==========================================================================================
(A) N = 3: a^2 = 8, b^2 = 3, caustic lambda = 72/25 (a''^2 = 128/25, b''^2 = 3/25)
  member 1: [(2*sqrt(2), 0), (-8*sqrt(2)/5, 3*sqrt(3)/5), (-8*sqrt(2)/5, -3*sqrt(3)/5)]
    certified on ellipse and all sides tangent to the caustic: True
    A                            = 54*sqrt(6)/25
                                  = 5.29089784441166469210613360137
    Adag                         = (sqrt(15) + 4*sqrt(6))/64
                                  = 0.213608473708439519968256276541
    Apol                         = (8*sqrt(15) + 32*sqrt(6))/3
                                  = 36.4558461795736780745824045296
    Adual                        = (27*sqrt(15) + 108*sqrt(6))/320
                                  = 1.15348575802557340782858389332
    AdualP                       = (27*sqrt(15) + 108*sqrt(6))/1280
                                  = 0.288371439506393351957145973330
    Apd_i                        = 3*sqrt(6)/800
                                  = 0.00918558653543691786823981528015
    Apd_ii                       = (8*sqrt(15) + 32*sqrt(6))/3
                                  = 36.4558461795736780745824045296
    Apd_iii                      = (-24*sqrt(15) + 96*sqrt(6))/625
                                  = 0.227519063997131347492219042523
    Aant                         = -9*sqrt(15) + 36*sqrt(6)
                                  = 53.3247806243276595684888380914
  member 2: [(0, sqrt(3)), (-8*sqrt(3)/5, -sqrt(3)/5), (8*sqrt(3)/5, -sqrt(3)/5)]
    certified on ellipse and all sides tangent to the caustic: True
    A                            = 144/25
                                  = 5.76000000000000000000000000000
    Adag                         = 9/64
                                  = 0.140625000000000000000000000000
    Apol                         = 24
                                  = 24.0000000000000000000000000000
    Adual                        = 243/320
                                  = 0.759375000000000000000000000000
    AdualP                       = 243/1280
                                  = 0.189843750000000000000000000000
    Apd_i                        = 27/3200
                                  = 0.00843750000000000000000000000000
    Apd_ii                       = 24
                                  = 24.0000000000000000000000000000
    Apd_iii                      = 216/625
                                  = 0.345600000000000000000000000000
    Aant                         = 96
                                  = 96.0000000000000000000000000000
  consistency (rows k813, k814, exact):
    k813 Apol/Adag           member1 = 512/3 ; member2 = 512/3 ; difference simplifies to 0
    k814 Apol/Adual          member1 = 2560/81 ; member2 = 2560/81 ; difference simplifies to 0
    k814 (pointwise dual)    member1 = 10240/81 ; member2 = 10240/81 ; difference simplifies to 0
  row k815 (odd N): products A^dag_ped * A_dual under each reading:
    Apd_i*Adual        member1 = 243*sqrt(10)/256000 + 243/32000  (0.01059544324773795381904582)
                       member2 = 6561/1024000  (0.006407226562500000000000000)
                       difference = 243/204800 + 243*sqrt(10)/256000  (0.004188216685237953819045825)
    Apd_i*AdualP       member1 = 243*sqrt(10)/1024000 + 243/128000  (0.002648860811934488454761456)
                       member2 = 6561/4096000  (0.001601806640625000000000000)
                       difference = 243/819200 + 243*sqrt(10)/1024000  (0.001047054171309488454761456)
    Apd_ii*Adual       member1 = 27*sqrt(10)/5 + 999/40  (42.05129936490924839279403)
                       member2 = 729/40  (18.22500000000000000000000)
                       difference = 27/4 + 27*sqrt(10)/5  (23.82629936490924839279402)
    Apd_ii*AdualP      member1 = 27*sqrt(10)/20 + 999/160  (10.51282484122731209819851)
                       member2 = 729/160  (4.556250000000000000000000)
                       difference = 27/16 + 27*sqrt(10)/20  (5.956574841227312098198506)
    Apd_iii*Adual      member1 = 6561/25000  (0.2624400000000000000000000)
                       member2 = 6561/25000  (0.2624400000000000000000000)
                       difference = 0  (0)
    Apd_iii*AdualP     member1 = 6561/100000  (0.06561000000000000000000000)
                       member2 = 6561/100000  (0.06561000000000000000000000)
                       difference = 0  (0)
    Adag/Aant    member1 = (8*sqrt(10) + 37)/15552 ; member2 = 3/2048
    Adag*Aant    member1 = 729/64 ; member2 = 27/2
    Aant/A       member1 = 50/3 - 25*sqrt(10)/12 ; member2 = 50/3
==========================================================================================
(B) N = 4: a^2 = 25, b^2 = 16, caustic lambda = 400/41 (a''^2 = 625/41, b''^2 = 256/41)
  member 1: [(5, 0), (0, 4), (-5, 0), (0, -4)]
    certified on ellipse and all sides tangent to the caustic: True
    A                            = 40
                                  = 40.0000000000000000000000000000
    Adag                         = 1/10
                                  = 0.100000000000000000000000000000
    Apol                         = 125/256
                                  = 0.488281250000000000000000000000
    Adual                        = 2048/125
                                  = 16.3840000000000000000000000000
    AdualP                       = 1024/125
                                  = 8.19200000000000000000000000000
    Apd_i                        = 4096/210125
                                  = 0.0194931588340273646638905413444
    Apd_ii                       = 125/256
                                  = 0.488281250000000000000000000000
    Apd_iii                      = 32000/1681
                                  = 19.0362879238548483045806067817
    Aant                         = 80
                                  = 80.0000000000000000000000000000
  member 2: [(25*sqrt(41)/41, 16*sqrt(41)/41), (-25*sqrt(41)/41, 16*sqrt(41)/41), (-25*sqrt(41)/41, -16*sqrt(41)/41), (25*sqrt(41)/41, -16*sqrt(41)/41)]
    certified on ellipse and all sides tangent to the caustic: True
    A                            = 1600/41
                                  = 39.0243902439024390243902439024
    Adag                         = 41/400
                                  = 0.102500000000000000000000000000
    Apol                         = 1025/2048
                                  = 0.500488281250000000000000000000
    Adual                        = 10496/625
                                  = 16.7936000000000000000000000000
    AdualP                       = 5248/625
                                  = 8.39680000000000000000000000000
    Apd_i                        = 328/15625
                                  = 0.0209920000000000000000000000000
    Apd_ii                       = 1025/2048
                                  = 0.500488281250000000000000000000
    Apd_iii                      = 800/41
                                  = 19.5121951219512195121951219512
    Aant                         = 3200/41
                                  = 78.0487804878048780487804878049
  consistency (rows k813, k814, exact):
    k813 Apol/Adag           member1 = 625/128 ; member2 = 625/128 ; difference simplifies to 0
    k814 Apol/Adual          member1 = 15625/524288 ; member2 = 15625/524288 ; difference simplifies to 0
    k814 (pointwise dual)    member1 = 15625/262144 ; member2 = 15625/262144 ; difference simplifies to 0
  row k816 (even N): ratios A^dag_ped / A_dual under each reading:
    Apd_i/Adual        member1 = 2/1681  (0.001189767995240928019036288)
                       member2 = 1/800  (0.001250000000000000000000000)
                       difference = -81/1344800  (-0.00006023200475907198096371208)
    Apd_i/AdualP       member1 = 4/1681  (0.002379535990481856038072576)
                       member2 = 1/400  (0.002500000000000000000000000)
                       difference = -81/672400  (-0.0001204640095181439619274242)
    Apd_ii/Adual       member1 = 15625/524288  (0.02980232238769531250000000)
                       member2 = 15625/524288  (0.02980232238769531250000000)
                       difference = 0  (0)
    Apd_ii/AdualP      member1 = 15625/262144  (0.05960464477539062500000000)
                       member2 = 15625/262144  (0.05960464477539062500000000)
                       difference = 0  (0)
    Apd_iii/Adual      member1 = 15625/13448  (1.161882807852468768590125)
                       member2 = 15625/13448  (1.161882807852468768590125)
                       difference = 0  (0)
    Apd_iii/AdualP     member1 = 15625/6724  (2.323765615704937537180250)
                       member2 = 15625/6724  (2.323765615704937537180250)
                       difference = 0  (0)
    Adag/Aant    member1 = 1/800 ; member2 = 1681/1280000
    Adag*Aant    member1 = 8 ; member2 = 8
    Aant/A       member1 = 2 ; member2 = 2
