Lemma 3.8 / Theorem C(iii) arithmetic, N<=200, all coprime tau: counterexamples = []

Family a=2 N=8 tau=1: lambda = 0.2677523569258437738377986 (geometry program printed its own lambda for comparison)
Family a=2 N=4 tau=1: lambda = 0.8 (geometry program printed its own lambda for comparison)
Family a=1.3 N=12 tau=5: lambda = 0.998745041392009646302928 (geometry program printed its own lambda for comparison)
Family a=3.5 N=16 tau=3: lambda = 0.7071488197394602110925973 (geometry program printed its own lambda for comparison)
Family a=2 N=6 tau=1: lambda = 0.4444444444444444444444444 (geometry program printed its own lambda for comparison)
Family a=2 N=5 tau=1: lambda = 0.594497789748865945281097 (geometry program printed its own lambda for comparison)
Family a=1.3 N=10 tau=3: lambda = 0.8047981989033840990603253 (geometry program printed its own lambda for comparison)
Family a=2 N=20 tau=7: lambda = 0.9927107931534176853142195 (geometry program printed its own lambda for comparison)
Family a=2 N=24 tau=7: lambda = 0.9217705580902056986775801 (geometry program printed its own lambda for comparison)

=== Lemma checks (random complex points) ===
Lemma checks for a=2.0 b=1.0 N=8 tau=1 lambda=0.2677523569258437738377986  (N h/(4K) = 1.0)
  L3.1 chord: m(t).P(t+-Delta) - 1                                       2.37e-41
  L3.1 tangency: (al-lam) m1^2 + (be-lam) m2^2 - 1                       2.31e-41
  L3.3: m.m - dn^2 t/(b^2 cn^2 Delta)                                    4.59e-41
  L3.4a: m(z+w)+m(z-w), P(z+w)+P(z-w), z in Pi                           4.41e-40
  L3.4b: m(Z+w)-m(Z-w), P(Z+w)-P(Z-w), Z in Pi'                          1.03e-39
  L3.4c: m(t+2K)+m(t)                                                    1.29e-41
  dn(Z) for Z in Pi' (zero of dn)                                        3.97e-41
  L3.12: P(u+2iK') - R P(u), m(t+2iK') - R m(t)                          2.79e-39
  ThC(ii): n.n at Pi'+-Delta                                             2.95e-41
  ThC(ii): |d(n.n)/dv| at Pi'+-Delta (must be >> 0: simple zero)         0.268
  ThC(i): V_0(u) - diag(e0/c0,1/c0) P(u+Delta)                           3.85e-40
  ThC(i): A'(u) - (e0/c0^2) A(u+Delta)                                   8.71e-40
  dn(K+iK'+eps)/eps = (-1.7970434e-27 + 0.44293884j)  (nonzero finite => simple zero; k' sc(eps)... expected i k' =  (0.0 + 0.44293884j) )
Lemma checks for a=1.300000000000000044408920985006261616945 b=1.0 N=12 tau=5 lambda=0.998745041392009646302928  (N h/(4K) = 5.0)
  L3.1 chord: m(t).P(t+-Delta) - 1                                       9.47e-41
  L3.1 tangency: (al-lam) m1^2 + (be-lam) m2^2 - 1                       4.32e-39
  L3.3: m.m - dn^2 t/(b^2 cn^2 Delta)                                    9.69e-38
  L3.4a: m(z+w)+m(z-w), P(z+w)+P(z-w), z in Pi                           1.03e-38
  L3.4b: m(Z+w)-m(Z-w), P(Z+w)-P(Z-w), Z in Pi'                          2.2e-39
  L3.4c: m(t+2K)+m(t)                                                    2.29e-38
  dn(Z) for Z in Pi' (zero of dn)                                        1.34e-41
  L3.12: P(u+2iK') - R P(u), m(t+2iK') - R m(t)                          8.18e-40
  ThC(ii): n.n at Pi'+-Delta                                             1.54e-39
  ThC(ii): |d(n.n)/dv| at Pi'+-Delta (must be >> 0: simple zero)         1.85
  ThC(i): V_0(u) - diag(e0/c0,1/c0) P(u+Delta)                           2.17e-40
  ThC(i): A'(u) - (e0/c0^2) A(u+Delta)                                   3.88e-37
  dn(K+iK'+eps)/eps = (1.3553224e-28 + 0.042608434j)  (nonzero finite => simple zero; k' sc(eps)... expected i k' =  (0.0 + 0.042608434j) )
Lemma checks for a=2.0 b=1.0 N=5 tau=1 lambda=0.594497789748865945281097  (N h/(4K) = 1.0)
  L3.1 chord: m(t).P(t+-Delta) - 1                                       2.37e-41
  L3.1 tangency: (al-lam) m1^2 + (be-lam) m2^2 - 1                       4.63e-41
  L3.3: m.m - dn^2 t/(b^2 cn^2 Delta)                                    8.28e-41
  L3.4a: m(z+w)+m(z-w), P(z+w)+P(z-w), z in Pi                           3.36e-39
  L3.4b: m(Z+w)-m(Z-w), P(Z+w)-P(Z-w), Z in Pi'                          1.1e-39
  L3.4c: m(t+2K)+m(t)                                                    5.31e-41
  dn(Z) for Z in Pi' (zero of dn)                                        3.5e-41
  L3.12: P(u+2iK') - R P(u), m(t+2iK') - R m(t)                          8.58e-41
  ThC(ii): n.n at Pi'+-Delta                                             4.81e-41
  ThC(ii): |d(n.n)/dv| at Pi'+-Delta (must be >> 0: simple zero)         0.418
  ThC(i): V_0(u) - diag(e0/c0,1/c0) P(u+Delta)                           1.95e-40
  ThC(i): A'(u) - (e0/c0^2) A(u+Delta)                                   2.76e-40
  dn(K+iK'+eps)/eps = (-2.0073491e-27 + 0.34506907j)  (nonzero finite => simple zero; k' sc(eps)... expected i k' =  (0.0 + 0.34506907j) )

=== Pole structure, cancellation, involutions, residues (N = 0 mod 4) ===
Pole structure a=2.0 N=8 tau=1 M=(mpf('0.8000000000000000000000000000000000000000023'), mpf('0.3000000000000000000000000000000000000000023')): u0 (vertex 7 at iK'), u1 (side 7 at K+iK')
  real-axis constant A*A_M(0.7) = 33.4063768502875027135279705346
  eps=1e-4: at u0 |A|*|eps|=2.0 |A_M|/|eps|=16.7 prod-cst=4.24e-35 | at u1 |A|/|eps|=15.4 |A_M|*|eps|=2.17 |A_M|*|eps|^2=0.000217 prod-cst=8.52e-32
  eps=1e-8: at u0 |A|*|eps|=2.0 |A_M|/|eps|=16.7 prod-cst=5.44e-31 | at u1 |A|/|eps|=15.4 |A_M|*|eps|=2.17 |A_M|*|eps|^2=2.17e-8 prod-cst=3.54e-24
  eps=1e-12: at u0 |A|*|eps|=2.0 |A_M|/|eps|=16.7 prod-cst=5.46e-27 | at u1 |A|/|eps|=15.4 |A_M|*|eps|=2.17 |A_M|*|eps|^2=2.17e-12 prod-cst=1.5e-16
  L3.13 residue: eps*A(u0+eps) = (-8.1114158901972223e-22 - 1.99931798677852209j)  formula = (0.0 - 1.99931798677852209j)  ratio = (1.0 - 4.05709144010006e-22j)
  L3.9: max_i |m_{rho(i)} - m_i| at u0 = 1.01e-39 ; A_M(u0+1e-20) = 1.67e-19
  L3.10: max_l |P(u_sigma(l)) - P(u_l)| at u1 = 2.32e-39 ; A(u1) = 1.78e-39
  L3.7: |Q_{i+1}-Q_{i-1}|/|eps| at u1+eps (eps=1e-6) = 17.5
  L3.7: |Q_{i+1}-Q_{i-1}|/|eps| at u1+eps (eps=1e-12) = 17.5
Pole structure a=2.0 N=4 tau=1 M=(mpf('0.8000000000000000000000000000000000000000023'), mpf('0.3000000000000000000000000000000000000000023')): u0 (vertex 0 at iK'), u1 (side 3 at K+iK')
  real-axis constant A*A_M(0.7) = 5.12
  eps=1e-4: at u0 |A|*|eps|=3.2 |A_M|/|eps|=1.6 prod-cst=1.66e-36 | at u1 |A|/|eps|=4.0 |A_M|*|eps|=1.28 |A_M|*|eps|^2=0.000128 prod-cst=1.76e-32
  eps=1e-8: at u0 |A|*|eps|=3.2 |A_M|/|eps|=1.6 prod-cst=1.58e-32 | at u1 |A|/|eps|=4.0 |A_M|*|eps|=1.28 |A_M|*|eps|^2=1.28e-8 prod-cst=1.05e-24
  eps=1e-12: at u0 |A|*|eps|=3.2 |A_M|/|eps|=1.6 prod-cst=1.57e-28 | at u1 |A|/|eps|=4.0 |A_M|*|eps|=1.28 |A_M|*|eps|^2=1.28e-12 prod-cst=5.19e-16
  L3.13 residue: eps*A(u0+eps) = (-1.87105164763024963e-22 - 3.2j)  formula = (0.0 - 3.2j)  ratio = (1.0 - 5.84703639884453e-23j)
  L3.9: max_i |m_{rho(i)} - m_i| at u0 = 4.36e-40 ; A_M(u0+1e-20) = 1.6e-20
  L3.10: max_l |P(u_sigma(l)) - P(u_l)| at u1 = 2.53e-40 ; A(u1) = 4.02e-40
  L3.7: |Q_{i+1}-Q_{i-1}|/|eps| at u1+eps (eps=1e-6) = 1.15
  L3.7: |Q_{i+1}-Q_{i-1}|/|eps| at u1+eps (eps=1e-12) = 1.15
Pole structure a=1.300000000000000044408920985006261616945 N=12 tau=5 M=(mpf('0.8000000000000000000000000000000000000000023'), mpf('0.3000000000000000000000000000000000000000023')): u0 (vertex 10 at iK'), u1 (side 1 at K+iK')
  real-axis constant A*A_M(0.7) = -0.0884149944140062429839728592243
  eps=1e-4: at u0 |A|*|eps|=1.66 |A_M|/|eps|=0.0532 prod-cst=5.41e-36 | at u1 |A|/|eps|=7.15 |A_M|*|eps|=0.0124 |A_M|*|eps|^2=1.24e-6 prod-cst=7.28e-33
  eps=1e-8: at u0 |A|*|eps|=1.66 |A_M|/|eps|=0.0532 prod-cst=5.27e-32 | at u1 |A|/|eps|=7.15 |A_M|*|eps|=0.0124 |A_M|*|eps|^2=1.24e-10 prod-cst=7.16e-25
  eps=1e-12: at u0 |A|*|eps|=1.66 |A_M|/|eps|=0.0532 prod-cst=5.18e-28 | at u1 |A|/|eps|=7.15 |A_M|*|eps|=0.0124 |A_M|*|eps|^2=1.24e-14 prod-cst=7.08e-17
  L3.13 residue: eps*A(u0+eps) = (5.28595526251699886e-22 - 1.6617911568513889j)  formula = (0.0 - 1.6617911568513889j)  ratio = (1.0 + 3.18087819923916e-22j)
  L3.9: max_i |m_{rho(i)} - m_i| at u0 = 2.36e-36 ; A_M(u0+1e-20) = 5.32e-22
  L3.10: max_l |P(u_sigma(l)) - P(u_l)| at u1 = 1.08e-37 ; A(u1) = 5.13e-38
  L3.7: |Q_{i+1}-Q_{i-1}|/|eps| at u1+eps (eps=1e-6) = 0.683
  L3.7: |Q_{i+1}-Q_{i-1}|/|eps| at u1+eps (eps=1e-12) = 0.683
Pole structure a=3.5 N=16 tau=3 M=(mpf('0.8000000000000000000000000000000000000000023'), mpf('0.3000000000000000000000000000000000000000023')): u0 (vertex 0 at iK'), u1 (side 11 at K+iK')
  real-axis constant A*A_M(0.7) = 330.341951809792699908993305893
  eps=1e-4: at u0 |A|*|eps|=5.71 |A_M|/|eps|=57.8 prod-cst=7.67e-34 | at u1 |A|/|eps|=86.1 |A_M|*|eps|=3.84 |A_M|*|eps|^2=0.000384 prod-cst=8.4e-30
  eps=1e-8: at u0 |A|*|eps|=5.71 |A_M|/|eps|=57.8 prod-cst=5.02e-30 | at u1 |A|/|eps|=86.1 |A_M|*|eps|=3.84 |A_M|*|eps|^2=3.84e-8 prod-cst=9.34e-22
  eps=1e-12: at u0 |A|*|eps|=5.71 |A_M|/|eps|=57.8 prod-cst=1.02e-25 | at u1 |A|/|eps|=86.1 |A_M|*|eps|=3.84 |A_M|*|eps|^2=3.84e-12 prod-cst=7.67e-14
  L3.13 residue: eps*A(u0+eps) = (-4.32731533386759613e-21 - 5.714022607077951j)  formula = (0.0 - 5.714022607077951j)  ratio = (1.0 - 7.57315053060406e-22j)
  L3.9: max_i |m_{rho(i)} - m_i| at u0 = 2.38e-38 ; A_M(u0+1e-20) = 5.78e-19
  L3.10: max_l |P(u_sigma(l)) - P(u_l)| at u1 = 3.29e-38 ; A(u1) = 2.75e-38
  L3.7: |Q_{i+1}-Q_{i-1}|/|eps| at u1+eps (eps=1e-6) = 2.37
  L3.7: |Q_{i+1}-Q_{i-1}|/|eps| at u1+eps (eps=1e-12) = 2.37
Pole structure a=2.0 N=20 tau=7 M=(mpf('0.8000000000000000000000000000000000000000023'), mpf('0.3000000000000000000000000000000000000000023')): u0 (vertex 15 at iK'), u1 (side 2 at K+iK')
  real-axis constant A*A_M(0.7) = 23.0286849139338366469562179235
  eps=1e-4: at u0 |A|*|eps|=3.46 |A_M|/|eps|=6.66 prod-cst=2.0e-35 | at u1 |A|/|eps|=44.1 |A_M|*|eps|=0.523 |A_M|*|eps|^2=5.23e-5 prod-cst=6.17e-30
  eps=1e-8: at u0 |A|*|eps|=3.46 |A_M|/|eps|=6.66 prod-cst=2.45e-31 | at u1 |A|/|eps|=44.1 |A_M|*|eps|=0.523 |A_M|*|eps|^2=5.23e-9 prod-cst=6.18e-22
  eps=1e-12: at u0 |A|*|eps|=3.46 |A_M|/|eps|=6.66 prod-cst=1.59e-26 | at u1 |A|/|eps|=44.1 |A_M|*|eps|=0.523 |A_M|*|eps|^2=5.23e-13 prod-cst=6.26e-14
  L3.13 residue: eps*A(u0+eps) = (-2.12336320896816099e-21 - 3.45564376275702534j)  formula = (0.0 - 3.45564376275702534j)  ratio = (1.0 - 6.14462414167968e-22j)
  L3.9: max_i |m_{rho(i)} - m_i| at u0 = 1.75e-36 ; A_M(u0+1e-20) = 6.66e-20
  L3.10: max_l |P(u_sigma(l)) - P(u_l)| at u1 = 7.36e-37 ; A(u1) = 9.78e-37
  L3.7: |Q_{i+1}-Q_{i-1}|/|eps| at u1+eps (eps=1e-6) = 0.419
  L3.7: |Q_{i+1}-Q_{i-1}|/|eps| at u1+eps (eps=1e-12) = 0.419
Pole structure a=2.0 N=8 tau=1 M=(mpc(real='0.4000000000000000000000000000000000000000011', imag='1.100000000000000000000000000000000000000005'), mpc(real='-0.6000000000000000000000000000000000000000046', imag='0.2000000000000000000000000000000000000000006')): u0 (vertex 2 at iK'), u1 (side 5 at K+iK')
  real-axis constant A*A_M(0.7) = (25.9428589310555981657196852082 + 3.27167908788247870588856343073j)
  eps=1e-4: at u0 |A|*|eps|=2.0 |A_M|/|eps|=13.1 prod-cst=6.63e-36 | at u1 |A|/|eps|=15.4 |A_M|*|eps|=1.7 |A_M|*|eps|^2=0.00017 prod-cst=6.67e-32
  eps=1e-8: at u0 |A|*|eps|=2.0 |A_M|/|eps|=13.1 prod-cst=9.11e-32 | at u1 |A|/|eps|=15.4 |A_M|*|eps|=1.7 |A_M|*|eps|^2=1.7e-8 prod-cst=2.77e-24
  eps=1e-12: at u0 |A|*|eps|=2.0 |A_M|/|eps|=13.1 prod-cst=2.87e-27 | at u1 |A|/|eps|=15.4 |A_M|*|eps|=1.7 |A_M|*|eps|^2=1.7e-12 prod-cst=1.17e-16
  L3.13 residue: eps*A(u0+eps) = (-8.11141589019722229e-22 - 1.99931798677852209j)  formula = (0.0 - 1.99931798677852209j)  ratio = (1.0 - 4.05709144010006e-22j)
  L3.9: max_i |m_{rho(i)} - m_i| at u0 = 1.01e-39 ; A_M(u0+1e-20) = 1.31e-19
  L3.10: max_l |P(u_sigma(l)) - P(u_l)| at u1 = 9.9e-40 ; A(u1) = 6.64e-41
  L3.7: |Q_{i+1}-Q_{i-1}|/|eps| at u1+eps (eps=1e-6) = 13.8
  L3.7: |Q_{i+1}-Q_{i-1}|/|eps| at u1+eps (eps=1e-12) = 13.8

=== Constancy on the complex u-plane ===
  prod        N= 8 tau=1 M=(0.0, 0.0)                   spread over complex u = 1.27e-40   value = 29.67461789067155044
  prod        N= 8 tau=1 M=(0.8, 0.3)                   spread over complex u = 1.9e-40   value = 33.406376850287502714
  prod        N= 8 tau=1 M=(-2.5, 1.7)                  spread over complex u = 7.96e-41   value = 76.398284864493199458
  prod        N= 8 tau=1 M=((0.4 + 1.1j), (-0.6 + 0.2j)) spread over complex u = 2.25e-40   value = (25.942858931055598166 + 3.2716790878824787059j)
  outer_ratio N= 8 tau=1 M=(0.8, 0.3)                   spread over complex u = 8.92e-41   value = 1.0514972257584532484
  outer_ratio N= 8 tau=1 M=(-2.5, 1.7)                  spread over complex u = 1.09e-40   value = 2.2563123894736516115
  prod        N= 4 tau=1 M=(0.0, 0.0)                   spread over complex u = 3.92e-40   value = 5.12
  prod        N= 4 tau=1 M=(0.8, 0.3)                   spread over complex u = 3.18e-40   value = 5.12
  prod        N= 4 tau=1 M=(-2.5, 1.7)                  spread over complex u = 1.08e-39   value = 5.12
  prod        N= 4 tau=1 M=((0.4 + 1.1j), (-0.6 + 0.2j)) spread over complex u = 4.35e-40   value = (5.12 + 2.053039793168989223e-40j)
  outer_ratio N= 4 tau=1 M=(0.8, 0.3)                   spread over complex u = 4.6e-41   value = 0.5
  outer_ratio N= 4 tau=1 M=(-2.5, 1.7)                  spread over complex u = 1.39e-40   value = 0.5
  prod        N=12 tau=5 M=(0.0, 0.0)                   spread over complex u = 2.1e-38   value = 1.8789292789394128884
  prod        N=12 tau=5 M=(0.8, 0.3)                   spread over complex u = 3.47e-37   value = -0.088414994414006242984
  prod        N=12 tau=5 M=(-2.5, 1.7)                  spread over complex u = 2.57e-38   value = -22.753299020307506099
  prod        N=12 tau=5 M=((0.4 + 1.1j), (-0.6 + 0.2j)) spread over complex u = 3.0e-38   value = (3.8462735522928320198 - 1.7247949793783400604j)
  outer_ratio N=12 tau=5 M=(0.8, 0.3)                   spread over complex u = 4.01e-38   value = 0.020675376477425652495
  outer_ratio N=12 tau=5 M=(-2.5, 1.7)                  spread over complex u = 1.11e-37   value = -0.031161213305014177133
  prod        N=16 tau=3 M=(0.0, 0.0)                   spread over complex u = 2.74e-39   value = 318.28151263179955734
  prod        N=16 tau=3 M=(0.8, 0.3)                   spread over complex u = 2.66e-39   value = 330.34195180979269991
  prod        N=16 tau=3 M=(-2.5, 1.7)                  spread over complex u = 2.92e-39   value = 469.28481960009726021
  prod        N=16 tau=3 M=((0.4 + 1.1j), (-0.6 + 0.2j)) spread over complex u = 2.31e-39   value = (306.22107345380641477 + 10.573535717692618144j)
  outer_ratio N=16 tau=3 M=(0.8, 0.3)                   spread over complex u = 3.89e-40   value = 0.79366959841989522753
  outer_ratio N=16 tau=3 M=(-2.5, 1.7)                  spread over complex u = 4.06e-40   value = 1.0502906157954731087
  prod        N=20 tau=7 M=(0.0, 0.0)                   spread over complex u = 7.65e-38   value = 29.7207319519769813
  prod        N=20 tau=7 M=(0.8, 0.3)                   spread over complex u = 7.58e-38   value = 23.028684913933836647
  prod        N=20 tau=7 M=(-2.5, 1.7)                  spread over complex u = 8.51e-38   value = -54.067363839412528463
  prod        N=20 tau=7 M=((0.4 + 1.1j), (-0.6 + 0.2j)) spread over complex u = 8.31e-38   value = (36.412778990020125953 - 5.8670001429419350381j)
  outer_ratio N=20 tau=7 M=(0.8, 0.3)                   spread over complex u = 3.26e-38   value = 0.10757620572928552294
  outer_ratio N=20 tau=7 M=(-2.5, 1.7)                  spread over complex u = 4.77e-37   value = 0.023449102256345593385
  prod        N=24 tau=7 M=(0.0, 0.0)                   spread over complex u = 6.91e-40   value = 104.16033160754583215
  prod        N=24 tau=7 M=(0.8, 0.3)                   spread over complex u = 1.01e-39   value = 93.468315679972939425
  prod        N=24 tau=7 M=(-2.5, 1.7)                  spread over complex u = 6.48e-39   value = -29.709566444531208281
  prod        N=24 tau=7 M=((0.4 + 1.1j), (-0.6 + 0.2j)) spread over complex u = 1.0e-39   value = (114.85234753511872488 - 9.3738221830776045817j)
  outer_ratio N=24 tau=7 M=(0.8, 0.3)                   spread over complex u = 3.11e-40   value = 0.33073756127133153777
  outer_ratio N=24 tau=7 M=(-2.5, 1.7)                  spread over complex u = 1.24e-39   value = 0.1803303150650504025
  -- controls --
  prod        N= 6 tau=1 M=(0.0, 0.0)                   spread over complex u = 0.296   value = 19.324024995785984236
  prod        N= 6 tau=1 M=(0.8, 0.3)                   spread over complex u = 0.296   value = 21.339244745346522592
  prod        N= 6 tau=1 M=(-2.5, 1.7)                  spread over complex u = 0.296   value = 44.555680490283683652
  prod        N= 6 tau=1 M=((0.4 + 1.1j), (-0.6 + 0.2j)) spread over complex u = 0.296   value = (17.30880524622544588 + 1.7667679996147185587j)
  ratio       N= 6 tau=1 M=(0.0, 0.0)                   spread over complex u = 1.34e-40   value = 0.77777777777777777778
  ratio       N= 6 tau=1 M=(0.8, 0.3)                   spread over complex u = 1.95e-40   value = 0.85888888888888888889
  ratio       N= 6 tau=1 M=(-2.5, 1.7)                  spread over complex u = 2.93e-40   value = 1.7933333333333333333
  ratio       N= 6 tau=1 M=((0.4 + 1.1j), (-0.6 + 0.2j)) spread over complex u = 1.45e-40   value = (0.69666666666666666667 + 0.071111111111111111111j)
  outer_prod  N= 6 tau=1 M=(0.8, 0.3)                   spread over complex u = 9.41e-41   value = 43.244444444444444444
  outer_prod  N= 6 tau=1 M=(-2.5, 1.7)                  spread over complex u = 1.84e-40   value = 80.622222222222222222
  prod        N=10 tau=3 M=(0.0, 0.0)                   spread over complex u = 0.945   value = 11.784873717975648167
  prod        N=10 tau=3 M=(0.8, 0.3)                   spread over complex u = 0.945   value = 8.9529916948703603434
  prod        N=10 tau=3 M=(-2.5, 1.7)                  spread over complex u = 0.945   value = -23.671840927479599379
  prod        N=10 tau=3 M=((0.4 + 1.1j), (-0.6 + 0.2j)) spread over complex u = 0.945   value = (14.616755741080935991 - 2.4827458832703893249j)
  ratio       N=10 tau=3 M=(0.0, 0.0)                   spread over complex u = 7.42e-40   value = 0.32833811224594005904
  ratio       N=10 tau=3 M=(0.8, 0.3)                   spread over complex u = 1.79e-39   value = 0.24943910833456652619
  ratio       N=10 tau=3 M=(-2.5, 1.7)                  spread over complex u = 1.83e-39   value = -0.65952064905536691391
  ratio       N=10 tau=3 M=((0.4 + 1.1j), (-0.6 + 0.2j)) spread over complex u = 1.76e-39   value = (0.40723711615731359189 - 0.069171729456546658937j)
  outer_prod  N=10 tau=3 M=(0.8, 0.3)                   spread over complex u = 1.75e-40   value = 109.29448800067858485
  outer_prod  N=10 tau=3 M=(-2.5, 1.7)                  spread over complex u = 9.83e-39   value = -5.0896409948626556886
  prod        N= 5 tau=1 M=(0.0, 0.0)                   spread over complex u = 4.03e-40   value = 12.392644786552070916
  prod        N= 5 tau=1 M=(0.8, 0.3)                   spread over complex u = 0.343   value = 13.308072372967537392
  prod        N= 5 tau=1 M=(-2.5, 1.7)                  spread over complex u = 0.697   value = 24.532120974598964146
  prod        N= 5 tau=1 M=((0.4 + 1.1j), (-0.6 + 0.2j)) spread over complex u = 0.367   value = (11.35561999159585155 + 0.76634593844058844338j)
  ratio       N= 5 tau=1 M=(0.0, 0.0)                   spread over complex u = 0.0343   value = 0.61819954734344373311
  ratio       N= 5 tau=1 M=(0.8, 0.3)                   spread over complex u = 0.147   value = 0.66386509568239472547
  ratio       N= 5 tau=1 M=(-2.5, 1.7)                  spread over complex u = 0.231   value = 1.2237699331404102396
  ratio       N= 5 tau=1 M=((0.4 + 1.1j), (-0.6 + 0.2j)) spread over complex u = 0.231   value = (0.56646819621801311435 + 0.038228701008726985559j)

=== Lemma 3.11 (c1 = 0 for even N) and Lemma 3.12 ===
  N= 8 tau=1: max|A_M - A_{-M}| = 4.26e-40 (L3.11, 0 iff c1=0) ; max|A_M(u+2iK') + A_{RM}(u)| = 9.4e-41 ; max|A(u+2iK')+A(u)| = 9.18e-41 (L3.12)
  N= 6 tau=1: max|A_M - A_{-M}| = 1.42e-40 (L3.11, 0 iff c1=0) ; max|A_M(u+2iK') + A_{RM}(u)| = 4.59e-40 ; max|A(u+2iK')+A(u)| = 1.22e-40 (L3.12)
  N=10 tau=3: max|A_M - A_{-M}| = 4.1e-40 (L3.11, 0 iff c1=0) ; max|A_M(u+2iK') + A_{RM}(u)| = 1.18e-40 ; max|A(u+2iK')+A(u)| = 1.84e-40 (L3.12)
  N= 5 tau=1: max|A_M - A_{-M}| = 1.19 (L3.11, 0 iff c1=0) ; max|A_M(u+2iK') + A_{RM}(u)| = 1.95e-40 ; max|A(u+2iK')+A(u)| = 3.44e-41 (L3.12)
  N=20 tau=7: max|A_M - A_{-M}| = 1.48e-38 (L3.11, 0 iff c1=0) ; max|A_M(u+2iK') + A_{RM}(u)| = 9.47e-41 ; max|A(u+2iK')+A(u)| = 1.84e-40 (L3.12)

=== Theorems B/C: residue ratios at several poles vs real-axis ratio ===
  ThB N=6 tau=1: residue ratios kappa(O) at 4 poles: ['(0.777772394342 + 4.69453836508e-21j)', '(0.777772394342 + 7.0441448418e-21j)', '(0.777772394342 + 1.40836524375e-20j)', '(0.777772394342 + 2.11324843741e-20j)'] ; real-axis A_O/A = 0.777777777778 ; kappa((1,0)) at 4 poles: ['(0.888882736391 + 3.42726516243e-21j)', '(0.888882736391 + 8.05045309842e-21j)', '(0.888882736391 + 1.02818068421e-20j)', '(0.888882736391 + 2.4151393364e-20j)'] ; real-axis = 0.888888888889
  ThB N=10 tau=3: residue ratios kappa(O) at 4 poles: ['(0.32835653643 + 3.21926134753e-21j)', '(0.32835653643 + 1.60961154855e-21j)', '(0.32835653643 + 2.68386276706e-20j)', '(0.32835653643 + 6.43639746141e-21j)'] ; real-axis A_O/A = 0.328338112246 ; kappa((1,0)) at 4 poles: ['(0.220269644366 + 2.15956013173e-21j)', '(0.220269644366 + 2.57742970613e-21j)', '(0.220269644366 + 3.39894780574e-20j)', '(0.220269644366 + 1.32041200272e-21j)'] ; real-axis = 0.22025728497
  ThC N=8 tau=1: residue ratios kappa(O) at 4 poles: ['(0.94691755043 + 1.41741863826e-27j)', '(0.94691755043 - 1.18650203795e-27j)', '(0.94691755043 + 6.07787727106e-27j)', '(0.94691755043 + 3.55242312873e-27j)'] ; real-axis A_O/A = 0.94691755043 ; kappa((1,0)) at 4 poles: ['(1.09017737965 + 1.68566705461e-27j)', '(1.09017737965 - 1.84629452437e-28j)', '(1.09017737965 + 6.18723564678e-27j)', '(1.09017737965 + 9.57925053661e-27j)'] ; real-axis = 1.09017737965
  ThC N=4 tau=1: residue ratios kappa(O) at 4 poles: ['(0.5 + 4.33700680394e-49j)', '(0.5 + 2.05680867659e-48j)', '(0.5 - 1.98785843738e-47j)', '(0.5 - 3.36259137679e-47j)'] ; real-axis A_O/A = 0.5 ; kappa((1,0)) at 4 poles: ['(0.5 - 9.60997063552e-28j)', '(0.5 + 1.02536374656e-41j)', '(0.5 - 9.73360045436e-46j)', '(0.5 - 2.90523033766e-42j)'] ; real-axis = 0.5
  ThC N=12 tau=5: residue ratios kappa(O) at 4 poles: ['(0.0251748664348 + 3.32512957609e-29j)', '(0.0251748664348 - 4.80631087027e-29j)', '(0.0251748664348 - 8.16743794925e-29j)', '(0.0251748664348 - 1.89626313204e-28j)'] ; real-axis A_O/A = 0.0251748664348 ; kappa((1,0)) at 4 poles: ['(0.0190111815617 - 2.23857840319e-30j)', '(0.0190111815617 + 1.27766543242e-29j)', '(0.0190111815617 + 5.85971670649e-29j)', '(0.0190111815617 - 1.30538675093e-28j)'] ; real-axis = 0.0190111815617
  ThC N=16 tau=3: residue ratios kappa(O) at 4 poles: ['(0.771394527946 + 1.59993566257e-27j)', '(0.771394527946 - 1.58811104975e-27j)', '(0.771394527946 - 2.8948336127e-28j)', '(0.771394527946 - 2.20701938453e-28j)'] ; real-axis A_O/A = 0.771394527946 ; kappa((1,0)) at 4 poles: ['(0.801908323116 + 2.34200813883e-27j)', '(0.801908323116 - 2.53400901084e-27j)', '(0.801908323116 - 1.19867685319e-28j)', '(0.801908323116 - 1.05647294541e-28j)'] ; real-axis = 0.801908323116
total time 10.5s
