check_note.py -- AMR-050-0022, -0033, -0047, -0063, -0066
mpmath 1.3.0
====================================================================================================
(i) reflection-law trajectories, 50 digits, 6 members per family
a=1.5 b=1 N= 3 tau=1 lambda=0.943379766653 selfcheck=2.4e-48 | k404:0.068 k605,a:2.3e-50* k806,a:6.7e-50* k904,a:2.9e-50* k907,a:8.0e-50* k907,a:5.0e-50*
a=1.5 b=1 N= 4 tau=1 lambda=0.692307692308 selfcheck=9.2e-50 | k404:0.12 k605,a:0.12 k806,a:2.7e-50* k904,a:0.12 k907,a:0.12 k907,a:0.12
a=1.5 b=1 N= 5 tau=1 lambda=0.491722668738 selfcheck=5.0e-50 | k404:0.074 k605,a:1.8e-50* k806,a:2.3e-50* k904,a:2.5e-50* k907,a:3.6e-50* k907,a:2.7e-50*
a=1.5 b=1 N= 5 tau=2 lambda=0.998682995904 selfcheck=3.0e-45 | k404:0.074 k605,a:4.7e-50* k806,a:1.3e-48* k904,a:5.6e-49* k907,a:3.2e-48* k907,a:3.6e-47*
a=1.5 b=1 N= 6 tau=1 lambda=0.36 selfcheck=6.0e-50 | k404:6.6e-51* k605,a:0.0032 k806,a:2.3e-50* k904,a:0.0032 k907,a:0.0032 k907,a:0.0032
a=1.5 b=1 N= 7 tau=1 lambda=0.272762145821 selfcheck=2.6e-49 | k404:0.01 k605,a:1.2e-50* k806,a:2.1e-50* k904,a:4.1e-50* k907,a:2.2e-50* k907,a:1.2e-50*
a=1.5 b=1 N= 7 tau=3 lambda=0.999971860034 selfcheck=8.7e-42 | k404:0.075 k605,a:2.5e-49* k806,a:1.7e-47* k904,a:4.4e-48* k907,a:7.9e-47* k907,a:1.3e-44*
a=1.5 b=1 N= 8 tau=1 lambda=0.212968439502 selfcheck=2.1e-49 | k404:8.0e-5 k605,a:8.0e-5 k806,a:1.8e-50* k904,a:8.0e-5 k907,a:8.0e-5 k907,a:8.0e-5
a=1.5 b=1 N= 8 tau=3 lambda=0.991120137081 selfcheck=9.2e-47 | k404:0.07 k605,a:0.068 k806,a:3.1e-49* k904,a:0.068 k907,a:0.068 k907,a:0.068
a=1.5 b=1 N= 9 tau=2 lambda=0.581951571809 selfcheck=8.6e-50 | k404:0.00059 k605,a:2.2e-50* k806,a:3.0e-50* k904,a:2.9e-50* k907,a:5.9e-50* k907,a:3.5e-50*
a=1.5 b=1 N= 9 tau=4 lambda=0.999999400936 selfcheck=1.8e-38 | k404:0.075 k605,a:1.4e-46* k806,a:1.7e-45* k904,a:2.8e-45* k907,a:2.9e-45* k907,a:3.9e-40*
a=1.5 b=1 N=10 tau=1 lambda=0.139429404037 selfcheck=6.1e-49 | k404:4.0e-50* k605,a:2.9e-6 k806,a:5.4e-50* k904,a:2.9e-6 k907,a:2.9e-6 k907,a:2.9e-6
a=1.5 b=1 N=10 tau=3 lambda=0.864220038425 selfcheck=5.5e-49 | k404:7.5e-50* k605,a:0.00053 k806,a:2.5e-50* k904,a:0.00053 k907,a:0.00053 k907,a:0.00053
a=1.5 b=1 N=11 tau=4 lambda=0.983367899998 selfcheck=4.5e-47 | k404:0.0082 k605,a:1.7e-49* k806,a:1.7e-49* k904,a:7.2e-50* k907,a:1.5e-49* k907,a:2.9e-49*
a=1.5 b=1 N=12 tau=5 lambda=0.999807237231 selfcheck=2.1e-43 | k404:0.078 k605,a:0.077 k806,a:3.0e-48* k904,a:0.077 k907,a:0.077 k907,a:0.077
a=1.5 b=1 N=14 tau=3 lambda=0.549734907216 selfcheck=8.7e-49 | k404:2.9e-50* k605,a:3.6e-7 k806,a:4.8e-50* k904,a:3.6e-7 k907,a:3.6e-7 k907,a:3.6e-7
a=2.2 b=1 N= 3 tau=1 lambda=0.988351863357 selfcheck=1.0e-46 | k404:0.31 k605,a:1.6e-50* k806,a:1.8e-49* k904,a:2.8e-50* k907,a:1.9e-49* k907,a:1.3e-49*
a=2.2 b=1 N= 4 tau=1 lambda=0.828767123288 selfcheck=1.2e-48 | k404:0.18 k605,a:0.2 k806,a:3.9e-50* k904,a:0.2 k907,a:0.2 k907,a:0.2
a=2.2 b=1 N= 5 tau=1 lambda=0.626737915125 selfcheck=3.0e-49 | k404:0.59 k605,a:6.8e-50* k806,a:7.0e-50* k904,a:1.0e-49* k907,a:8.7e-50* k907,a:7.6e-50*
a=2.2 b=1 N= 5 tau=2 lambda=0.9999603815 selfcheck=1.7e-41 | k404:0.23 k605,a:2.7e-49* k806,a:3.4e-47* k904,a:9.5e-48* k907,a:7.3e-47* k907,a:1.1e-44*
a=2.2 b=1 N= 6 tau=1 lambda=0.47265625 selfcheck=2.7e-49 | k404:1.3e-49* k605,a:0.017 k806,a:5.1e-50* k904,a:0.017 k907,a:0.017 k907,a:0.017
a=2.2 b=1 N= 7 tau=1 lambda=0.363965724314 selfcheck=1.2e-49 | k404:0.12 k605,a:1.9e-50* k806,a:2.7e-50* k904,a:7.1e-50* k907,a:9.3e-50* k907,a:5.1e-50*
a=2.2 b=1 N= 7 tau=3 lambda=0.999999867645 selfcheck=1.3e-36 | k404:0.23 k605,a:2.6e-46* k806,a:6.8e-44* k904,a:1.0e-44* k907,a:4.3e-44* k907,a:3.6e-39*
a=2.2 b=1 N= 8 tau=1 lambda=0.286989200573 selfcheck=3.8e-49 | k404:0.0014 k605,a:0.0014 k806,a:4.8e-50* k904,a:0.0014 k907,a:0.0014 k907,a:0.0014
a=2.2 b=1 N= 8 tau=3 lambda=0.999315432113 selfcheck=3.9e-44 | k404:0.32 k605,a:0.32 k806,a:7.8e-48* k904,a:0.32 k907,a:0.32 k907,a:0.32
a=2.2 b=1 N= 9 tau=2 lambda=0.723411053912 selfcheck=7.7e-49 | k404:0.072 k605,a:5.9e-50* k806,a:1.2e-49* k904,a:1.4e-49* k907,a:2.1e-49* k907,a:1.6e-49*
a=2.2 b=1 N= 9 tau=4 lambda=0.999999999558 selfcheck=1.0e-31 | k404:0.23 k605,a:3.1e-44* k806,a:7.7e-42* k904,a:1.2e-42* k907,a:1.3e-41* k907,a:1.2e-34*
a=2.2 b=1 N=10 tau=1 lambda=0.189952997529 selfcheck=1.9e-49 | k404:5.2e-50* k605,a:5.2e-5 k806,a:1.9e-50* k904,a:5.2e-5 k907,a:5.2e-5 k907,a:5.2e-5
a=2.2 b=1 N=10 tau=3 lambda=0.953538028432 selfcheck=1.1e-47 | k404:7.1e-49* k605,a:0.0094 k806,a:2.6e-49* k904,a:0.0094 k907,a:0.0094 k907,a:0.0094
a=2.2 b=1 N=11 tau=4 lambda=0.998232882349 selfcheck=7.2e-45 | k404:0.1 k605,a:2.0e-48* k806,a:6.0e-48* k904,a:2.4e-48* k907,a:4.0e-48* k907,a:1.1e-47*
a=2.2 b=1 N=12 tau=5 lambda=0.999997709979 selfcheck=4.9e-39 | k404:0.32 k605,a:0.32 k806,a:4.0e-45* k904,a:0.32 k907,a:0.32 k907,a:0.32
a=2.2 b=1 N=14 tau=3 lambda=0.689853906957 selfcheck=8.7e-49 | k404:2.7e-49* k605,a:1.9e-5 k806,a:6.3e-50* k904,a:1.9e-5 k907,a:1.9e-5 k907,a:1.9e-5
a=2.6 b=1.7 N= 3 tau=1 lambda=2.7387983123 selfcheck=2.6e-48 | k404:0.08 k605,a:1.1e-50* k806,a:3.1e-50* k904,a:2.9e-50* k907,a:4.4e-50* k907,a:3.3e-50*
a=2.6 b=1.7 N= 4 tau=1 lambda=2.02449740933 selfcheck=1.4e-49 | k404:0.12 k605,a:0.13 k806,a:1.2e-50* k904,a:0.13 k907,a:0.13 k907,a:0.13
a=2.6 b=1.7 N= 5 tau=1 lambda=1.44158654664 selfcheck=6.5e-50 | k404:0.084 k605,a:1.4e-50* k806,a:1.4e-50* k904,a:2.9e-50* k907,a:2.4e-50* k907,a:1.7e-50*
a=2.6 b=1.7 N= 5 tau=2 lambda=2.88684933706 selfcheck=6.1e-45 | k404:0.08 k605,a:5.7e-50* k806,a:2.9e-48* k904,a:9.9e-49* k907,a:1.7e-48* k907,a:7.2e-47*
a=2.6 b=1.7 N= 6 tau=1 lambda=1.05659275284 selfcheck=4.5e-49 | k404:1.2e-50* k605,a:0.0037 k806,a:4.3e-50* k904,a:0.0037 k907,a:0.0037 k907,a:0.0037
a=2.6 b=1.7 N= 7 tau=1 lambda=0.80102718294 selfcheck=2.8e-49 | k404:0.013 k605,a:8.2e-51* k806,a:1.5e-50* k904,a:4.1e-50* k907,a:1.8e-50* k907,a:2.0e-50*
a=2.6 b=1.7 N= 7 tau=3 lambda=2.8899393355 selfcheck=9.9e-42 | k404:0.081 k605,a:3.0e-48* k806,a:1.2e-46* k904,a:5.8e-47* k907,a:1.4e-46* k907,a:2.2e-43*
a=2.6 b=1.7 N= 8 tau=1 lambda=0.625652770416 selfcheck=5.2e-49 | k404:9.7e-5 k605,a:9.7e-5 k806,a:3.0e-50* k904,a:9.7e-5 k907,a:9.7e-5 k907,a:9.7e-5
a=2.6 b=1.7 N= 8 tau=3 lambda=2.86758497643 selfcheck=7.8e-47 | k404:0.071 k605,a:0.069 k806,a:5.0e-49* k904,a:0.069 k907,a:0.069 k907,a:0.069
a=2.6 b=1.7 N= 9 tau=2 lambda=1.70444256182 selfcheck=2.8e-49 | k404:0.00071 k605,a:4.4e-51* k806,a:1.2e-50* k904,a:4.6e-50* k907,a:4.3e-50* k907,a:2.2e-50*
a=2.6 b=1.7 N= 9 tau=4 lambda=2.88999883546 selfcheck=3.2e-38 | k404:0.081 k605,a:2.5e-46* k806,a:8.4e-45* k904,a:4.8e-45* k907,a:5.8e-45* k907,a:9.8e-40*
a=2.6 b=1.7 N=10 tau=1 lambda=0.409772737626 selfcheck=4.2e-49 | k404:9.8e-51* k605,a:3.4e-6 k806,a:2.0e-50* k904,a:3.4e-6 k907,a:3.4e-6 k907,a:3.4e-6
a=2.6 b=1.7 N=10 tau=3 lambda=2.51724279624 selfcheck=1.3e-48 | k404:2.1e-49* k605,a:0.00073 k806,a:8.8e-50* k904,a:0.00073 k907,a:0.00073 k907,a:0.00073
a=2.6 b=1.7 N=11 tau=4 lambda=2.84723216776 selfcheck=2.7e-47 | k404:0.012 k605,a:1.2e-49* k806,a:4.8e-49* k904,a:1.8e-49* k907,a:2.4e-49* k907,a:6.8e-49*
a=2.6 b=1.7 N=12 tau=5 lambda=2.88956233138 selfcheck=2.1e-43 | k404:0.081 k605,a:0.079 k806,a:5.7e-48* k904,a:0.079 k907,a:0.079 k907,a:0.079
a=2.6 b=1.7 N=14 tau=3 lambda=1.6106878335 selfcheck=6.6e-49 | k404:7.4e-51* k605,a:5.1e-7 k806,a:1.4e-50* k904,a:5.1e-7 k907,a:5.1e-7 k907,a:5.1e-7
(* = row in its class of N)
largest self-check error (closure, tangency, reflection, outer vertices, polar/dual constructions): 1.04e-31
PASS  (i) self-checks below 1e-30
   in class : A*_f1 = A*_f2 (even N) largest relative spread 7.26e-35
PASS  (i) A*_f1 = A*_f2 (even N) constant in its class
   in class : Abar_1 Abar_2 (k604,a) largest relative spread 3.59e-34
PASS  (i) Abar_1 Abar_2 (k604,a) constant in its class
   in class : k404 (f1)              largest relative spread 7.06e-49
PASS  (i) k404 (f1) constant in its class
   in class : k404 (f2)              largest relative spread 5.45e-49
PASS  (i) k404 (f2) constant in its class
   in class : k605,a                 largest relative spread 3.06e-44
PASS  (i) k605,a constant in its class
   in class : k806,a (f1)            largest relative spread 7.71e-42
PASS  (i) k806,a (f1) constant in its class
   in class : k806,a (f2)            largest relative spread 9.75e-42
PASS  (i) k806,a (f2) constant in its class
   in class : k806,a f1 vs f2        largest relative spread 8.42e-42
PASS  (i) k806,a f1 vs f2 constant in its class
   in class : k904,a                 largest relative spread 1.17e-42
PASS  (i) k904,a constant in its class
   in class : k907,a pointwise       largest relative spread 1.24e-34
PASS  (i) k907,a pointwise constant in its class
   in class : k907,a polar (third reading) largest relative spread 4.47e-33
PASS  (i) k907,a polar (third reading) constant in its class
   in class : k907,a reciprocal      largest relative spread 1.27e-41
PASS  (i) k907,a reciprocal constant in its class
   in class : kappa = Abar_1/A       largest relative spread 1.14e-49
PASS  (i) kappa = Abar_1/A constant in its class
   in class : kappa* = A*_f1/A       largest relative spread 2.76e-40
PASS  (i) kappa* = A*_f1/A constant in its class
   controls : Abar_1 Abar_2 (k604,a) smallest relative spread outside the class 3.56e-7
PASS  (i) Abar_1 Abar_2 (k604,a) varies outside its class
   controls : k404 (f1)              smallest relative spread outside the class 7.98e-5
PASS  (i) k404 (f1) varies outside its class
   controls : k404 (f2)              smallest relative spread outside the class 7.98e-5
PASS  (i) k404 (f2) varies outside its class
   controls : k605,a                 smallest relative spread outside the class 3.56e-7
PASS  (i) k605,a varies outside its class
   controls : k904,a                 smallest relative spread outside the class 3.56e-7
PASS  (i) k904,a varies outside its class
   controls : k907,a pointwise       smallest relative spread outside the class 3.56e-7
PASS  (i) k907,a pointwise varies outside its class
   controls : k907,a polar (third reading) smallest relative spread outside the class 3.56e-7
PASS  (i) k907,a polar (third reading) varies outside its class
   controls : k907,a reciprocal      smallest relative spread outside the class 3.56e-7
PASS  (i) k907,a reciprocal varies outside its class
   controls : kappa = Abar_1/A       smallest relative spread outside the class 4.99e-5
PASS  (i) kappa = Abar_1/A varies outside its class
   controls : kappa* = A*_f1/A       smallest relative spread outside the class 0.000639
PASS  (i) kappa* = A*_f1/A varies outside its class
   value k806,a (f1) at a=1.5 b=1 N=4 tau=1: 0.5
   value k806,a (f1) at a=1.5 b=1 N=5 tau=1: 0.581896864694968412865096332088
   value k806,a (f1) at a=1.5 b=1 N=6 tau=1: 0.65
   value k404 (f1) at a=1.5 b=1 N=6 tau=1: 0.813802083333333333333333333333
PASS  (i) N=4, a=1.5, b=1: A'dag_j/Adag_j = 1/2 at all members, both foci
PASS  (i) N=4, a=1.5, b=1: kappa* = A*_f1/A = 2
PASS  (i) N=4, a=2.2, b=1: A'dag_j/Adag_j = 1/2 at all members, both foci
PASS  (i) N=4, a=2.2, b=1: kappa* = A*_f1/A = 2
PASS  (i) N=4, a=2.6, b=1.7: A'dag_j/Adag_j = 1/2 at all members, both foci
PASS  (i) N=4, a=2.6, b=1.7: kappa* = A*_f1/A = 2
time 5.0 s
====================================================================================================
(ii) N = 4, the rhombus member, exact (sympy; symbolic b, c > 0 and a = sqrt(b^2 + c^2))
PASS  (ii) the four sides are tangent to the caustic lambda = a^2 b^2/(a^2 + b^2)
PASS  (ii) law of reflection at the four vertices
    outer polygon: [(-sqrt(b**2 + c**2), b), (-sqrt(b**2 + c**2), -b), (sqrt(b**2 + c**2), -b), (sqrt(b**2 + c**2), b)]
    focus f1: A = 2*b*sqrt(b**2 + c**2), Adag = 2/(b*sqrt(b**2 + c**2)), A'dag = 1/(b*sqrt(b**2 + c**2)), A'dag/Adag = 1/2, A* = 4*b*sqrt(b**2 + c**2), A*/A = 2
PASS  (ii) focus f1: A'dag/Adag = 1/2 exactly
PASS  (ii) focus f1: kappa* = A*/A = 2 exactly
    focus f2: A = 2*b*sqrt(b**2 + c**2), Adag = 2/(b*sqrt(b**2 + c**2)), A'dag = 1/(b*sqrt(b**2 + c**2)), A'dag/Adag = 1/2, A* = 4*b*sqrt(b**2 + c**2), A*/A = 2
PASS  (ii) focus f2: A'dag/Adag = 1/2 exactly
PASS  (ii) focus f2: kappa* = A*/A = 2 exactly
PASS  (ii) the closed form R of [Lemma 4 of doi:10.5281/zenodo.23079798] equals 2 for N = 4
time 5.4 s
====================================================================================================
(iii-a) N = 6, a = 2, b = 1, lambda = 4/9: the member through (2, 0), exact in Q(sqrt3, sqrt5)
PASS  (iii-a) the six vertices lie on x^2/4 + y^2 = 1
PASS  (iii-a) the six chords are tangent to the caustic with lambda = 4/9
PASS  (iii-a) law of reflection at the six vertices (a 6-periodic of winding 1)
    focus f1: A = 20*sqrt(5)/9, focal pedal area = 200*sqrt(5)/81 (ratio 10/9), focal antipedal area = 0, 6 distinct antipedal vertices
PASS  (iii-a) focus f1: antipedal area exactly 0, six distinct antipedal vertices
    focus f2: A = 20*sqrt(5)/9, focal pedal area = 200*sqrt(5)/81 (ratio 10/9), focal antipedal area = 0, 6 distinct antipedal vertices
PASS  (iii-a) focus f2: antipedal area exactly 0, six distinct antipedal vertices
PASS  (iii-a) the closed form R of [Lemma 4 of doi:10.5281/zenodo.23079798] vanishes for a = 2, b = 1, lambda = 4/9
(iii-b) N = 6, a = 2, b = 1 at six members, reflection-law trajectories, 50 digits
    lambda = 0.4444444444444444444444444444444444444444 (4/9 = 0.4444444444444444444444444444444444444444)
PASS  (iii-b) the caustic parameter is 4/9
    theta0=0.137 A=4.99227525349798 Abar1=5.54697250388664 A*_f1=-5.35e-50 A*_f2=-4.81e-50
    theta0=0.9 A=4.97027710893432 Abar1=5.52253012103813 A*_f1=-2.67e-50 A*_f2=-5.08e-50
    theta0=1.7 A=5.0244497089778 Abar1=5.58272189886422 A*_f1=-1.07e-50 A*_f2=-4.28e-50
    theta0=2.45 A=4.97785759161053 Abar1=5.53095287956726 A*_f1=-8.89e-50 A*_f2=1.87e-50
    theta0=3.3 A=4.99826099963226 Abar1=5.55362333292473 A*_f1=0.0 A*_f2=-7.48e-50
    theta0=4.6 A=5.02537748993703 Abar1=5.5837527665967 A*_f1=9.49e-50 A*_f2=-5.61e-50
PASS  (iii-b) A*_fj/A below 1e-45 at all members, focal pedal areas positive
(iii-c) kappa* = A*_f1/A for even N (40 digits, two members) and the closed form R (relative differences)
    N= 4 tau=1: 1.3:2.0 1.6:2.0 1.7:2.0 1.8:2.0 1.9:2.0 2:2.0 2.5:2.0 3:2.0
    N= 6 tau=1: 1.3:1.02861 1.6:0.64381 1.7:0.497045 1.8:0.34087 1.9:0.175208 2:-2.37467e-40 2.5:-1.02083 3:-2.28571   sign change below a/b = 2
    N= 8 tau=1: 1.3:0.839166 1.6:0.421605 1.7:0.263049 1.8:0.0947113 1.9:-0.0834482 2:-0.271464 2.5:-1.36019 3:-2.69796   sign change below a/b = 1.9
    N=10 tau=1: 1.3:0.766354 1.6:0.340171 1.7:0.178486 1.8:0.00692501 1.9:-0.174537 2:-0.365922 2.5:-1.4722 3:-2.82816   sign change below a/b = 1.9
    N=12 tau=1: 1.3:0.730054 1.6:0.300413 1.7:0.137453 1.8:-0.0354234 1.9:-0.218236 2:-0.410998 2.5:-1.52441 3:-2.88768   sign change below a/b = 1.8
    N= 8 tau=3: 1.3:32.3403 1.6:121.989 1.7:178.669 1.8:255.345 1.9:357.172 2:490.239 2.5:1916.64 3:5777.6
    N=10 tau=3: 1.3:3.93625 1.6:5.60877 1.7:6.33274 1.8:7.14951 1.9:8.0649 2:9.08479 2.5:15.9585 3:26.3354
    N=12 tau=5: 1.3:634.03 1.6:7968.04 1.7:15894.3 1.8:30084.2 1.9:54491.8 2:95063.5 2.5:1.01059e+6 3:6.67012e+6
PASS  (iii-c) kappa* agrees at two members (largest relative difference 1.4e-33)
PASS  (iii-c) kappa* equals the closed form R (largest relative difference 5.9e-34)
(iii-d) the families with kappa* = 0 for N = 8 and N = 12, winding 1 (root in a/b, 40 digits)
    N= 8: a/b = 1.853847304381821014624039441, lambda = 0.2527893504891401958462889, R = 1.92e-40
           max |A*_fj|/|A| over 6 members = 4.26e-40; relative spread of A = 0.000216; k404 ratio in [9.64e-41, 3.08e-40]
PASS  (iii-d) N=8: A*_fj vanishes at all members while A varies
           control a/b = root + 0.05: relative spread of A*_f1/Abar_1 = 0.000494
PASS  (iii-d) N=8: the k404 ratio varies at a/b = root + 0.05
    N=12: a/b = 1.77996941624483732972770821, lambda = 0.1134099358639112809097636, R = -1.72e-33
           max |A*_fj|/|A| over 6 members = 1.72e-33; relative spread of A = 1.34e-6; k404 ratio in [-1.09e-33, -1.09e-33]
PASS  (iii-d) N=12: A*_fj vanishes at all members while A varies
           control a/b = root + 0.05: relative spread of A*_f1/Abar_1 = 3.61e-6
PASS  (iii-d) N=12: the k404 ratio varies at a/b = root + 0.05
time 11.3 s
====================================================================================================
(iv) positivity of the focal pedal areas, term by term, on star families (30 digits, 4 members)
    smallest Abar_j/A = 2.3328e-12; smallest term (Q_l - f_j) x (Q_l+1 - f_j) = 2.3484e-19; smallest turning cross product = 0.00013518
PASS  (iv) every focal pedal area and every term of its shoelace sum is positive (16 star families)
====================================================================================================
(v) k907,a: the readings of "dual" (part (i)) and three literal daggered alternatives, odd N
    a=1.5 b=1 N= 3 tau=1: reciprocal 8.0e-50, pointwise 5.0e-50, polar 5.9e-49 | R2a 0.53, R2b 0.043, R3 0.4
    a=1.5 b=1 N= 5 tau=1: reciprocal 3.6e-50, pointwise 2.7e-50, polar 4.7e-50 | R2a 0.00028, R2b 5.9e-7, R3 0.029
    a=1.5 b=1 N= 5 tau=2: reciprocal 3.2e-48, pointwise 3.6e-47, polar 4.8e-46 | R2a 0.61, R2b 0.048, R3 0.08
    a=1.5 b=1 N= 7 tau=1: reciprocal 2.2e-50, pointwise 1.2e-50, polar 2.4e-50 | R2a 0.00034, R2b 3.3e-8, R3 7.0e-5
    a=1.5 b=1 N= 7 tau=3: reciprocal 7.9e-47, pointwise 1.3e-44, polar 4.8e-43 | R2a 0.61, R2b 0.048, R3 0.075
    a=1.5 b=1 N= 9 tau=2: reciprocal 5.9e-50, pointwise 3.5e-50, polar 9.7e-50 | R2a 4.9e-5, R2b 3.8e-9, R3 0.0015
    a=1.5 b=1 N= 9 tau=4: reciprocal 2.9e-45, pointwise 3.9e-40, polar 3.0e-39 | R2a 0.61, R2b 0.048, R3 0.075
    a=1.5 b=1 N=11 tau=4: reciprocal 1.5e-49, pointwise 2.9e-49, polar 6.6e-48 | R2a 0.00022, R2b 6.1e-7, R3 0.00061
    a=2.2 b=1 N= 3 tau=1: reciprocal 1.9e-49, pointwise 1.3e-49, polar 7.5e-48 | R2a 0.14, R2b 0.088, R3 0.37
    a=2.2 b=1 N= 5 tau=1: reciprocal 8.7e-50, pointwise 7.6e-50, polar 1.6e-49 | R2a 0.022, R2b 0.0007, R3 0.073
    a=2.2 b=1 N= 5 tau=2: reciprocal 7.3e-47, pointwise 1.1e-44, polar 1.6e-43 | R2a 0.14, R2b 0.09, R3 0.058
    a=2.2 b=1 N= 7 tau=1: reciprocal 9.3e-50, pointwise 5.1e-50, polar 1.1e-49 | R2a 0.0044, R2b 8.8e-8, R3 0.0003
    a=2.2 b=1 N= 7 tau=3: reciprocal 4.3e-44, pointwise 3.6e-39, polar 9.8e-38 | R2a 0.14, R2b 0.09, R3 0.057
    a=2.2 b=1 N= 9 tau=2: reciprocal 2.1e-49, pointwise 1.6e-49, polar 4.7e-49 | R2a 0.004, R2b 8.8e-8, R3 0.07
    a=2.2 b=1 N= 9 tau=4: reciprocal 1.3e-41, pointwise 1.2e-34, polar 4.5e-33 | R2a 0.14, R2b 0.09, R3 0.057
    a=2.2 b=1 N=11 tau=4: reciprocal 4.0e-48, pointwise 1.1e-47, polar 7.3e-46 | R2a 0.031, R2b 0.00023, R3 0.0042
    a=2.6 b=1.7 N= 3 tau=1: reciprocal 4.4e-50, pointwise 3.3e-50, polar 2.4e-49 | R2a 0.55, R2b 0.049, R3 0.41
    a=2.6 b=1.7 N= 5 tau=1: reciprocal 2.4e-50, pointwise 1.7e-50, polar 2.9e-50 | R2a 0.00047, R2b 1.0e-6, R3 0.032
    a=2.6 b=1.7 N= 5 tau=2: reciprocal 1.7e-48, pointwise 7.2e-47, polar 1.2e-45 | R2a 0.63, R2b 0.047, R3 0.082
    a=2.6 b=1.7 N= 7 tau=1: reciprocal 1.8e-50, pointwise 2.0e-50, polar 2.3e-50 | R2a 0.00037, R2b 5.1e-8, R3 8.0e-5
    a=2.6 b=1.7 N= 7 tau=3: reciprocal 1.4e-46, pointwise 2.2e-43, polar 2.8e-42 | R2a 0.63, R2b 0.048, R3 0.078
    a=2.6 b=1.7 N= 9 tau=2: reciprocal 4.3e-50, pointwise 2.2e-50, polar 7.4e-50 | R2a 5.1e-5, R2b 5.5e-9, R3 0.0018
    a=2.6 b=1.7 N= 9 tau=4: reciprocal 5.8e-45, pointwise 9.8e-40, polar 1.1e-38 | R2a 0.63, R2b 0.048, R3 0.078
    a=2.6 b=1.7 N=11 tau=4: reciprocal 2.4e-49, pointwise 6.8e-49, polar 1.5e-47 | R2a 0.00042, R2b 1.0e-6, R3 0.00085
PASS  (v) the three daggered alternatives vary (smallest spread 3.79e-9)
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(vi) identities of Sections 2-4 at random complex points and the pole map of Section 5 (Jacobi functions, 60 digits)
    a=1.5 N= 3 tau=1: S1[Adag1:-1.0 A'dag1:-1.0 Adual1:-1.0 Adual~1:-1.0 Abar2:-1.0 Abar'2:-1.0 Adag2:1.0 Abar1:1.0 A*1:3.0 A:1.0 A':1.0] PA[A:-1.0 A*1:-2.0 Abar1:-5.5e-11 Adag1:5.5e-11]
    a=1.5 N= 4 tau=1: S1[Adag1:-1.0 A'dag1:-1.0 Adual1:-1.0 Adual~1:-1.0 Abar2:-1.0 Abar'2:-1.0 Adag2:-1.0 Abar1:-1.0 A*1:1.0 A:1.0 A':-1.0] PA[A:-1.0 A*1:-1.0 Abar1:1.0 Adag1:1.0]
    a=2.2 N= 5 tau=2: S1[Adag1:-1.0 A'dag1:-1.0 Adual1:-1.0 Adual~1:-1.0 Abar2:-1.0 Abar'2:-1.0 Adag2:1.0 Abar1:1.0 A*1:1.0 A:1.0 A':1.0] PA[A:-1.0 A*1:-2.0 Abar1:-4.2e-11 Adag1:4.2e-11]
    a=1.5 N= 6 tau=1: S1[Adag1:-1.0 A'dag1:-1.0 Adual1:-1.0 Adual~1:-1.0 Abar2:-1.0 Abar'2:-1.0 Adag2:-1.0 Abar1:-1.0 A*1:-1.0 A:-1.0 A':1.0] PA[A:-1.0 A*1:-1.0 Abar1:-1.0 Adag1:-1.0]
    a=2.2 N= 7 tau=3: S1[Adag1:-1.0 A'dag1:-1.0 Adual1:-1.0 Adual~1:-1.0 Abar2:-1.0 Abar'2:-1.0 Adag2:1.0 Abar1:1.0 A*1:1.0 A:1.0 A':1.0] PA[A:-1.0 A*1:-2.0 Abar1:-4.2e-11 Adag1:4.2e-11]
    a=1.5 N= 8 tau=3: S1[Adag1:-1.0 A'dag1:-1.0 Adual1:-1.0 Adual~1:-1.0 Abar2:-1.0 Abar'2:-1.0 Adag2:-1.0 Abar1:-1.0 A*1:1.0 A:1.0 A':-1.0] PA[A:-1.0 A*1:-1.0 Abar1:1.0 Adag1:1.0]
    a=2.2 N=10 tau=3: S1[Adag1:-1.0 A'dag1:-1.0 Adual1:-1.0 Adual~1:-1.0 Abar2:-1.0 Abar'2:-1.0 Adag2:-1.0 Abar1:-1.0 A*1:-1.0 A:-1.0 A':1.0] PA[A:-1.0 A*1:-1.0 Abar1:-1.0 Adag1:-1.0]
    largest identity residual (Lemmas 2.1, 2.3, 3.2, 4.4, r1, Theorems at complex phases): 4.09e-48
PASS  (vi) identities of Sections 2-4 and the theorems at complex phases
PASS  (vi) measured orders at S1, S2, P_A respect the bounds of the table in Section 5
====================================================================================================
TOTAL failures: 0
time 15.8 s
