## (W1) wrong caustic parameter: lambda*(1+1e-6) for N=7, tau=2 -> orbit does not close;
  correct lambda   closure error=2.04e-49  spreads: k605a=4.16e-50, k806a=1.93e-50, k904a=3.08e-50, k907aR=8.61e-50
  wrong lambda     closure error=2.6e-5  spreads: k605a=2.2e-6, k806a=3.51e-6, k904a=2.44e-6, k907aR=2.2e-6
## (W2) non-confocal "caustic" (b''^2 scaled by 0.97): reflection-law orbit must fail the tangency check
  tangency error = 0.00374  reflection-law error = 6.68e-51  closure = 0.0564
## (W3) wrong pedal point: antipedal/pedal ratio w.r.t. O (not a focus) and w.r.t. a random point, N=6 (k404 says f_1, f_2 only)
  M=O          spread=0.0549 value0=1.9507637882184693098
  M=(0.3,0.2)  spread=0.046 value0=1.825663605321373536
  M=f_1        spread=1.81e-50 value0=0.81380208333333333333
## (W4) wrong N-class fed as if in class: k404 at N=8 (tau=1,3) and N=12, k605a/k904a/k907a at N=6, 8, 10
  N=8 tau=1: k404=9.34e-5, k605a=9.34e-5, k904a=9.34e-5, k907aR=9.34e-5
  N=8 tau=3: k404=0.0695, k605a=0.0681, k904a=0.0681, k907aR=0.0681
  N=12 tau=1: k404=2.65e-7, k605a=2.65e-7, k904a=2.65e-7, k907aR=2.65e-7
  N=6 tau=1: k404=1.81e-50, k605a=0.00351, k904a=0.00351, k907aR=0.00351
  N=10 tau=3: k404=2.17e-49, k605a=0.000833, k904a=0.000833, k907aR=0.000833
## (W5) radius independence: inversion radius 1 vs 2 (k806a ratio unchanged; products scale by r^8), N=5
  r^2=1.0: A'dag/Adag=0.5818968646949684128650963  A'dag1*A'dag2=2.110352515142222031955923
  r^2=4.0: A'dag/Adag=0.5818968646949684128650963  A'dag1*A'dag2=540.2502438764088401807164
## (W6) signs: A, A', Adag_1, A'dag_1, Abar_1, A*_1, Apol_1, AdualR_1 for N=5 tau=1 and N=5 tau=2 (a=1.5) at one phase
  N=5 tau=1: A=3.4912348, Ap=5.5395726, Adag1=2.5041081, Adagp1=1.4571327, Abar1=2.6536406, Abarp1=4.1006416, Astar1=3.8639028, Apol1=10.334365, AdualR1=0.8149481, AdualP1=0.48665649
  N=5 tau=2: A=1.7920475, Ap=66.216092, Adag1=0.33955038, Adagp1=0.1034918, Abar1=0.004689854, Abarp1=2.2902927, Astar1=1220.2221, Apol1=1520.2588, AdualR1=0.25436562, AdualP1=0.0025900297
  N=7 tau=3: A=1.793562, Ap=453.6135, Adag1=0.33896093, Adagp1=0.10229528, Abar1=0.00010027351, Abarp1=2.29077, Astar1=57146.592, Apol1=71230.362, AdualR1=0.25453208, AdualP1=5.6382495e-5
  N=6 tau=1: A=3.8400502, Ap=5.2499314, Adag1=3.555602, Adagp1=2.3111413, Abar1=3.6864481, Abarp1=4.8000627, Astar1=3.0000392, Apol1=9.0001176, AdualR1=0.94816053, AdualP1=0.63210702
  N=4 tau=1: A=2.8364433, Ap=6.3459755, Adag1=1.4102168, Adagp1=0.70510839, Abar1=1.3518054, Abarp1=3.1729878, Astar1=5.6728867, Apol1=14.278445, AdualR1=0.62676302, AdualP1=0.31338151
## (W7) N = 4 value of k806,a over ellipses a/b in {1.05, 1.3, 1.7, 2.5, 4, 8}: ratio A'dag_j/Adag_j
  a=1.05: ratio j=1 0.5 j=2 0.5 spread 1.07e-50; lambda-check lambda=0.52437574316290130797 a^2b^2/(a^2+b^2)=0.52437574316290130797
  a=1.3: ratio j=1 0.5 j=2 0.5 spread 9.35e-51; lambda-check lambda=0.62825278810408921933 a^2b^2/(a^2+b^2)=0.62825278810408921933
  a=1.7: ratio j=1 0.5 j=2 0.5 spread 3.34e-50; lambda-check lambda=0.74293059125964010283 a^2b^2/(a^2+b^2)=0.74293059125964010283
  a=2.5: ratio j=1 0.5 j=2 0.5 spread 5.48e-50; lambda-check lambda=0.86206896551724137931 a^2b^2/(a^2+b^2)=0.86206896551724137931
  a=4: ratio j=1 0.5 j=2 0.5 spread 1.41e-48; lambda-check lambda=0.94117647058823529412 a^2b^2/(a^2+b^2)=0.94117647058823529412
  a=8: ratio j=1 0.5 j=2 0.5 spread 6.53e-47; lambda-check lambda=0.98461538461538461538 a^2b^2/(a^2+b^2)=0.98461538461538461538
## (W8) claim's formula A_{1,dual} = (b^4/(2a^2)) sum sin(angle P_l f_1 P_{l+1}) (reading R, unit circle at O_p), N=5,7
  N=5 tau=1: A_dual=0.8099815205862892428489736  formula=0.8099815205862892428489736
  N=7 tau=2: A_dual=0.9792059127296728388212717  formula=0.9792059127296728388212717
