=0.268321994664 : closure 4.38e-49, definitions 8.9e-50; constant k808,k810,k103,k109,k112,k113 max spread 1.34e-50; varying k110,k807,k809,k106,k111 min spread 5.05e-10
  [PASS] a=2 b=1.2 N=10 tau=1 lam=0.219608278796 : closure 4.42e-49, definitions 8.73e-50; constant k110,k807,k809,k106,k113 max spread 6.49e-51; varying k808,k810,k103,k109,k112,k111 min spread 1.52e-5
  [PASS] a=2 b=1.2 N=5 tau=2 lam=1.43930728718 : closure 1.4e-46, definitions 5.12e-47; constant k808,k810,k103,k109,k112,k113 max spread 6.73e-47; varying k110,k807,k809,k106,k111 min spread 0.0902
  [PASS] a=2 b=1.2 N=7 tau=3 lam=1.43999143649 : closure 1.33e-44, definitions 4.94e-45; constant k808,k810,k103,k109,k112,k113 max spread 7.17e-45; varying k110,k807,k809,k106,k111 min spread 0.0901
  [PASS] a=2 b=1.2 N=8 tau=3 lam=1.43381631277 : closure 2.49e-47, definitions 1.09e-47; constant k110,k807,k809,k106,k113 max spread 2.16e-48; varying k808,k810,k103,k109,k112,k111 min spread 0.102
  [PASS] a=2 b=1.2 N=9 tau=4 lam=1.43999989428 : closure 8.36e-43, definitions 3.01e-43; constant k808,k810,k103,k109,k112,k113 max spread 4.3e-43; varying k110,k807,k809,k106,k111 min spread 0.09
  [PASS] a=2 b=1.2 N=10 tau=3 lam=1.29275808943 : closure 3.35e-48, definitions 1.61e-48; constant k110,k807,k809,k106,k113 max spread 4.97e-50; varying k808,k810,k103,k109,k112,k111 min spread 0.00152
  [PASS] a=2 b=1.2 N=12 tau=5 lam=1.43992294084 : closure 2.21e-45, definitions 7.98e-46; constant k110,k807,k809,k106,k113 max spread 3.24e-46; varying k808,k810,k103,k109,k112,k111 min spread 0.101
  [PASS] a=1.7 b=0.3 N=3 tau=1 lam=0.0899778436087 : closure 2.26e-46, definitions 2.73e-47; constant k808,k810,k103,k109,k112,k113 max spread 1.39e-46; varying k110,k807,k809,k106,k111 min spread 0.961
  [PASS] a=1.7 b=0.3 N=4 tau=1 lam=0.0872818791946 : closure 1.79e-48, definitions 4.56e-48; constant k110,k807,k809,k106,k113 max spread 4.26e-50; varying k808,k810,k103,k109,k112,k111 min spread 0.804
  [PASS] a=1.7 b=0.3 N=5 tau=1 lam=0.0775422941161 : closure 4.47e-49, definitions 1.42e-49; constant k808,k810,k103,k109,k112,k113 max spread 1.14e-49; varying k110,k807,k809,k106,k111 min spread 0.0207
  [PASS] a=1.7 b=0.3 N=6 tau=1 lam=0.065025 : closure 6.74e-49, definitions 3.03e-49; constant k110,k807,k809,k106,k113 max spread 2.82e-50; varying k808,k810,k103,k109,k112,k111 min spread 0.163
  [PASS] a=1.7 b=0.3 N=7 tau=1 lam=0.0535233276092 : closure 9.49e-49, definitions 2.43e-49; constant k808,k810,k103,k109,k112,k113 max spread 1.68e-49; varying k110,k807,k809,k106,k111 min spread 0.000447
  [PASS] a=1.7 b=0.3 N=8 tau=1 lam=0.0440753718619 : closure 8.34e-49, definitions 3.76e-49; constant k110,k807,k809,k106,k113 max spread 4.45e-50; varying k808,k810,k103,k109,k112,k111 min spread 0.0272
  [PASS] a=1.7 b=0.3 N=9 tau=1 lam=0.0365846653578 : closure 2.31e-49, definitions 5.48e-50; constant k808,k810,k103,k109,k112,k113 max spread 1.87e-50; varying k110,k807,k809,k106,k111 min spread 1.61e-5
  [PASS] a=1.7 b=0.3 N=10 tau=1 lam=0.0306849946782 : closure 1.21e-48, definitions 3.45e-49; constant k110,k807,k809,k106,k113 max spread 4.33e-50; varying k808,k810,k103,k109,k112,k111 min spread 0.00451
  [PASS] a=1.7 b=0.3 N=5 tau=2 lam=0.0899999986135 : closure 3.43e-42, definitions 5.54e-43; constant k808,k810,k103,k109,k112,k113 max spread 2.15e-42; varying k110,k807,k809,k106,k111 min spread 0.961
  [PASS] a=1.7 b=0.3 N=7 tau=2 lam=0.0894456628989 : closure 2.62e-47, definitions 8.04e-48; constant k808,k810,k103,k109,k112,k113 max spread 7.41e-48; varying k110,k807,k809,k106,k111 min spread 0.0187
  [PASS] a=1.7 b=0.3 N=8 tau=3 lam=0.0899998246984 : closure 8.98e-45, definitions 2.77e-44; constant k110,k807,k809,k106,k113 max spread 8.97e-45; varying k808,k810,k103,k109,k112,k111 min spread 0.8
  [PASS] a=1.7 b=0.3 N=9 tau=2 lam=0.083125738348 : closure 1.75e-48, definitions 9.69e-49; constant k808,k810,k103,k109,k112,k113 max spread 5.85e-49; varying k110,k807,k809,k106,k111 min spread 0.000325
  [PASS] a=1.7 b=0.3 N=10 tau=3 lam=0.0897516339243 : closure 1.87e-47, definitions 1.91e-47; constant k110,k807,k809,k106,k113 max spread 1.81e-48; varying k808,k810,k103,k109,k112,k111 min spread 0.143
  [PASS] a=1.7 b=0.3 N=11 tau=4 lam=0.0899991201593 : closure 8.84e-45, definitions 3.05e-45; constant k808,k810,k103,k109,k112,k113 max spread 1.66e-45; varying k110,k807,k809,k106,k111 min spread 0.0186
  summary (i): 56 families, 392 polygons; largest closure defect 3.43e-42, largest residual of the definitions 1.69e-42, largest relative spread of a constant row 3.12e-42, smallest relative spread of a row of the wrong parity 2.95e-13

(ii) constants against the closed forms; (iv) conjugate-billiard relation on the same orbits
  [PASS] a=1 b=0.5 N=3 tau=1: (ii) Theorem B gamma = -1.0, 2+gamma = 1.0, k808 = k810 = 35.0277563773199 : G - gamma F 2.19e-50, period Delta 5.4e-50, rows and three forms 1.0e-49
  [PASS] a=1 b=0.5 N=3 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 3.23e-50, largest deviation 9.16e-49
  [PASS] a=1 b=0.5 N=4 tau=1: (ii) Theorem A constants c1 = 0.8, c2 = 0.0 : complex phases 2.22e-50, rows vs closed forms 1.8e-49
  [PASS] a=1 b=0.5 N=4 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 3.05e-50, largest deviation 5.92e-50
  [PASS] a=1 b=0.5 N=5 tau=1: (ii) Theorem B gamma = 0.641234989147323, 2+gamma = 2.641234989, k808 = k810 = 2.57748264084281 : G - gamma F 9.55e-51, period Delta 3.36e-51, rows and three forms 8.34e-51
  [PASS] a=1 b=0.5 N=5 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 1.71e-50, largest deviation 5.52e-50
  [PASS] a=1 b=0.5 N=6 tau=1: (ii) Theorem A constants c1 = 1.56161859761401, c2 = 1.56161859761401 : complex phases 9.42e-50, rows vs closed forms 2.25e-49
  [PASS] a=1 b=0.5 N=6 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 2.26e-49, largest deviation 3.04e-49
  [PASS] a=1 b=0.5 N=7 tau=1: (ii) Theorem B gamma = 1.22871575114579, 2+gamma = 3.228715751, k808 = k810 = 1.59504821199 : G - gamma F 6.22e-50, period Delta 2.25e-50, rows and three forms 3.35e-50
  [PASS] a=1 b=0.5 N=7 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 1.58e-49, largest deviation 2.17e-49
  [PASS] a=1 b=0.5 N=8 tau=1: (ii) Theorem A constants c1 = 1.92090980254623, c2 = 2.66340473239864 : complex phases 1.41e-49, rows vs closed forms 1.47e-49
  [PASS] a=1 b=0.5 N=8 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 3.4e-49, largest deviation 4.31e-49
  [PASS] a=1 b=0.5 N=9 tau=1: (ii) Theorem B gamma = 1.50077459718467, 2+gamma = 3.500774597, k808 = k810 = 1.32609608696602 : G - gamma F 9.4e-50, period Delta 2.05e-50, rows and three forms 3.43e-50
  [PASS] a=1 b=0.5 N=9 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 2.86e-49, largest deviation 3.1e-49
  [PASS] a=1 b=0.5 N=10 tau=1: (ii) Theorem A constants c1 = 2.10616867034395, c2 = 3.34092443365156 : complex phases 1.76e-49, rows vs closed forms 1.16e-49
  [PASS] a=1 b=0.5 N=10 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 4.33e-49, largest deviation 4.81e-49
  [PASS] a=1 b=0.5 N=5 tau=2: (ii) Theorem B gamma = -1.14206515520185, 2+gamma = 0.8579348448, k808 = k810 = 562.651681832112 : G - gamma F 3.49e-49, period Delta 1.87e-48, rows and three forms 1.57e-47
  [PASS] a=1 b=0.5 N=5 tau=2: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 1.75e-48, largest deviation 4.46e-45
  [PASS] a=1 b=0.5 N=7 tau=3: (ii) Theorem B gamma = -1.14285305830925, 2+gamma = 0.8571469417, k808 = k810 = 7843.08021493967 : G - gamma F 5.42e-46, period Delta 2.94e-45, rows and three forms 1.65e-45
  [PASS] a=1 b=0.5 N=7 tau=3: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 1.29e-45, largest deviation 1.51e-41
  [PASS] a=1 b=0.5 N=8 tau=3: (ii) Theorem A constants c1 = 1.06051416745349, c2 = -1.20038425795391 : complex phases 7.12e-49, rows vs closed forms 1.08e-45
  [PASS] a=1 b=0.5 N=8 tau=3: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 4.71e-49, largest deviation 6.57e-47
  [PASS] a=1 b=0.5 N=9 tau=4: (ii) Theorem B gamma = -1.14285712180215, 2+gamma = 0.8571428782, k808 = k810 = 109240.442379646 : G - gamma F 4.54e-43, period Delta 2.46e-42, rows and three forms 5.51e-43
  [PASS] a=1 b=0.5 N=9 tau=4: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 5.44e-43, largest deviation 2.03e-38
  [PASS] a=1 b=0.5 N=10 tau=3: (ii) Theorem A constants c1 = 3.93910595347379, c2 = -2.71033831752938 : complex phases 1.39e-49, rows vs closed forms 4.72e-48
  [PASS] a=1 b=0.5 N=10 tau=3: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 4.84e-49, largest deviation 1.93e-48
  [PASS] a=1 b=0.5 N=12 tau=5: (ii) Theorem A constants c1 = 1.0622330361267, c2 = -1.21392018405041 : complex phases 5.05e-46, rows vs closed forms 6.05e-41
  [PASS] a=1 b=0.5 N=12 tau=5: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 2.2e-46, largest deviation 2.07e-43
  [PASS] a=1 b=0.8 N=3 tau=1: (ii) Theorem B gamma = -1.0, 2+gamma = 1.0, k808 = k810 = 17.8020020017791 : G - gamma F 1.52e-49, period Delta 6.47e-50, rows and three forms 2.81e-49
  [PASS] a=1 b=0.8 N=3 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 1.75e-49, largest deviation 7.55e-49
  [PASS] a=1 b=0.8 N=4 tau=1: (ii) Theorem A constants c1 = 2.49756097560976, c2 = 0.0 : complex phases 3.43e-50, rows vs closed forms 2.73e-49
  [PASS] a=1 b=0.8 N=4 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 1.16e-49, largest deviation 2.02e-49
  [PASS] a=1 b=0.8 N=5 tau=1: (ii) Theorem B gamma = 0.620532621969501, 2+gamma = 2.620532622, k808 = k810 = 2.35922654005618 : G - gamma F 1.24e-49, period Delta 1.77e-50, rows and three forms 9.29e-50
  [PASS] a=1 b=0.8 N=5 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 2.08e-49, largest deviation 2.78e-49
  [PASS] a=1 b=0.8 N=6 tau=1: (ii) Theorem A constants c1 = 4.28441969813316, c2 = 4.28441969813316 : complex phases 1.25e-50, rows vs closed forms 4.91e-50
  [PASS] a=1 b=0.8 N=6 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 9.44e-50, largest deviation 1.01e-49
  [PASS] a=1 b=0.8 N=7 tau=1: (ii) Theorem B gamma = 1.24499315246412, 2+gamma = 3.244993152, k808 = k810 = 1.52562713207873 : G - gamma F 1.35e-49, period Delta 1.07e-50, rows and three forms 7.18e-50
  [PASS] a=1 b=0.8 N=7 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 3.64e-49, largest deviation 3.76e-49
  [PASS] a=1 b=0.8 N=8 tau=1: (ii) Theorem A constants c1 = 5.09822521273348, c2 = 7.19466395791987 : complex phases 7.08e-50, rows vs closed forms 8.99e-50
  [PASS] a=1 b=0.8 N=8 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 4.68e-49, largest deviation 4.49e-49
  [PASS] a=1 b=0.8 N=9 tau=1: (ii) Theorem B gamma = 1.52870203267776, 2+gamma = 3.528702033, k808 = k810 = 1.28704333903517 : G - gamma F 1.32e-49, period Delta 7.37e-51, rows and three forms 6.02e-50
  [PASS] a=1 b=0.8 N=9 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 4.94e-49, largest deviation 4.78e-49
  [PASS] a=1 b=0.8 N=10 tau=1: (ii) Theorem A constants c1 = 5.51338042409207, c2 = 8.90195499808631 : complex phases 4.16e-50, rows vs closed forms 3.35e-50
  [PASS] a=1 b=0.8 N=10 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 2.9e-49, largest deviation 2.72e-49
  [PASS] a=1 b=0.8 N=5 tau=2: (ii) Theorem B gamma = -1.43474648771135, 2+gamma = 0.5652535123, k808 = k810 = 126.619659552589 : G - gamma F 2.74e-49, period Delta 1.99e-49, rows and three forms 7.13e-49
  [PASS] a=1 b=0.8 N=5 tau=2: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 3.79e-49, largest deviation 1.35e-47
  [PASS] a=1 b=0.8 N=7 tau=3: (ii) Theorem B gamma = -1.46828432899947, 2+gamma = 0.531715671, k808 = k810 = 535.420232267997 : G - gamma F 3.58e-48, period Delta 2.73e-48, rows and three forms 6.91e-48
  [PASS] a=1 b=0.8 N=7 tau=3: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 3.91e-48, largest deviation 5.64e-46
  [PASS] a=1 b=0.8 N=8 tau=3: (ii) Theorem A constants c1 = 4.53231943702993, c2 = -6.05300048917743 : complex phases 3.87e-49, rows vs closed forms 3.15e-47
  [PASS] a=1 b=0.8 N=8 tau=3: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 9.85e-49, largest deviation 8.99e-48
  [PASS] a=1 b=0.8 N=9 tau=4: (ii) Theorem B gamma = -1.47044385743369, 2+gamma = 0.5295561426, k808 = k810 = 2148.72809591503 : G - gamma F 8.79e-47, period Delta 6.75e-47, rows and three forms 7.18e-47
  [PASS] a=1 b=0.8 N=9 tau=4: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 7.49e-47, largest deviation 3.58e-44
  [PASS] a=1 b=0.8 N=10 tau=3: (ii) Theorem A constants c1 = 13.8239670506249, c2 = -8.66497217503759 : complex phases 6.34e-51, rows vs closed forms 9.73e-50
  [PASS] a=1 b=0.8 N=10 tau=3: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 4.05e-50, largest deviation 1.97e-49
  [PASS] a=1 b=0.8 N=12 tau=5: (ii) Theorem A constants c1 = 4.71938708946278, c2 = -6.8970783009376 : complex phases 2.23e-49, rows vs closed forms 7.72e-46
  [PASS] a=1 b=0.8 N=12 tau=5: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 4.1e-49, largest deviation 8.22e-47
  [PASS] a=2 b=1.2 N=3 tau=1: (ii) Theorem B gamma = -1.0, 2+gamma = 1.0, k808 = k810 = 25.8076127072299 : G - gamma F 2.07e-49, period Delta 3.59e-49, rows and three forms 4.18e-49
  [PASS] a=2 b=1.2 N=3 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 3.13e-49, largest deviation 3.28e-48
  [PASS] a=2 b=1.2 N=4 tau=1: (ii) Theorem A constants c1 = 20.3294117647059, c2 = 0.0 : complex phases 5.54e-50, rows vs closed forms 3.87e-49
  [PASS] a=2 b=1.2 N=4 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 9.57e-50, largest deviation 4.09e-49
  [PASS] a=2 b=1.2 N=5 tau=1: (ii) Theorem B gamma = 0.630882535378537, 2+gamma = 2.630882535, k808 = k810 = 2.46550323461174 : G - gamma F 5.72e-50, period Delta 1.88e-50, rows and three forms 4.34e-50
  [PASS] a=2 b=1.2 N=5 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 9.56e-50, largest deviation 2.7e-49
  [PASS] a=2 b=1.2 N=6 tau=1: (ii) Theorem A constants c1 = 37.253940149206, c2 = 37.253940149206 : complex phases 6.9e-50, rows vs closed forms 1.57e-49
  [PASS] a=2 b=1.2 N=6 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 2.19e-49, largest deviation 6.25e-49
  [PASS] a=2 b=1.2 N=7 tau=1: (ii) Theorem B gamma = 1.23681541479136, 2+gamma = 3.236815415, k808 = k810 = 1.55963987738811 : G - gamma F 9.59e-50, period Delta 2.37e-50, rows and three forms 5.48e-50
  [PASS] a=2 b=1.2 N=7 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 2.44e-49, largest deviation 6.17e-49
  [PASS] a=2 b=1.2 N=8 tau=1: (ii) Theorem A constants c1 = 45.0716603440914, c2 = 63.0474090294047 : complex phases 1.3e-49, rows vs closed forms 1.31e-49
  [PASS] a=2 b=1.2 N=8 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 3.92e-49, largest deviation 9.3e-49
  [PASS] a=2 b=1.2 N=9 tau=1: (ii) Theorem B gamma = 1.51471113447616, 2+gamma = 3.514711134, k808 = k810 = 1.30624720953332 : G - gamma F 2.18e-50, period Delta 2.82e-51, rows and three forms 1.02e-50
  [PASS] a=2 b=1.2 N=9 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 7.28e-50, largest deviation 3.27e-49
  [PASS] a=2 b=1.2 N=10 tau=1: (ii) Theorem A constants c1 = 49.0766286547808, c2 = 78.543749618714 : complex phases 1.92e-50, rows vs closed forms 1.96e-50
  [PASS] a=2 b=1.2 N=10 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 9.42e-50, largest deviation 2.91e-49
  [PASS] a=2 b=1.2 N=5 tau=2: (ii) Theorem B gamma = -1.21648419678105, 2+gamma = 0.7835158032, k808 = k810 = 290.916349086531 : G - gamma F 9.17e-48, period Delta 2.85e-47, rows and three forms 1.07e-47
  [PASS] a=2 b=1.2 N=5 tau=2: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 1.56e-47, largest deviation 1.67e-45
  [PASS] a=2 b=1.2 N=7 tau=3: (ii) Theorem B gamma = -1.21947473509684, 2+gamma = 0.7805252649, k808 = k810 = 2626.86439575192 : G - gamma F 4.71e-46, period Delta 1.48e-45, rows and three forms 2.94e-46
  [PASS] a=2 b=1.2 N=7 tau=3: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 5.77e-46, largest deviation 9.94e-43
  [PASS] a=2 b=1.2 N=8 tau=3: (ii) Theorem A constants c1 = 29.9478999786771, c2 = -35.7169902244774 : complex phases 3.43e-48, rows vs closed forms 7.02e-46
  [PASS] a=2 b=1.2 N=8 tau=3: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 2.99e-48, largest deviation 1.67e-46
  [PASS] a=2 b=1.2 N=9 tau=4: (ii) Theorem B gamma = -1.21951173263729, 2+gamma = 0.7804882674, k808 = k810 = 23642.6561557258 : G - gamma F 5.68e-44, period Delta 1.79e-43, rows and three forms 4.13e-44
  [PASS] a=2 b=1.2 N=9 tau=4: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 5.41e-44, largest deviation 5.49e-40
  [PASS] a=2 b=1.2 N=10 tau=3: (ii) Theorem A constants c1 = 104.99420563142, c2 = -69.2576838347406 : complex phases 1.48e-49, rows vs closed forms 3.7e-48
  [PASS] a=2 b=1.2 N=10 tau=3: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 6.86e-49, largest deviation 4.16e-48
  [PASS] a=2 b=1.2 N=12 tau=5: (ii) Theorem A constants c1 = 30.1125435869347, c2 = -36.7124643525892 : complex phases 2.11e-46, rows vs closed forms 2.36e-42
  [PASS] a=2 b=1.2 N=12 tau=5: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 1.22e-46, largest deviation 6.47e-44
  [PASS] a=1.7 b=0.3 N=3 tau=1: (ii) Theorem B gamma = -1.0, 2+gamma = 1.0, k808 = k810 = 259.000092851572 : G - gamma F 2.79e-48, period Delta 7.45e-47, rows and three forms 6.89e-48
  [PASS] a=1.7 b=0.3 N=3 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 2.44e-47, largest deviation 1.54e-45
  [PASS] a=1.7 b=0.3 N=4 tau=1: (ii) Theorem A constants c1 = 0.356110067114094, c2 = 0.0 : complex phases 1.99e-49, rows vs closed forms 8.15e-48
  [PASS] a=1.7 b=0.3 N=4 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 1.32e-49, largest deviation 5.33e-48
  [PASS] a=1.7 b=0.3 N=5 tau=1: (ii) Theorem B gamma = 0.73608425028948, 2+gamma = 2.73608425, k808 = k810 = 3.98326008604806 : G - gamma F 1.2e-49, period Delta 1.99e-49, rows and three forms 1.22e-49
  [PASS] a=1.7 b=0.3 N=5 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 1.98e-49, largest deviation 7.21e-49
  [PASS] a=1.7 b=0.3 N=6 tau=1: (ii) Theorem A constants c1 = 1.15281545211423, c2 = 1.15281545211423 : complex phases 4.25e-50, rows vs closed forms 1.71e-49
  [PASS] a=1.7 b=0.3 N=6 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 5.12e-50, largest deviation 1.77e-48
  [PASS] a=1.7 b=0.3 N=7 tau=1: (ii) Theorem B gamma = 1.1579906243297, 2+gamma = 3.157990624, k808 = k810 = 2.00827007382793 : G - gamma F 4.68e-50, period Delta 6.51e-50, rows and three forms 1.6e-50
  [PASS] a=1.7 b=0.3 N=7 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 1.56e-49, largest deviation 5.94e-49
  [PASS] a=1.7 b=0.3 N=8 tau=1: (ii) Theorem A constants c1 = 1.65483656491794, c2 = 2.11436311594022 : complex phases 1.03e-49, rows vs closed forms 1.76e-49
  [PASS] a=1.7 b=0.3 N=8 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 1.11e-49, largest deviation 8.18e-49
  [PASS] a=1.7 b=0.3 N=9 tau=1: (ii) Theorem B gamma = 1.37495561794142, 2+gamma = 3.374955618, k808 = k810 = 1.54826975180975 : G - gamma F 7.94e-51, period Delta 2.11e-51, rows and three forms 5.83e-51
  [PASS] a=1.7 b=0.3 N=9 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 6.31e-51, largest deviation 3.4e-49
  [PASS] a=1.7 b=0.3 N=10 tau=1: (ii) Theorem A constants c1 = 1.94499288069545, c2 = 2.83136050858055 : complex phases 2.2e-49, rows vs closed forms 2.36e-49
  [PASS] a=1.7 b=0.3 N=10 tau=1: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 2.63e-49, largest deviation 1.22e-48
  [PASS] a=1.7 b=0.3 N=5 tau=2: (ii) Theorem B gamma = -1.01581622588272, 2+gamma = 0.9841837741, k808 = k810 = 33267.2896084293 : G - gamma F 5.18e-44, period Delta 2.75e-42, rows and three forms 3.75e-44
  [PASS] a=1.7 b=0.3 N=5 tau=2: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 5.42e-43, largest deviation 4.76e-39
  [PASS] a=1.7 b=0.3 N=7 tau=2: (ii) Theorem B gamma = -0.68067564819737, 2+gamma = 1.319324352, k808 = k810 = 39.2436698981155 : G - gamma F 4.44e-48, period Delta 1.08e-47, rows and three forms 3.81e-48
  [PASS] a=1.7 b=0.3 N=7 tau=2: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 6.9e-48, largest deviation 7.99e-47
  [PASS] a=1.7 b=0.3 N=8 tau=3: (ii) Theorem A constants c1 = 0.370057745126195, c2 = -0.375864370809154 : complex phases 9.36e-45, rows vs closed forms 2.19e-39
  [PASS] a=1.7 b=0.3 N=8 tau=3: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 3.09e-45, largest deviation 3.07e-42
  [PASS] a=1.7 b=0.3 N=9 tau=2: (ii) Theorem B gamma = 0.478008682156369, 2+gamma = 2.478008682, k808 = k810 = 5.92656204576793 : G - gamma F 3.39e-49, period Delta 5.17e-49, rows and three forms 3.68e-49
  [PASS] a=1.7 b=0.3 N=9 tau=2: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 6.62e-49, largest deviation 5.08e-48
  [PASS] a=1.7 b=0.3 N=10 tau=3: (ii) Theorem A constants c1 = 1.68944769970623, c2 = -1.43801610675385 : complex phases 6.55e-48, rows vs closed forms 1.36e-45
  [PASS] a=1.7 b=0.3 N=10 tau=3: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 6.25e-48, largest deviation 1.33e-46
  [PASS] a=1.7 b=0.3 N=11 tau=4: (ii) Theorem B gamma = -1.01518453330372, 2+gamma = 0.9848154667, k808 = k810 = 1319.75772018377 : G - gamma F 6.46e-47, period Delta 3.44e-45, rows and three forms 4.13e-46
  [PASS] a=1.7 b=0.3 N=11 tau=4: (iv) orbit = P(u+lh), P' = sigma P(.+Delta), P'' = sigma^-1 P(.+Delta), area formulas : 2N Delta - 4 tau K: 1.62e-45, largest deviation 8.44e-43
  summary (ii): largest relative deviation 2.19e-39; (iv): largest relative deviation 2.03e-38

(iii) exact N = 4 (sympy)
  rhombus member: A = 2*a*b , A' = 4*a*b , A'' = 4*a**3*b**3/(a**2 + b**2)**2 , A^x = 2*a**3*b**3/(a**2 + b**2)**2 , A'^- = a*b
  [PASS] rhombus: sides tangent to the caustic with lambda = a^2 b^2/(a^2+b^2)
  [PASS] rhombus: k110 = 8a^4b^4/(a^2+b^2)^2, k807 = 4a^4b^4/(a^2+b^2)^2, k809 = 4a^2b^2, k106 = 8a^2b^2
  [PASS] family: P1 on the billiard ellipse
  [PASS] family: all four sides tangent to the caustic
  [PASS] family: reflection law at P1 (squared form)
  [PASS] family: k110 = A A'' = 8a^4b^4/(a^2+b^2)^2 at every member
  [PASS] family: k807 = A A^x = 4a^4b^4/(a^2+b^2)^2 at every member
  [PASS] family: k809 = A' A'^- = 4a^2b^2 at every member
  [PASS] family: k106 = A A' = 8a^2b^2 and k113 = A'/A'' = (a^2+b^2)^2/(a^2b^2) at every member
  [PASS] family: the printed value 2a^4b^4/(a^2+b^2)^2 is not k110 (difference not identically 0)
  [PASS] Lemma 6.1: duplication form of gamma = lambda-form, 2 + gamma = 2q/(a^4b^4 - c^4 lambda^2), q(b^2) = b^6c^2, and the N = 3 root of 2 + gamma = 1 (exact)
  [PASS] Proposition 7.4: common points (x0^2, y0^2) = (a^2a''^2, -b^2b''^2)/c^2 for (E, C) and (a''^4, -b''^4)/c^2 for (C, E*) (exact)
  (part (iii) took 1 s)

(v) further statements
  [PASS] a=1 b=0.5 N=5 tau=1: residue of F at iK' = i ab (dn h - 1)/(k^2 sn h) = (0.0 - 0.355717895334j); oddness at S0, S1 : residue error 3.3e-40, oddness 4.85e-49
  [PASS] a=1 b=0.5 N=4 tau=1: residue of F at iK' = 2 i ab (dn h - 1)/(k^2 sn h) = (0.0 - 0.8j); oddness at S0, S1 : residue error 2.25e-31, oddness 9.53e-49
  [PASS] a=2 b=1.2 N=6 tau=1: residue of F at iK' = 2 i ab (dn h - 1)/(k^2 sn h) = (0.0 - 2.78107443266j); oddness at S0, S1 : residue error 2.25e-31, oddness 5.37e-48
  [PASS] a=1 b=0.8 N=7 tau=3: residue of F at iK' = i ab (dn h - 1)/(k^2 sn h) = (0.0 - 0.480102122787j); oddness at S0, S1 : residue error 1.71e-40, oddness 4.35e-46
  [PASS] a=1 b=0.5 N=8 tau=3: residue of F at iK' = 2 i ab (dn h - 1)/(k^2 sn h) = (0.0 - 0.865657906288j); oddness at S0, S1 : residue error 8.44e-31, oddness 3.2e-47
  [PASS] a=2 b=1.2 N=9 tau=4: residue of F at iK' = i ab (dn h - 1)/(k^2 sn h) = (0.0 - 1.91999996916j); oddness at S0, S1 : residue error 6.84e-41, oddness 1.01e-41
  [PASS] a=1.0 b=0.5: sn(K - Delta) = a''/a; as lambda -> b^2, K -> infinity and K - Delta -> arcsinh(c/b) : identity 2.82e-32; at 1 - lambda/b^2 = 1e-10, 1e-15, 1e-20: K = 13.4485, 19.205, 24.9615, K - Delta - arcsinh(c/b) = 3.98e-11, 3.98e-16, 3.98e-21
  [PASS] a=2.0 b=1.2: sn(K - Delta) = a''/a; as lambda -> b^2, K -> infinity and K - Delta -> arcsinh(c/b) : identity 5.41e-32; at 1 - lambda/b^2 = 1e-10, 1e-15, 1e-20: K = 13.1869, 18.9434, 24.6998, K - Delta - arcsinh(c/b) = 4.67e-11, 4.67e-16, 4.67e-21
  [PASS] a=1.7 b=0.3: sn(K - Delta) = a''/a; as lambda -> b^2, K -> infinity and K - Delta -> arcsinh(c/b) : identity 8.76e-32; at 1 - lambda/b^2 = 1e-10, 1e-15, 1e-20: K = 14.618, 20.3745, 26.1309, K - Delta - arcsinh(c/b) = 2.73e-11, 2.73e-16, 2.73e-21
  [PASS] numerator and denominator of 2 + gamma (lambda-form) positive on [0, b^2] for four ellipses : smallest normalised value 9.999e-5
  [PASS] a=1 b=0.5: N=3 gamma = -1, k808 = 4 kappa; N=6 k807/k110 = k809/k106 = 3/4; printed N=6 values of k106, k110 confirmed; N=4 k110 = 4 x printed value : 2.14e-50, 2.0e-50, 5.35e-51, 1.2e-50, 9.5e-50, 1.6e-49
  [PASS] a=1 b=0.8: N=3 gamma = -1, k808 = 4 kappa; N=6 k807/k110 = k809/k106 = 3/4; printed N=6 values of k106, k110 confirmed; N=4 k110 = 4 x printed value : 1.52e-49, 2.67e-51, 6.68e-51, 5.58e-51, 3.18e-50, 3.63e-49
  [PASS] a=2 b=1.2: N=3 gamma = -1, k808 = 4 kappa; N=6 k807/k110 = k809/k106 = 3/4; printed N=6 values of k106, k110 confirmed; N=4 k110 = 4 x printed value : 2.07e-49, 9.35e-51, 2.67e-51, 4.98e-51, 8.57e-50, 3.74e-49
  [PASS] a=1.7 b=0.3: N=3 gamma = -1, k808 = 4 kappa; N=6 k807/k110 = k809/k106 = 3/4; printed N=6 values of k106, k110 confirmed; N=4 k110 = 4 x printed value : 2.79e-48, 2.0e-50, 8.02e-51, 1.45e-50, 6.68e-51, 1.73e-48

TOTAL checks: 194, failures: 0
running time: 13 s
