n= 1 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=2.000000  R_on_axis=2*2^(-1/n)=1.000000
n= 2 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=2.828427  R_on_axis=2*2^(-1/n)=1.414214
n= 3 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.174802  R_on_axis=2*2^(-1/n)=1.587401
n= 4 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.363586  R_on_axis=2*2^(-1/n)=1.681793
n= 5 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.482202  R_on_axis=2*2^(-1/n)=1.741101
n= 6 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.563595  R_on_axis=2*2^(-1/n)=1.781797
n= 7 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.622895  R_on_axis=2*2^(-1/n)=1.811447
n= 8 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.668016  R_on_axis=2*2^(-1/n)=1.834008
n= 9 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.703499  R_on_axis=2*2^(-1/n)=1.851749
n=10 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.732132  R_on_axis=2*2^(-1/n)=1.866066
n=11 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.755724  R_on_axis=2*2^(-1/n)=1.877862
n=12 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.775497  R_on_axis=2*2^(-1/n)=1.887749
n=13 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.792310  R_on_axis=2*2^(-1/n)=1.896155
n=14 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.806781  R_on_axis=2*2^(-1/n)=1.903390
n=15 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.819366  R_on_axis=2*2^(-1/n)=1.909683
n=16 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.830413  R_on_axis=2*2^(-1/n)=1.915207
n=17 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.840187  R_on_axis=2*2^(-1/n)=1.920093
n=18 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.848895  R_on_axis=2*2^(-1/n)=1.924448
n=19 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.856704  R_on_axis=2*2^(-1/n)=1.928352
n=20 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.863745  R_on_axis=2*2^(-1/n)=1.931873
n=21 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.870127  R_on_axis=2*2^(-1/n)=1.935064
n=22 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.875938  R_on_axis=2*2^(-1/n)=1.937969
n=23 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.881251  R_on_axis=2*2^(-1/n)=1.940626
n=24 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.886128  R_on_axis=2*2^(-1/n)=1.943064
n=25 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.890620  R_on_axis=2*2^(-1/n)=1.945310
n=26 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.894771  R_on_axis=2*2^(-1/n)=1.947385
n=27 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.898618  R_on_axis=2*2^(-1/n)=1.949309
n=28 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.902195  R_on_axis=2*2^(-1/n)=1.951097
n=29 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.905527  R_on_axis=2*2^(-1/n)=1.952764
n=30 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.908640  R_on_axis=2*2^(-1/n)=1.954320
n=31 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.911554  R_on_axis=2*2^(-1/n)=1.955777
n=32 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.914288  R_on_axis=2*2^(-1/n)=1.957144
n=33 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.916858  R_on_axis=2*2^(-1/n)=1.958429
n=34 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.919279  R_on_axis=2*2^(-1/n)=1.959639
n=35 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.921562  R_on_axis=2*2^(-1/n)=1.960781
n=36 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.923720  R_on_axis=2*2^(-1/n)=1.961860
n=37 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.925763  R_on_axis=2*2^(-1/n)=1.962881
n=38 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.927699  R_on_axis=2*2^(-1/n)=1.963849
n=39 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.929536  R_on_axis=2*2^(-1/n)=1.964768
n=40 ok=True pell=True deriv=True T_squarefree=True T_roots_in_open=True U_squarefree=True U_roots_in_open=True parity=True ends=True monic=True mean_zero=True  axis_length=4*2^(-1/n)=3.931282  R_on_axis=2*2^(-1/n)=1.965641
PROPOSITION 2 EXACT CERTIFICATES n=1..40: ALL PASSED
n=2 section 4(b): midpoint product = a^2 = d^2/4: True; axis extreme sqrt(1+d^2/4) on boundary: True; transversal F(0,sigma) = (sigma^2+a^2)^2: True; sqrt(1+1)=sqrt(2) at d=2
Lemma B equality case: Joukowski boundary identity True; 1/Koebe expansion True
WRONG INPUT perturbed T (+s^2/1000): failed checks = ['pell', 'deriv', 'parity', 'ends']
WRONG INPUT rho=2^(-1/(n+1)): failed checks = ['monic']
CONTROL unchanged T: failed checks = []
WRONG INPUT root at 3/2: failed checks = ['pell', 'deriv', 'T_roots_in_open', 'parity', 'ends', 'monic', 'mean_zero']
