check_note.py -- mpmath 50 digits; sympy 1.14.0

=== (i) reflection-law trajectories: k813, k814 (both duals), identifications, N = 4 constants
  a=1 b=0.6 N=3 tau=1: lambda=0.34676101402462327616 k^2=0.97973332 closure 7.9e-49 tangency 8.8e-49; spreads k813 1.9e-47, k814 1.7e-47, k814 point-wise 1.7e-47; identities 1.4e-49
  ok    (i) a=1 b=0.6 N=3 tau=1: closed, every chord tangent to the caustic
  ok    (i) a=1 b=0.6 N=3 tau=1: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1 b=0.6 N=3 tau=1: identifications of Sec. 3, positivity
  ok    (i) a=1 b=0.6 N=3: A_dual^P = A_dual/4
  a=1 b=0.6 N=4 tau=1: lambda=0.26470588235294117647 k^2=0.8704 closure 3.0e-49 tangency 3.4e-49; spreads k813 1.1e-48, k814 6.6e-49, k814 point-wise 6.3e-49; identities 4.0e-50
  ok    (i) a=1 b=0.6 N=4 tau=1: closed, every chord tangent to the caustic
  ok    (i) a=1 b=0.6 N=4 tau=1: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1 b=0.6 N=4 tau=1: identifications of Sec. 3, positivity
  ok    (i) a=1 b=0.6 N=4: caustic lambda = a^2b^2/(a^2+b^2)
    N=4 constants: rel. errors 8.7e-49, 5.2e-49, 2.0e-49; A_dual^P/A_dual - 1/2: 1.5e-50; A A^dag - 4: 1.6e-48
  ok    (i) a=1 b=0.6 N=4: 2a^4/b^4, 2a^6/b^10, 2a^6/(b^2(a^2+b^2)^2), A_dual^P = A_dual/2, A A^dag = 4
  a=1 b=0.6 N=5 tau=1: lambda=0.19081540761318619605 k^2=0.79091966 closure 5.2e-49 tangency 4.1e-49; spreads k813 9.7e-49, k814 4.6e-49, k814 point-wise 5.0e-49; identities 4.7e-50
  ok    (i) a=1 b=0.6 N=5 tau=1: closed, every chord tangent to the caustic
  ok    (i) a=1 b=0.6 N=5 tau=1: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1 b=0.6 N=5 tau=1: identifications of Sec. 3, positivity
  a=1 b=0.6 N=5 tau=2: lambda=0.35982682179615615467 k^2=0.99972948 closure 1.5e-46 tangency 8.7e-47; spreads k813 2.3e-43, k814 2.3e-43, k814 point-wise 2.4e-43; identities 1.2e-47
  ok    (i) a=1 b=0.6 N=5 tau=2: closed, every chord tangent to the caustic
  ok    (i) a=1 b=0.6 N=5 tau=2: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1 b=0.6 N=5 tau=2: identifications of Sec. 3, positivity
  a=1 b=0.6 N=6 tau=1: lambda=0.140625 k^2=0.74472727 closure 3.4e-50 tangency 3.1e-50; spreads k813 7.2e-50, k814 5.0e-50, k814 point-wise 6.7e-50; identities 3.2e-50
  ok    (i) a=1 b=0.6 N=6 tau=1: closed, every chord tangent to the caustic
  ok    (i) a=1 b=0.6 N=6 tau=1: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1 b=0.6 N=6 tau=1: identifications of Sec. 3, positivity
  ok    (i) a=1 b=0.6 N=6: caustic lambda = a^2b^2/(a+b)^2
  a=1 b=0.6 N=7 tau=2: lambda=0.30896561068026663975 k^2=0.92614783 closure 6.3e-49 tangency 7.4e-49; spreads k813 2.1e-48, k814 1.5e-48, k814 point-wise 1.5e-48; identities 7.0e-50
  ok    (i) a=1 b=0.6 N=7 tau=2: closed, every chord tangent to the caustic
  ok    (i) a=1 b=0.6 N=7 tau=2: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1 b=0.6 N=7 tau=2: identifications of Sec. 3, positivity
  a=1 b=0.6 N=8 tau=1: lambda=0.083662740249749384921 k^2=0.6984328 closure 3.2e-49 tangency 1.9e-49; spreads k813 2.0e-49, k814 1.2e-49, k814 point-wise 1.5e-49; identities 1.5e-49
  ok    (i) a=1 b=0.6 N=8 tau=1: closed, every chord tangent to the caustic
  ok    (i) a=1 b=0.6 N=8 tau=1: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1 b=0.6 N=8 tau=1: identifications of Sec. 3, positivity
  a=1 b=0.6 N=8 tau=3: lambda=0.3584540781928348293 k^2=0.99759032 closure 3.1e-47 tangency 3.8e-47; spreads k813 4.1e-45, k814 4.1e-45, k814 point-wise 4.2e-45; identities 1.7e-48
  ok    (i) a=1 b=0.6 N=8 tau=3: closed, every chord tangent to the caustic
  ok    (i) a=1 b=0.6 N=8 tau=3: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1 b=0.6 N=8 tau=3: identifications of Sec. 3, positivity
  a=1 b=0.6 N=9 tau=2: lambda=0.22453527531273672427 k^2=0.82531156 closure 9.3e-49 tangency 7.6e-49; spreads k813 1.3e-48, k814 7.1e-49, k814 point-wise 7.3e-49; identities 5.0e-50
  ok    (i) a=1 b=0.6 N=9 tau=2: closed, every chord tangent to the caustic
  ok    (i) a=1 b=0.6 N=9 tau=2: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1 b=0.6 N=9 tau=2: identifications of Sec. 3, positivity
  a=1 b=0.6 N=10 tau=3: lambda=0.323189522357789849 k^2=0.94561184 closure 2.6e-48 tangency 2.2e-48; spreads k813 9.0e-48, k814 7.1e-48, k814 point-wise 7.1e-48; identities 4.2e-50
  ok    (i) a=1 b=0.6 N=10 tau=3: closed, every chord tangent to the caustic
  ok    (i) a=1 b=0.6 N=10 tau=3: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1 b=0.6 N=10 tau=3: identifications of Sec. 3, positivity
  a=1 b=0.6 N=12 tau=5: lambda=0.35998073520981635599 k^2=0.9999699 closure 7.1e-46 tangency 3.9e-46; spreads k813 6.7e-42, k814 6.7e-42, k814 point-wise 6.9e-42; identities 1.3e-46
  ok    (i) a=1 b=0.6 N=12 tau=5: closed, every chord tangent to the caustic
  ok    (i) a=1 b=0.6 N=12 tau=5: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1 b=0.6 N=12 tau=5: identifications of Sec. 3, positivity
  a=1.5 b=1 N=3 tau=1: lambda=0.94337976665279116377 k^2=0.95666665 closure 1.3e-48 tangency 9.9e-49; spreads k813 1.2e-47, k814 1.0e-47, k814 point-wise 1.0e-47; identities 9.8e-50
  ok    (i) a=1.5 b=1 N=3 tau=1: closed, every chord tangent to the caustic
  ok    (i) a=1.5 b=1 N=3 tau=1: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1.5 b=1 N=3 tau=1: identifications of Sec. 3, positivity
  ok    (i) a=1.5 b=1 N=3: A_dual^P = A_dual/4
  a=1.5 b=1 N=4 tau=1: lambda=0.69230769230769230769 k^2=0.80246914 closure 4.9e-49 tangency 3.7e-49; spreads k813 1.1e-48, k814 7.3e-49, k814 point-wise 7.5e-49; identities 3.7e-50
  ok    (i) a=1.5 b=1 N=4 tau=1: closed, every chord tangent to the caustic
  ok    (i) a=1.5 b=1 N=4 tau=1: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1.5 b=1 N=4 tau=1: identifications of Sec. 3, positivity
  ok    (i) a=1.5 b=1 N=4: caustic lambda = a^2b^2/(a^2+b^2)
    N=4 constants: rel. errors 8.8e-49, 5.8e-49, 2.4e-49; A_dual^P/A_dual - 1/2: 6.7e-51; A A^dag - 4: 1.7e-48
  ok    (i) a=1.5 b=1 N=4: 2a^4/b^4, 2a^6/b^10, 2a^6/(b^2(a^2+b^2)^2), A_dual^P = A_dual/2, A A^dag = 4
  a=1.5 b=1 N=5 tau=1: lambda=0.49172266873827466865 k^2=0.71092312 closure 1.4e-49 tangency 6.9e-50; spreads k813 1.5e-49, k814 7.3e-50, k814 point-wise 6.4e-50; identities 6.0e-50
  ok    (i) a=1.5 b=1 N=5 tau=1: closed, every chord tangent to the caustic
  ok    (i) a=1.5 b=1 N=5 tau=1: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1.5 b=1 N=5 tau=1: identifications of Sec. 3, positivity
  a=1.5 b=1 N=5 tau=2: lambda=0.99868299590383558918 k^2=0.99894751 closure 4.1e-47 tangency 2.0e-47; spreads k813 1.4e-44, k814 1.4e-44, k814 point-wise 1.5e-44; identities 4.2e-48
  ok    (i) a=1.5 b=1 N=5 tau=2: closed, every chord tangent to the caustic
  ok    (i) a=1.5 b=1 N=5 tau=2: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1.5 b=1 N=5 tau=2: identifications of Sec. 3, positivity
  a=1.5 b=1 N=6 tau=1: lambda=0.36 k^2=0.66137566 closure 5.9e-49 tangency 2.8e-49; spreads k813 4.5e-49, k814 2.1e-49, k814 point-wise 2.4e-49; identities 3.7e-50
  ok    (i) a=1.5 b=1 N=6 tau=1: closed, every chord tangent to the caustic
  ok    (i) a=1.5 b=1 N=6 tau=1: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1.5 b=1 N=6 tau=1: identifications of Sec. 3, positivity
  ok    (i) a=1.5 b=1 N=6: caustic lambda = a^2b^2/(a+b)^2
  a=1.5 b=1 N=7 tau=2: lambda=0.82037298999395891005 k^2=0.87435393 closure 2.2e-48 tangency 1.5e-48; spreads k813 4.1e-48, k814 3.0e-48, k814 point-wise 2.9e-48; identities 4.1e-50
  ok    (i) a=1.5 b=1 N=7 tau=2: closed, every chord tangent to the caustic
  ok    (i) a=1.5 b=1 N=7 tau=2: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1.5 b=1 N=7 tau=2: identifications of Sec. 3, positivity
  a=1.5 b=1 N=8 tau=1: lambda=0.21296843950232320948 k^2=0.61363801 closure 1.3e-49 tangency 2.7e-50; spreads k813 4.7e-50, k814 4.0e-50, k814 point-wise 5.0e-50; identities 4.0e-50
  ok    (i) a=1.5 b=1 N=8 tau=1: closed, every chord tangent to the caustic
  ok    (i) a=1.5 b=1 N=8 tau=1: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1.5 b=1 N=8 tau=1: identifications of Sec. 3, positivity
  a=1.5 b=1 N=8 tau=3: lambda=0.99112013708084732346 k^2=0.99294622 closure 2.8e-47 tangency 2.7e-47; spreads k813 1.0e-45, k814 9.9e-46, k814 point-wise 1.0e-45; identities 4.1e-49
  ok    (i) a=1.5 b=1 N=8 tau=3: closed, every chord tangent to the caustic
  ok    (i) a=1.5 b=1 N=8 tau=3: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1.5 b=1 N=8 tau=3: identifications of Sec. 3, positivity
  a=1.5 b=1 N=9 tau=2: lambda=0.5819515718090468775 k^2=0.74937872 closure 1.1e-48 tangency 7.0e-49; spreads k813 8.0e-49, k814 4.5e-49, k814 point-wise 4.5e-49; identities 3.2e-50
  ok    (i) a=1.5 b=1 N=9 tau=2: closed, every chord tangent to the caustic
  ok    (i) a=1.5 b=1 N=9 tau=2: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1.5 b=1 N=9 tau=2: identifications of Sec. 3, positivity
  a=1.5 b=1 N=10 tau=3: lambda=0.86422003842542711469 k^2=0.9020191 closure 2.0e-48 tangency 1.5e-48; spreads k813 3.6e-48, k814 2.8e-48, k814 point-wise 2.8e-48; identities 6.3e-50
  ok    (i) a=1.5 b=1 N=10 tau=3: closed, every chord tangent to the caustic
  ok    (i) a=1.5 b=1 N=10 tau=3: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1.5 b=1 N=10 tau=3: identifications of Sec. 3, positivity
  a=1.5 b=1 N=12 tau=5: lambda=0.99980723723103774661 k^2=0.99984581 closure 1.1e-46 tangency 5.9e-47; spreads k813 2.1e-43, k814 2.1e-43, k814 point-wise 2.2e-43; identities 2.9e-47
  ok    (i) a=1.5 b=1 N=12 tau=5: closed, every chord tangent to the caustic
  ok    (i) a=1.5 b=1 N=12 tau=5: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=1.5 b=1 N=12 tau=5: identifications of Sec. 3, positivity
  a=2.6 b=0.9 N=3 tau=1: lambda=0.80692839119997622914 k^2=0.99948403 closure 5.5e-47 tangency 1.7e-47; spreads k813 3.0e-45, k814 2.9e-45, k814 point-wise 2.9e-45; identities 1.8e-48
  ok    (i) a=2.6 b=0.9 N=3 tau=1: closed, every chord tangent to the caustic
  ok    (i) a=2.6 b=0.9 N=3 tau=1: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=2.6 b=0.9 N=3 tau=1: identifications of Sec. 3, positivity
  ok    (i) a=2.6 b=0.9 N=3: A_dual^P = A_dual/4
  a=2.6 b=0.9 N=4 tau=1: lambda=0.72332892998678996037 k^2=0.98564257 closure 5.8e-48 tangency 1.8e-48; spreads k813 1.4e-47, k814 8.9e-48, k814 point-wise 8.8e-48; identities 1.2e-49
  ok    (i) a=2.6 b=0.9 N=4 tau=1: closed, every chord tangent to the caustic
  ok    (i) a=2.6 b=0.9 N=4 tau=1: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=2.6 b=0.9 N=4 tau=1: identifications of Sec. 3, positivity
  ok    (i) a=2.6 b=0.9 N=4: caustic lambda = a^2b^2/(a^2+b^2)
    N=4 constants: rel. errors 1.2e-47, 7.9e-48, 1.9e-48; A_dual^P/A_dual - 1/2: 2.7e-50; A A^dag - 4: 1.9e-47
  ok    (i) a=2.6 b=0.9 N=4: 2a^4/b^4, 2a^6/b^10, 2a^6/(b^2(a^2+b^2)^2), A_dual^P = A_dual/2, A A^dag = 4
  a=2.6 b=0.9 N=5 tau=1: lambda=0.57610071555365845779 k^2=0.96217608 closure 3.8e-48 tangency 1.3e-48; spreads k813 3.5e-48, k814 1.9e-48, k814 point-wise 1.9e-48; identities 1.5e-49
  ok    (i) a=2.6 b=0.9 N=5 tau=1: closed, every chord tangent to the caustic
  ok    (i) a=2.6 b=0.9 N=5 tau=1: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=2.6 b=0.9 N=5 tau=1: identifications of Sec. 3, positivity
  a=2.6 b=0.9 N=5 tau=2: lambda=0.80999685668343208122 k^2=0.99999947 closure 1.5e-44 tangency 1.6e-45; spreads k813 4.5e-40, k814 4.5e-40, k814 point-wise 4.5e-40; identities 3.5e-45
  ok    (i) a=2.6 b=0.9 N=5 tau=2: closed, every chord tangent to the caustic
  ok    (i) a=2.6 b=0.9 N=5 tau=2: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=2.6 b=0.9 N=5 tau=2: identifications of Sec. 3, positivity
  a=2.6 b=0.9 N=6 tau=1: lambda=0.44698775510204081633 k^2=0.94249778 closure 2.2e-48 tangency 6.6e-49; spreads k813 1.1e-48, k814 5.1e-49, k814 point-wise 5.7e-49; identities 1.5e-49
  ok    (i) a=2.6 b=0.9 N=6 tau=1: closed, every chord tangent to the caustic
  ok    (i) a=2.6 b=0.9 N=6 tau=1: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=2.6 b=0.9 N=6 tau=1: identifications of Sec. 3, positivity
  ok    (i) a=2.6 b=0.9 N=6: caustic lambda = a^2b^2/(a+b)^2
  a=2.6 b=0.9 N=7 tau=2: lambda=0.78058254053298780526 k^2=0.99508021 closure 1.3e-47 tangency 4.3e-48; spreads k813 4.4e-47, k814 3.5e-47, k814 point-wise 3.5e-47; identities 3.8e-49
  ok    (i) a=2.6 b=0.9 N=7 tau=2: closed, every chord tangent to the caustic
  ok    (i) a=2.6 b=0.9 N=7 tau=2: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=2.6 b=0.9 N=7 tau=2: identifications of Sec. 3, positivity
  a=2.6 b=0.9 N=8 tau=1: lambda=0.27871874785215549612 k^2=0.91802836 closure 6.2e-49 tangency 1.4e-49; spreads k813 1.8e-49, k814 2.4e-49, k814 point-wise 5.2e-49; identities 1.4e-48
  ok    (i) a=2.6 b=0.9 N=8 tau=1: closed, every chord tangent to the caustic
  ok    (i) a=2.6 b=0.9 N=8 tau=1: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=2.6 b=0.9 N=8 tau=1: identifications of Sec. 3, positivity
  a=2.6 b=0.9 N=8 tau=3: lambda=0.80990147819289804142 k^2=0.99998344 closure 6.7e-45 tangency 3.7e-46; spreads k813 8.2e-42, k814 8.2e-42, k814 point-wise 8.2e-42; identities 1.2e-46
  ok    (i) a=2.6 b=0.9 N=8 tau=3: closed, every chord tangent to the caustic
  ok    (i) a=2.6 b=0.9 N=8 tau=3: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=2.6 b=0.9 N=8 tau=3: identifications of Sec. 3, positivity
  a=2.6 b=0.9 N=9 tau=2: lambda=0.65041267565024778599 k^2=0.9738792 closure 6.5e-48 tangency 2.2e-48; spreads k813 3.8e-48, k814 2.2e-48, k814 point-wise 2.2e-48; identities 3.0e-49
  ok    (i) a=2.6 b=0.9 N=9 tau=2: closed, every chord tangent to the caustic
  ok    (i) a=2.6 b=0.9 N=9 tau=2: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=2.6 b=0.9 N=9 tau=2: identifications of Sec. 3, positivity
  a=2.6 b=0.9 N=10 tau=3: lambda=0.79315348948770306715 k^2=0.99717665 closure 5.4e-47 tangency 1.9e-47; spreads k813 2.5e-46, k814 2.1e-46, k814 point-wise 2.1e-46; identities 7.5e-49
  ok    (i) a=2.6 b=0.9 N=10 tau=3: closed, every chord tangent to the caustic
  ok    (i) a=2.6 b=0.9 N=10 tau=3: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=2.6 b=0.9 N=10 tau=3: identifications of Sec. 3, positivity
  a=2.6 b=0.9 N=12 tau=5: lambda=0.80999989973636640139 k^2=0.99999998 closure 4.5e-42 tangency 2.6e-43; spreads k813 6.1e-36, k814 6.1e-36, k814 point-wise 6.2e-36; identities 1.1e-43
  ok    (i) a=2.6 b=0.9 N=12 tau=5: closed, every chord tangent to the caustic
  ok    (i) a=2.6 b=0.9 N=12 tau=5: A_pol/A^dag, A_pol/A_dual, A_pol/A_dual^P constant
  ok    (i) a=2.6 b=0.9 N=12 tau=5: identifications of Sec. 3, positivity

=== (ii) exact N = 3 certificate for row k815 (a^2 = 8, b^2 = 3, lambda = 72/25)
  ok    (ii) M1: exact 3-periodic of the family (on e, sides tangent, reflection law)
  ok    (ii) M1: polar polygon = antipedal of P^dag = poles of the sides; sides touch (O_p, a/b^2)
    M1 A       = 54*sqrt(6)/25
    M1 Adag    = (sqrt(15) + 4*sqrt(6))/64
    M1 Apol    = (8*sqrt(15) + 32*sqrt(6))/3
    M1 Adual   = (27*sqrt(15) + 108*sqrt(6))/320
    M1 AdualP  = (27*sqrt(15) + 108*sqrt(6))/1280
    M1 Aped    = (-24*sqrt(15) + 96*sqrt(6))/625
    M1 Aant    = -9*sqrt(15) + 36*sqrt(6)
    M1 Ai      = 3*sqrt(6)/800
  ok    (ii) M2: exact 3-periodic of the family (on e, sides tangent, reflection law)
  ok    (ii) M2: polar polygon = antipedal of P^dag = poles of the sides; sides touch (O_p, a/b^2)
    M2 A       = 144/25
    M2 Adag    = 9/64
    M2 Apol    = 24
    M2 Adual   = 243/320
    M2 AdualP  = 243/1280
    M2 Aped    = 216/625
    M2 Aant    = 96
    M2 Ai      = 27/3200
    k813 : M1 512/3   M2 512/3
    k814 : M1 2560/81   M2 2560/81
    k814P: M1 10240/81   M2 10240/81
    i    : M1 243*sqrt(10)/256000 + 243/32000   M2 6561/1024000
    iP   : M1 243*sqrt(10)/1024000 + 243/128000   M2 6561/4096000
    ii   : M1 27*sqrt(10)/5 + 999/40   M2 729/40
    iii  : M1 6561/25000   M2 6561/25000
    iiiP : M1 6561/100000   M2 6561/100000
    difference under reading (i): 243/204800 + 243*sqrt(10)/256000 = 0.0041882166852379538190;  under reading (ii): 27/4 + 27*sqrt(10)/5 = 23.826299364909248393
  ok    (ii) k813 = 512/3, k814 = 2560/81 on both members
  ok    (ii) k814 with the point-wise dual equal on both members
  ok    (ii) k815 reading (i): 243 sqrt10/256000 + 243/32000 vs 6561/1024000 -- not constant
  ok    (ii) k815 reading (i), point-wise dual: not constant
  ok    (ii) k815 reading (ii) (A_pol A_dual): 27 sqrt10/5 + 999/40 vs 729/40 -- not constant
  ok    (ii) k815 reading (iii) (Abar A_dual): 6561/25000 on both members (both duals equal)

=== (iii) exact N = 4 certificate for row k816 (a = 5, b = 4, lambda = 400/41); N = 4 constants; N = 6, 8
  ok    (iii) R1 (rhombus): exact 4-periodic of the family
  ok    (iii) R1 (rhombus): polar polygon two ways; sides touch (O_p, a/b^2)
    R1 (rhombus) A       = 40
    R1 (rhombus) Adag    = 1/10
    R1 (rhombus) Apol    = 125/256
    R1 (rhombus) Adual   = 2048/125
    R1 (rhombus) AdualP  = 1024/125
    R1 (rhombus) Aped    = 32000/1681
    R1 (rhombus) Aant    = 80
    R1 (rhombus) Ai      = 4096/210125
  ok    (iii) R1 (rhombus): A^dag_ped = 4b^2(hk)^3/(a^6(a^2+b^2)^2), the N = 4 formula of 10.5281/zenodo.23073120
  ok    (iii) R2 (rectangle): exact 4-periodic of the family
  ok    (iii) R2 (rectangle): polar polygon two ways; sides touch (O_p, a/b^2)
    R2 (rectangle) A       = 1600/41
    R2 (rectangle) Adag    = 41/400
    R2 (rectangle) Apol    = 1025/2048
    R2 (rectangle) Adual   = 10496/625
    R2 (rectangle) AdualP  = 5248/625
    R2 (rectangle) Aped    = 800/41
    R2 (rectangle) Aant    = 3200/41
    R2 (rectangle) Ai      = 328/15625
  ok    (iii) R2 (rectangle): A^dag_ped = 4b^2(hk)^3/(a^6(a^2+b^2)^2), the N = 4 formula of 10.5281/zenodo.23073120
    i     : R1 2/1681   R2 1/800
    iP    : R1 4/1681   R2 1/400
    ii    : R1 15625/524288   R2 15625/524288
    iii   : R1 15625/13448   R2 15625/13448
    AAd   : R1 4   R2 4
    AdAant: R1 8   R2 8
    AantA : R1 2   R2 2
  ok    (iii) k816 reading (i), N = 4: 2/1681 vs 1/800 -- not constant
  ok    (iii) k816 reading (i), point-wise dual: 4/1681 vs 1/400 -- not constant
  ok    (iii) k816 readings (ii), (iii): 15625/524288 and 15625/13448 on both members
  ok    (iii) N = 4: A A^dag = 4, A^dag A_ant = 8, A_ant/A = 2 on both members
  the rhombus (a,0),(0,b),(-a,0),(0,-b) with symbolic a > b > 0, lambda = a^2 b^2/(a^2+b^2):
  ok    (iii) the rhombus: all four sides tangent to the caustic lambda = a^2b^2/(a^2+b^2)
    A       = 2*a*b
    A^dag   = 2/(a*b)
    A_pol   = 4*a**3/b**5
    A_dual  = 2*b**5/a**3
    Abar    = 4*a**3*b**3/(a**2 + b**2)**2
    A_ant   = 4*a*b
    A_pol/A^dag = 2*a**4/b**4;  A_pol/A_dual = 2*a**6/b**10;  Abar/A_dual = 2*a**6/(b**2*(a**2 + b**2)**2)
  ok    (iii) rhombus: A_pol/A^dag = 2a^4/b^4, A_pol/A_dual = 2a^6/b^10, Abar/A_dual = 2a^6/(b^2(a^2+b^2)^2)
  ok    (iii) rhombus: A A^dag = 4, A_ant/A = 2 = R(a, b, a^2b^2/(a^2+b^2))
  k816 for N = 6, 8 under every reading (reflection-law trajectories), and N = 4 numerically:
    a=1 b=0.6 N=4 tau=1: spreads i 0.22, iP 0.22, ii 6.7e-49, iii 3.7e-49, iiiP 3.3e-49
  ok    (iii) a=1 b=0.6 N=4: k816 varies under reading (i), constant under (ii), (iii)
    a=1 b=0.6 N=6 tau=1: spreads i 9.9e-50, iP 8.2e-50, ii 3.9e-50, iii 6.0e-50, iiiP 7.8e-50
  ok    (iii) a=1 b=0.6 N=6 tau=1: k816 constant under (i), (ii), (iii), both duals
    a=1 b=0.6 N=8 tau=1: spreads i 4.0e-49, iP 3.8e-49, ii 1.2e-49, iii 1.4e-49, iiiP 1.7e-49
  ok    (iii) a=1 b=0.6 N=8 tau=1: k816 constant under (i), (ii), (iii), both duals
    a=1 b=0.6 N=8 tau=3: spreads i 6.8e-47, iP 3.5e-47, ii 4.1e-45, iii 2.0e-46, iiiP 1.3e-46
  ok    (iii) a=1 b=0.6 N=8 tau=3: k816 constant under (i), (ii), (iii), both duals
    a=2.6 b=0.9 N=4 tau=1: spreads i 0.61, iP 0.61, ii 8.8e-48, iii 3.7e-48, iiiP 3.7e-48
  ok    (iii) a=2.6 b=0.9 N=4: k816 varies under reading (i), constant under (ii), (iii)
    a=2.6 b=0.9 N=6 tau=1: spreads i 1.7e-48, iP 1.6e-48, ii 5.1e-49, iii 7.6e-49, iiiP 8.2e-49
  ok    (iii) a=2.6 b=0.9 N=6 tau=1: k816 constant under (i), (ii), (iii), both duals
    a=2.6 b=0.9 N=8 tau=1: spreads i 5.5e-49, iP 8.6e-49, ii 2.7e-49, iii 1.8e-49, iiiP 5.0e-49
  ok    (iii) a=2.6 b=0.9 N=8 tau=1: k816 constant under (i), (ii), (iii), both duals
    a=2.6 b=0.9 N=8 tau=3: spreads i 1.0e-44, iP 2.1e-44, ii 8.2e-42, iii 7.0e-44, iiiP 5.4e-44
  ok    (iii) a=2.6 b=0.9 N=8 tau=3: k816 constant under (i), (ii), (iii), both duals

=== (iv) row k818: A_ant = R A, A^dag = kappa_b A for N = 2 mod 4; the exceptional families
    a=1 b=0.8 N=6 tau=1: R = 1.08482142857143, A_ant/A - R: 5.9e-50, spread A^dag/A 2.0e-49, A^dag/A_ant = 2.1875 (spread 2.3e-49)
  ok    (iv) a=1 b=0.8 N=6 tau=1: A_ant = R A, A^dag/A and A^dag/A_ant constant
    a=1.7 b=0.3 N=6 tau=1: R = -13.2132132132132, A_ant/A - R: 2.6e-49, spread A^dag/A 7.8e-49, A^dag/A_ant = -5.89551118759342 (spread 9.4e-49)
  ok    (iv) a=1.7 b=0.3 N=6 tau=1: A_ant = R A, A^dag/A and A^dag/A_ant constant
    a=1 b=0.5 N=10 tau=1: R = -0.365922242006809, A_ant/A - R: 8.7e-51, spread A^dag/A 6.1e-50, A^dag/A_ant = -74.4175010170852 (spread 8.0e-50)
  ok    (iv) a=1 b=0.5 N=10 tau=1: A_ant = R A, A^dag/A and A^dag/A_ant constant
    a=1 b=0.5 N=10 tau=3: R = 9.08479156859737, A_ant/A - R: 4.1e-48, spread A^dag/A 4.8e-48, A^dag/A_ant = 0.345230234619224 (spread 8.9e-48)
  ok    (iv) a=1 b=0.5 N=10 tau=3: A_ant = R A, A^dag/A and A^dag/A_ant constant
    a=2.6 b=0.9 N=10 tau=1: R = -2.50522307081226, A_ant/A - R: 2.0e-49, spread A^dag/A 7.1e-49, A^dag/A_ant = -1.23147783981507 (spread 7.9e-49)
  ok    (iv) a=2.6 b=0.9 N=10 tau=1: A_ant = R A, A^dag/A and A^dag/A_ant constant
    a=1 b=0.6 N=14 tau=1: R = 0.169832752877993, A_ant/A - R: 3.5e-50, spread A^dag/A 2.4e-49, A^dag/A_ant = 82.7544431286178 (spread 2.8e-49)
  ok    (iv) a=1 b=0.6 N=14 tau=1: A_ant = R A, A^dag/A and A^dag/A_ant constant
  exceptional family N = 6, a = 1, b = 1/2 (exact):
  ok    (iv) H: exact 6-periodic (on e, sides tangent to lambda = 1/9, reflection law, 6 distinct vertices)
    H A     = 5*sqrt(5)/9
    H Adag  = 15*sqrt(5)/2
    H Apol  = 32*sqrt(5)
    H Adual = 5*sqrt(5)/96
    H Aped  = 50*sqrt(5)/81
    H Aant  = 0
    H Ai    = 25*sqrt(5)/36
  ok    (iv) H: focal antipedal area exactly 0, A = 5 sqrt5/9, and R(1, 1/2, 1/9) = 0 exactly
    H: A_pol/A^dag = 64/15, A^dag_ped/A_dual = 40/3, Abar/A_dual = 320/27
  ok    (iv) H: A_pol/A^dag = 64/15, A^dag_ped/A_dual = 40/3, Abar/A_dual = 320/27
    N = 6 with symbolic a, b: R(a, b, a^2b^2/(a+b)^2) = -(a - 2*b)*(a + b)**2/(b**2*(2*a + b))
  ok    (iv) N = 6: the hexagon (+-a,0), (+-a^2/(a+b), +-y0) is inscribed and tangent to lambda = a^2b^2/(a+b)^2; R = (2b-a)(a+b)^2/(b^2(2a+b))
    numerically: lambda - 1/9 = 4.68e-51, max |A_ant/A| over 7 members = 9.47e-50
  ok    (iv) N = 6, b/a = 1/2: lambda = 1/9 and A_ant = 0 at 7 members
  exceptional aspect ratios (a = 1, winding 1): roots of R(b) = 0 with lambda = lambda_N(b)
  ok    (iv) N=10: R changes sign between b/a = 0.50 and 0.5773
    N=10: b*/a = 0.5543486017301 (quoted 0.5543486017...), max |A_ant/A| at 3 members = 3.3e-51
  ok    (iv) N=10: exceptional aspect ratio 0.5543486017..., A_ant vanishes along the family
  ok    (iv) N=14: R changes sign between b/a = 0.50 and 0.5773
    N=14: b*/a = 0.5661108682363 (quoted 0.5661108682...), max |A_ant/A| at 3 members = 2.6e-41
  ok    (iv) N=14: exceptional aspect ratio 0.5661108682..., A_ant vanishes along the family
  ok    (iv) N=18: R changes sign between b/a = 0.50 and 0.5773
    N=18: b*/a = 0.5706649669806 (quoted 0.5706649669...), max |A_ant/A| at 3 members = 2.2e-44
  ok    (iv) N=18: exceptional aspect ratio 0.5706649669..., A_ant vanishes along the family

=== (v) controls: the rows outside their classes of N, an unclosed orbit, a repeated 3-periodic
    a=1 b=0.6 N=5 tau=1: k815(i) 0.12, k816(i) 0.085, k816(iii) 0.042, k818 0.23, A_ant/A 0.25
  ok    (v) a=1 b=0.6 N=5 tau=1: k815(i), k816(i), k816(iii), k818, A_ant/A vary
    a=1 b=0.6 N=7 tau=2: k815(i) 0.23, k816(i) 0.25, k816(iii) 0.024, k818 0.037, A_ant/A 0.049
  ok    (v) a=1 b=0.6 N=7 tau=2: k815(i), k816(i), k816(iii), k818, A_ant/A vary
    a=1 b=0.6 N=6 tau=1: k815(i) 0.0084, k815(iii) 0.0084
  ok    (v) a=1 b=0.6 N=6 tau=1: k815(i), k815(iii) vary
    a=1 b=0.6 N=8 tau=1: k815(i) 0.00043, k815(iii) 0.00043, k818 0.00043
  ok    (v) a=1 b=0.6 N=8 tau=1: k815(i), k815(iii), k818 vary
    a=1 b=0.6 N=12 tau=5: k815(i) 0.15, k815(iii) 0.15, k818 0.15
  ok    (v) a=1 b=0.6 N=12 tau=5: k815(i), k815(iii), k818 vary
    a=1.5 b=1 N=5 tau=1: k815(i) 0.074, k816(i) 0.051, k816(iii) 0.024, k818 0.11, A_ant/A 0.12
  ok    (v) a=1.5 b=1 N=5 tau=1: k815(i), k816(i), k816(iii), k818, A_ant/A vary
    a=1.5 b=1 N=7 tau=2: k815(i) 0.088, k816(i) 0.098, k816(iii) 0.011, k818 0.016, A_ant/A 0.021
  ok    (v) a=1.5 b=1 N=7 tau=2: k815(i), k816(i), k816(iii), k818, A_ant/A vary
    a=1.5 b=1 N=6 tau=1: k815(i) 0.0043, k815(iii) 0.0043
  ok    (v) a=1.5 b=1 N=6 tau=1: k815(i), k815(iii) vary
    a=1.5 b=1 N=8 tau=1: k815(i) 0.00017, k815(iii) 0.00017, k818 0.00017
  ok    (v) a=1.5 b=1 N=8 tau=1: k815(i), k815(iii), k818 vary
    a=1.5 b=1 N=12 tau=5: k815(i) 0.087, k815(iii) 0.087, k818 0.087
  ok    (v) a=1.5 b=1 N=12 tau=5: k815(i), k815(iii), k818 vary
    unclosed pentagons (J perturbed by 1e-3): spread of A_pol/A^dag = 0.0208
  ok    (v) unclosed orbits: A_pol/A^dag varies
    3-periodic listed twice: spreads k816(i) 0.576, k816(iii) 0.818
  ok    (v) a 3-periodic listed twice does not satisfy the even-N rows

checks: 171, failures: 0
TOTAL failures: 0   (running time 14 s)
