check_note.py  (AMR-022-4023)  numpy 2.5.2, scipy-optimize, sympy 1.14.0, mpmath 1.3.0
====================================================================================================
Part (i)  EXACT (sympy): Proposition 1.2, collinear Chebyshev configurations, n = 1..12
  identity |cos(a+ib)|^2 = cos^2 a + sinh^2 b : True
  three-term recurrences of T_k, U_k (k <= 13), hence T_n(cos z) = cos(nz): True
 n   2*2^(-1/n)   zeros  monic  mean0  ends  Pell  T'=nU  U-roots  crit|T|=1  sin-id  |U|<=n  sinh-coef  max|w| on {|T_n|=1} - 1
 1   1.0000000000   True   True   True   True  True  True   True     True       True    True    True       +2.22e-16
 2   1.4142135624   True   True   True   True  True  True   True     True       True    True    True       -1.11e-16
 3   1.5874010520   True   True   True   True  True  True   True     True       True    True    True       +4.44e-16
 4   1.6817928305   True   True   True   True  True  True   True     True       True    True    True       -1.11e-16
 5   1.7411011266   True   True   True   True  True  True   True     True       True    True    True       +2.22e-15
 6   1.7817974363   True   True   True   True  True  True   True     True       True    True    True       +3.11e-15
 7   1.8114473285   True   True   True   True  True  True   True     True       True    True    True       +3.55e-15
 8   1.8340080864   True   True   True   True  True  True   True     True       True    True    True       +4.44e-16
 9   1.8517494246   True   True   True   True  True  True   True     True       True    True    True       +4.44e-15
10   1.8660659831   True   True   True   True  True  True   True     True       True    True    True       +2.22e-15
11   1.8778618213   True   True   True   True  True  True   True     True       True    True    True       -6.88e-15
12   1.8877486254   True   True   True   True  True  True   True     True       True    True    True       -6.55e-15
  consequences (proof of Proposition 1.2): the real lemniscate of q_n is [-2rho, 2rho]; |q_n(+-2rho)| = 1;
  {|T_n| <= 1} lies in the closed unit disc, so max_{x in E} |x - c| = 2 rho = 2*2^(-1/n) exactly.
  Part (i): ALL PASSED   (2.2 s)
====================================================================================================
Part (ii)  NUMERICAL SANITY CHECK (evidence only) of Theorem 1.1 in R^2, R^3, R^4, seed 2026
  connectedness: Thales test (sufficient); R(c) by ray casting; T(u) <= B(u) < 2 for all sampled u
  m  n  shape             lambda*   R(c)      max_u B(u)  max ratio T/B   hull max|q_u|  (m=2: crit |q|, R from boundary)
  2  2  random            0.92808   1.364305  1.364305    1.000000000    0.861328  0.8613  1.364305
  2  3  random            1.05007   1.351853  1.351930    0.999942865    0.475348  0.4758  1.351853
  2  3  near-collinear    1.03354   1.330956  1.331004    0.999963914    0.525344  0.5257  1.330956
  2  3  random            1.00414   1.310973  1.316221    0.996068071    0.459638  0.4929  1.310973
  2  3  random            1.02780   1.324100  1.324107    0.999994547    0.541143  0.5412  1.324100
  2  3  near-collinear    1.01954   1.316869  1.318820    0.998521043    0.528687  0.5436  1.316869
  2  3  random            0.96459   1.277475  1.293233    0.988047843    0.406262  0.4996  1.277475
  2  3  regular polygon   0.91649   1.209599  1.256559    0.962627977    0.384900  0.7698  1.209599
  2  4  random            1.08381   1.314916  1.315509    0.999548837    0.375954  0.3797  1.314916
  2  4  near-collinear    0.99424   1.337635  1.342137    0.997640287    0.570838  0.5787  1.337635
  2  4  random            1.04246   1.270276  1.278699    0.993416669    0.379275  0.4327  1.270276
  2  4  random            0.94803   1.192220  1.231645    0.968542324    0.241722  0.4228  1.192220
  2  4  near-collinear    1.00230   1.237591  1.258356    0.983723579    0.313222  0.4304  1.237591
  2  4  random            0.93371   1.275752  1.293502    0.992016872    0.416827  0.4676  1.275752
  2  4  cross-polytope    0.91973   1.144461  1.228299    0.931744842    0.178888  0.7156  1.144461
  2  4  regular polygon   0.91973   1.144461  1.228299    0.931744842    0.178888  0.7156  1.144461
  2  5  random            0.90814   1.258112  1.271430    0.989525096    0.207873  0.2794  1.258112
  2  5  near-collinear    1.02235   1.252001  1.279166    0.981608240    0.139758  0.2236  1.252001
  2  5  random            1.01005   1.199710  1.246139    0.969635856    0.179075  0.4053  1.199710
  2  5  random            1.05367   1.251263  1.267117    0.987651084    0.236837  0.3018  1.251263
  2  5  near-collinear    1.09164   1.299827  1.302643    0.997856575    0.210893  0.2245  1.299827
  2  5  random            1.06337   1.252498  1.258160    0.995499209    0.305462  0.3330  1.252497
  2  5  regular polygon   0.92608   1.109484  1.220931    0.908719082    0.085143  0.6811  1.109484
  2  6  random            1.06797   1.263171  1.271708    0.994137094    0.127329  0.1582  1.263171
  2  6  near-collinear    1.07972   1.252055  1.258808    0.994658062    0.199719  0.2239  1.252055
  2  6  random            1.02627   1.307182  1.314061    0.994888746    0.127617  0.1694  1.307182
  2  6  random            1.01202   1.184702  1.219811    0.971580255    0.150666  0.2727  1.184702
  2  6  near-collinear    1.07625   1.265460  1.269947    0.996598562    0.150842  0.1686  1.265460
  2  6  random            1.04445   1.212661  1.227581    0.988014568    0.212582  0.2780  1.212661
  2  6  regular polygon   0.93266   1.087940  1.220112    0.891671799    0.041136  0.6582  1.087940
  2  8  random            1.07491   1.210910  1.223703    0.989729007    0.148161  0.1941  1.210910
  2  8  near-collinear    1.10106   1.281509  1.292172    0.993316731    0.054502  0.0786  1.281509
  2  8  random            1.06246   1.198733  1.222894    0.980410686    0.117606  0.1982  1.198733
  2  8  random            1.02253   1.162838  1.195067    0.973601218    0.101031  0.2029  1.162838
  2  8  near-collinear    1.03820   1.178572  1.204329    0.979593421    0.114261  0.2047  1.178572
  2  8  random            1.03457   1.164910  1.209803    0.962891654    0.093866  0.2450  1.164910
  2  8  regular polygon   0.94208   1.062194  1.222348    0.868978683    0.009694  0.6204  1.062194
  3  3  random            0.92051   1.224976  1.259668    0.972438803    0.386514
  3  3  near-collinear    1.03136   1.332430  1.333130    0.999470577    0.496626
  3  3  near-planar       1.02911   1.329385  1.330077    0.999475791    0.504935
  3  3  random            0.91283   1.215928  1.254542    0.969218797    0.379970
  3  3  near-collinear    1.02035   1.319268  1.320805    0.998833751    0.518883
  3  4  random            1.03108   1.268414  1.276732    0.993446881    0.338746
  3  4  near-collinear    1.04563   1.277839  1.286565    0.993213177    0.352678
  3  4  near-planar       0.97387   1.220298  1.255645    0.971788169    0.227635
  3  4  random            0.99863   1.247191  1.265161    0.985768059    0.268694
  3  4  near-collinear    1.06221   1.299070  1.301398    0.998195560    0.350812
  3  4  regular simplex   0.87701   1.109115  1.186322    0.934919025    0.197196
  3  5  random            0.97838   1.206985  1.225651    0.984496173    0.185237
  3  5  near-collinear    1.06695   1.263880  1.266946    0.997553958    0.280647
  3  5  near-planar       0.98564   1.189656  1.232299    0.965290527    0.160503
  3  5  random            0.99882   1.215826  1.243504    0.978056775    0.158132
  3  5  near-collinear    1.08938   1.281209  1.282807    0.998754420    0.290561
  3  6  random            0.96230   1.137298  1.194383    0.970939282    0.120153
  3  6  near-collinear    1.08852   1.260651  1.262588    0.998444900    0.214145
  3  6  near-planar       1.00708   1.207768  1.235139    0.977292070    0.095210
  3  6  random            0.92331   1.157266  1.201383    0.963907825    0.099545
  3  6  near-collinear    1.02656   1.207378  1.231892    0.980284092    0.148741
  3  6  cross-polytope    0.85958   1.025190  1.147238    0.893615654    0.059760
  3  8  random            0.98067   1.114445  1.159748    0.961377330    0.101655
  3  8  near-collinear    1.03703   1.200976  1.222731    0.982107045    0.060582
  3  8  near-planar       1.02837   1.205522  1.257640    0.981947159    0.063846
  3  8  random            0.97151   1.107304  1.155003    0.958635232    0.093571
  3  8  near-collinear    1.10054   1.237877  1.240570    0.997784998    0.164545
  3  8  cube              0.85253   0.984926  1.136210    0.866852526    0.018373
  4  3  random            0.98282   1.298777  1.306533    0.993977436    0.407550
  4  3  near-collinear    0.98046   1.288280  1.296835    0.993382878    0.455372
  4  3  near-planar       0.98702   1.292626  1.298507    0.995470681    0.480335
  4  3  random            0.96764   1.277974  1.290066    0.990587630    0.431486
  4  4  random            0.96306   1.239013  1.267674    0.977482869    0.250069
  4  4  near-collinear    0.97989   1.223614  1.245472    0.982415850    0.282407
  4  4  near-planar       0.98247   1.227846  1.249305    0.982751479    0.274924
  4  4  random            1.00558   1.247308  1.259535    0.990256374    0.315785
  4  5  random            0.94397   1.146009  1.191426    0.961844758    0.176320
  4  5  near-collinear    1.06558   1.266649  1.268821    0.998270792    0.270653
  4  5  near-planar       1.01935   1.226113  1.241934    0.987147995    0.212570
  4  5  random            0.87515   1.093627  1.175705    0.929886584    0.091539
  4  5  regular simplex   0.83878   1.040993  1.140064    0.913100634    0.103797
  4  6  random            0.97132   1.144348  1.186761    0.964235329    0.142767
  4  6  near-collinear    1.05773   1.235015  1.242248    0.994139735    0.193395
  4  6  near-planar       1.00540   1.182046  1.206676    0.979533521    0.165774
  4  6  random            0.93963   1.112650  1.163675    0.956119231    0.134687
  4  8  random            0.96183   1.105903  1.153609    0.958711239    0.082881
  4  8  near-collinear    1.00238   1.257023  1.283157    0.983947609    0.035632
  4  8  near-planar       1.02065   1.173021  1.206499    0.972096503    0.076137
  4  8  random            0.90767   1.055348  1.139833    0.926082262    0.042613
  4  8  cross-polytope    0.81671   0.958174  1.104097    0.867834469    0.020876
  certified-connected configurations: 87 of 87 shapes; largest R(c) = 1.364305 (< 2)
  control 1: two far clusters in R^3 -> Thales test fails (must fail)
  control 2: {P <= 3} for 3^(1/10) x Chebyshev (n = 10) in R^3 (connected; = 3^(1/10) x the level-1 set) -> R(c) = 2.0828 (must exceed 2; predicted 3^(1/10) 2^(9/10) = 2.0828)
  Part (ii): no violation   (33.9 s)
====================================================================================================
Part (iii)  Lemma 2.2 on explicit polynomials
  z^2 - 1 (mean of zeros 0): connected True, max_L |z| = 2^(1/2) = 1.414214 (exact)
  z^3 - 1 (mean of zeros 0): connected True, max_L |z| = 2^(1/3) = 1.259921 (exact)
  z^4 - 1 (mean of zeros 0): connected True, max_L |z| = 2^(1/4) = 1.189207 (exact)
  z^5 - 1 (mean of zeros 0): connected True, max_L |z| = 2^(1/5) = 1.148698 (exact)
  z^6 - 1 (mean of zeros 0): connected True, max_L |z| = 2^(1/6) = 1.122462 (exact)
  z^7 - 1 (mean of zeros 0): connected True, max_L |z| = 2^(1/7) = 1.104090 (exact)
  z^8 - 1 (mean of zeros 0): connected True, max_L |z| = 2^(1/8) = 1.090508 (exact)
  name                                              max crit |q| (50 digits)   max_{|q|=1} |z - c_q|
  z^3 - z                                           0.384900179460             1.324718
  z^2 (z - 3/2)                                     0.500000000000             1.306444
  zeros 0.9, -0.9, 0.9i                             0.561184461652             1.256221
  0.99 x cube roots of unity                        0.970299000000             1.253653
  zeros 0, 1, 1+i, i scaled 0.8                     0.102400000000             1.024672
  0.99 x Chebyshev zeros, n = 9                     0.913517247484             1.835342
  zeros 1.2, 0.2+0.7i, -1.0+0.3i, -0.4-0.8i, 0.5-0.5i  0.930107433624             1.284067
  Part (iii): ALL PASSED   (34.4 s)
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Part (iv)  EXACT side statements
  n = 2: q(sqrt(1+d^2/4)) = 1: True; value sqrt(2) at d = 2: True
  Remark 3.3: P^2 = prod_k((|x|^2+a^2)^2 - 4a^2 x_k^2): True; vertex value identity: True; value 32a^6/27 at r = a/sqrt(3): True
  Remark 3.3, numerical (a^6 = 0.9): P(0) = 0.9; min of P over 200000 points of |x| = a/sqrt(3): 1.066667 (32*0.9/27 = 1.066667)
  Part (iv): ALL PASSED   (34.6 s)
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TOTAL: ALL CHECKS PASSED  (34.6 s)
