r2_remarks.py (verification run 2, AMR-022-4023)  seed=2031  numpy 2.5.2 mpmath 1.3.0
(A) Section 1.1: two points at distance d > 2
  d = 2.50: min P on the bisecting plane (sample) 1.5631 >= d^2/4 = 1.5625; diam E = 3.2015621187, (d^2+4)^(1/2) = 3.2015621187; diam > 4: False (d > 2 sqrt 3 = 3.4641: False)
  d = 3.60: min P on the bisecting plane (sample) 3.2407 >= d^2/4 = 3.2400; diam E = 4.1182520564, (d^2+4)^(1/2) = 4.1182520564; diam > 4: True (d > 2 sqrt 3 = 3.4641: True)
  d = 4.00: min P on the bisecting plane (sample) 4.0007 >= d^2/4 = 4.0000; diam E = 4.4721359550, (d^2+4)^(1/2) = 4.4721359550; diam > 4: True (d > 2 sqrt 3 = 3.4641: True)
  (A) ALL PASSED  (1.4 s)
(B) Remark 3.4
  (B1) Jung's radius 4*2^(-1/n) sqrt(m/(2m+2)): first n with radius > 2
      m=2: n>=5, m=3: n>=4, m=4: n>=3, m=5: n>=3, m=6: n>=3, m=7: n>=3, m=8: n>=3, m=9: n>=3, m=10: n>=3; for every m >= 2 the limit as n -> oo is 4 sqrt(m/(2m+2)) > 2
      m = 2: radius at n = 4: 1.941967, at n = 5: 2.010450
  (B2) min over 2000 random monic polynomials of max|p| / (2((b-a)/4)^n) = 1.016348 (>= 1); scaled T_6 on [-1, 3]: max = 2.000000 = M = 2.000000, leading coefficient 1.000000000000
  (B3) widths of connected voxel sets in R^3 (h = 0.03)
      n = 3: scale 1.3138, voxels 43236, max width 2.8563 <= 4*2^(-1/n) + 2h = 3.2348: True
      n = 4: scale 1.1112, voxels 38389, max width 2.8336 <= 4*2^(-1/n) + 2h = 3.4236: True
      n = 5: scale 1.4723, voxels 56154, max width 2.3097 <= 4*2^(-1/n) + 2h = 3.5422: True
      n = 6: scale 0.8866, voxels 70604, max width 2.7715 <= 4*2^(-1/n) + 2h = 3.6236: True
  (B) ALL PASSED  (9.1 s)
(C) Remark 3.2: other norms
  l^1: norming functional: max(|l(y)| - N(y), |l(v) - N(v)|) over samples = 8.9e-16
      m = 2 n = 3: scale 0.6705, max N(x - c) over the connected voxel set = 1.5126 (< 2: True)
      m = 2 n = 4: scale 1.1787, max N(x - c) over the connected voxel set = 1.3509 (< 2: True)
      m = 2 n = 5: scale 0.6071, max N(x - c) over the connected voxel set = 1.4295 (< 2: True)
      m = 3 n = 3: scale 0.9186, max N(x - c) over the connected voxel set = 1.3478 (< 2: True)
      m = 3 n = 4: scale 0.8159, max N(x - c) over the connected voxel set = 1.4426 (< 2: True)
      m = 3 n = 5: scale 0.5059, max N(x - c) over the connected voxel set = 1.3723 (< 2: True)
  l^3: norming functional: max(|l(y)| - N(y), |l(v) - N(v)|) over samples = 1.3e-15
      m = 2 n = 3: scale 0.7959, max N(x - c) over the connected voxel set = 1.5321 (< 2: True)
      m = 2 n = 4: scale 2.9998, max N(x - c) over the connected voxel set = 1.2482 (< 2: True)
      m = 2 n = 5: scale 0.4244, max N(x - c) over the connected voxel set = 1.4327 (< 2: True)
      m = 3 n = 3: scale 0.9798, max N(x - c) over the connected voxel set = 1.3696 (< 2: True)
      m = 3 n = 4: scale 1.1634, max N(x - c) over the connected voxel set = 1.4661 (< 2: True)
      m = 3 n = 5: scale 1.6075, max N(x - c) over the connected voxel set = 1.3453 (< 2: True)
  l^inf: norming functional: max(|l(y)| - N(y), |l(v) - N(v)|) over samples = 0.0e+00
      m = 2 n = 3: scale 1.0321, max N(x - c) over the connected voxel set = 1.5080 (< 2: True)
      m = 2 n = 4: scale 2.9998, max N(x - c) over the connected voxel set = 1.3006 (< 2: True)
      m = 2 n = 5: scale 1.1176, max N(x - c) over the connected voxel set = 1.6235 (< 2: True)
      m = 3 n = 3: scale 0.9289, max N(x - c) over the connected voxel set = 1.4338 (< 2: True)
      m = 3 n = 4: scale 1.5670, max N(x - c) over the connected voxel set = 1.5305 (< 2: True)
      m = 3 n = 5: scale 1.0271, max N(x - c) over the connected voxel set = 1.4329 (< 2: True)
  (C) ALL PASSED  (66.1 s)
(D) Remark 2.3: Chebyshev polynomials
  n = 2: grid components of {|q_n| < 1 - 1e-3}: 2 (expected n = 2), of {|q_n| < 1 + 1e-3}: 1 (expected 1); critical values ['-1.0']
  n = 3: grid components of {|q_n| < 1 - 1e-3}: 3 (expected n = 3), of {|q_n| < 1 + 1e-3}: 1 (expected 1); critical values ['-1.0', '1.0']
  n = 4: grid components of {|q_n| < 1 - 1e-3}: 4 (expected n = 4), of {|q_n| < 1 + 1e-3}: 1 (expected 1); critical values ['-1.0', '1.0', '-1.0']
  n = 5: grid components of {|q_n| < 1 - 1e-3}: 5 (expected n = 5), of {|q_n| < 1 + 1e-3}: 1 (expected 1); critical values ['-1.0', '1.0', '-1.0', '1.0']
  n = 6: grid components of {|q_n| < 1 - 1e-3}: 6 (expected n = 6), of {|q_n| < 1 + 1e-3}: 1 (expected 1); critical values ['-1.0', '1.0', '-1.0', '1.0', '-1.0']
  n = 7: grid components of {|q_n| < 1 - 1e-3}: 7 (expected n = 7), of {|q_n| < 1 + 1e-3}: 1 (expected 1); critical values ['-1.0', '1.0', '-1.0', '1.0', '-1.0', '1.0']
  scaling identity: monic True, sets agree on 4000 points True
  (D) ALL PASSED  (66.3 s)
(E) Theorem 1.1 / Remark 3.1 in R^3 (voxel component containing all x_i, spacing 0.03)
  n = 3 random         scale 1.1517  components 1  max|x - c| over the component of the x_i 1.3870  ratio to 2*2^(-1/n) 0.8738  projection violations 0
  n = 3 near-collinear scale 1.2648  components 1  max|x - c| over the component of the x_i 1.4842  ratio to 2*2^(-1/n) 0.9350  projection violations 0
  n = 3 near-planar    scale 1.1313  components 1  max|x - c| over the component of the x_i 1.3674  ratio to 2*2^(-1/n) 0.8614  projection violations 0
  n = 3 random         scale 1.2407  components 1  max|x - c| over the component of the x_i 1.4654  ratio to 2*2^(-1/n) 0.9232  projection violations 0
  n = 3 near-collinear scale 1.2537  components 1  max|x - c| over the component of the x_i 1.4753  ratio to 2*2^(-1/n) 0.9294  projection violations 0
  n = 3 near-planar    scale 1.1888  components 1  max|x - c| over the component of the x_i 1.4190  ratio to 2*2^(-1/n) 0.8939  projection violations 0
  n = 4 random         scale 1.2370  components 1  max|x - c| over the component of the x_i 1.3824  ratio to 2*2^(-1/n) 0.8220  projection violations 0
  n = 4 near-collinear scale 1.3780  components 1  max|x - c| over the component of the x_i 1.5110  ratio to 2*2^(-1/n) 0.8984  projection violations 0
  n = 4 near-planar    scale 1.2036  components 1  max|x - c| over the component of the x_i 1.3490  ratio to 2*2^(-1/n) 0.8021  projection violations 0
  n = 4 random         scale 1.1275  components 1  max|x - c| over the component of the x_i 1.2877  ratio to 2*2^(-1/n) 0.7657  projection violations 0
  n = 4 near-collinear scale 1.3094  components 1  max|x - c| over the component of the x_i 1.4457  ratio to 2*2^(-1/n) 0.8596  projection violations 0
  n = 4 near-planar    scale 1.3298  components 1  max|x - c| over the component of the x_i 1.4684  ratio to 2*2^(-1/n) 0.8731  projection violations 0
  n = 5 random         scale 1.2333  components 1  max|x - c| over the component of the x_i 1.3390  ratio to 2*2^(-1/n) 0.7691  projection violations 0
  n = 5 near-collinear scale 1.5339  components 1  max|x - c| over the component of the x_i 1.6343  ratio to 2*2^(-1/n) 0.9386  projection violations 0
  n = 5 near-planar    scale 1.2166  components 1  max|x - c| over the component of the x_i 1.3340  ratio to 2*2^(-1/n) 0.7662  projection violations 0
  n = 5 random         scale 1.2370  components 1  max|x - c| over the component of the x_i 1.3566  ratio to 2*2^(-1/n) 0.7792  projection violations 0
  n = 5 near-collinear scale 1.4745  components 1  max|x - c| over the component of the x_i 1.5789  ratio to 2*2^(-1/n) 0.9068  projection violations 0
  n = 5 near-planar    scale 1.3465  components 1  max|x - c| over the component of the x_i 1.4465  ratio to 2*2^(-1/n) 0.8308  projection violations 0
  n = 6 random         scale 1.2853  components 1  max|x - c| over the component of the x_i 1.3631  ratio to 2*2^(-1/n) 0.7650  projection violations 0
  n = 6 near-collinear scale 1.3539  components 1  max|x - c| over the component of the x_i 1.4926  ratio to 2*2^(-1/n) 0.8377  projection violations 0
  n = 6 near-planar    scale 1.3762  components 1  max|x - c| over the component of the x_i 1.4570  ratio to 2*2^(-1/n) 0.8177  projection violations 0
  n = 6 random         scale 1.1943  components 1  max|x - c| over the component of the x_i 1.2714  ratio to 2*2^(-1/n) 0.7136  projection violations 0
  n = 6 near-collinear scale 1.4355  components 1  max|x - c| over the component of the x_i 1.5071  ratio to 2*2^(-1/n) 0.8459  projection violations 0
  n = 6 near-planar    scale 1.4059  components 1  max|x - c| over the component of the x_i 1.4760  ratio to 2*2^(-1/n) 0.8284  projection violations 0
  n = 7 random         scale 1.2834  components 1  max|x - c| over the component of the x_i 1.3384  ratio to 2*2^(-1/n) 0.7389  projection violations 0
  n = 7 near-collinear scale 1.4764  components 1  max|x - c| over the component of the x_i 1.5052  ratio to 2*2^(-1/n) 0.8309  projection violations 0
  n = 7 near-planar    scale 1.2092  components 1  max|x - c| over the component of the x_i 1.2689  ratio to 2*2^(-1/n) 0.7005  projection violations 0
  n = 7 random         scale 1.2834  components 1  max|x - c| over the component of the x_i 1.3521  ratio to 2*2^(-1/n) 0.7464  projection violations 0
  n = 7 near-collinear scale 1.3836  components 1  max|x - c| over the component of the x_i 1.4278  ratio to 2*2^(-1/n) 0.7882  projection violations 0
  n = 7 near-planar    scale 1.2481  components 1  max|x - c| over the component of the x_i 1.3032  ratio to 2*2^(-1/n) 0.7194  projection violations 0
  largest ratio 0.9386 (voxel resolution 0.03)
  (E) no violation  (105.6 s)
TOTAL: ALL CHECKS PASSED  (105.6 s)
