N=6: sign changes of kappa(k) on (0,1): 1; roots k = ['0.9185586535436917868239815']; b/a = ['0.5']
[PASS] N = 6: unique zero of kappa at b/a = 1/2 (route 1) -- |b/a - 1/2| = 0.0
N=8: sign changes of kappa(k) on (0,1): 1; roots k = ['0.8748259114969211889848562']; b/a = ['0.539418752362378268812242616665']
[PASS] N = 8: unique zero, b/a agrees with the 20 printed digits 0.53941875236237826881 -- route 1 gives 0.539418752362378268812242616665
N=10: sign changes of kappa(k) on (0,1): 1; roots k = ['0.8539121735093071690863726']; b/a = ['0.554348601730070456191078297154']
[PASS] N = 10: unique zero, b/a agrees with the 20 printed digits 0.55434860173007045619 -- route 1 gives 0.554348601730070456191078297154
N=12: sign changes of kappa(k) on (0,1): 1; roots k = ['0.8424839460133789468339111']; b/a = ['0.561807405719182635080997446885']
[PASS] N = 12: unique zero, b/a agrees with the 20 printed digits 0.56180740571918263508 -- route 1 gives 0.561807405719182635080997446885
N=16: sign changes of kappa(k) on (0,1): 1; roots k = ['0.8311072771101641157161578']; b/a = ['0.568831376037339918578421216694']
[PASS] N = 16: unique zero, b/a agrees with the 20 printed digits 0.56883137603733991857 -- route 1 gives 0.568831376037339918578421216694
[PASS] N = 8: route 2 (reflection law, a = 1) agrees with route 1 to 20 digits -- route 2 gives 0.539418752362378268812243
[PASS] N = 10: route 2 (reflection law, a = 1) agrees with route 1 to 20 digits -- route 2 gives 0.554348601730070456191078
[PASS] N = 12: route 2 (reflection law, a = 1) agrees with route 1 to 20 digits -- route 2 gives 0.561807405719182635080997
[PASS] N = 16: route 2 (reflection law, a = 1) agrees with route 1 to 20 digits -- route 2 gives 0.568831376037339918578421

9 of 9 checks pass; ALL PASS: True  (elapsed 14.3 s)
