r2_component_example.py (verification run 2, AMR-022-4023)  seed=2028  numpy 2.5.2 sympy 1.14.0
(A) EXACT
  (A1) P^2 = prod_k((|x|^2+a^2)^2 - 4a^2 xi_k^2) at 60 random rational points: True
  (A2) identity prod(A - B y_k) - (A - B r^2)A^2 = B^2 (A e2 - B e3) on sum y = r^2: True
       r^2 e2 - 3 e3 = sum y_i y_j (r^2 - y_k) >= 0: True;  A - B r^2/3 = (r^2-a^2)^2 + (8/3)a^2 r^2 > 0: True
       (A - B r^2) A^2 = (a^2 - r^2)^2 (a^2 + r^2)^4: True   => P >= |a^2 - r^2| (a^2 + r^2)^2 on |x| = r
  (A3) value at r = a/sqrt(3) equals 32 a^6/27: True; window 27/32 < a^6 <= 1 non-empty: True
  (A) ALL PASSED  (0.0 s)
(B) NUMERICAL: minimum of P on the sphere |x| = a/sqrt(3)
  a^6    P(0)      min P (lattice)  min P (refined)  32a^6/27       argmin (refined)
  0.85   0.850000  1.0074085775     1.0074074074     1.0074074074   (0.56192, -0.00000, -0.00000)
  0.90   0.900000  1.0666679055     1.0666666667     1.0666666667   (0.56730, -0.00000, -0.00000)
  0.95   0.950000  1.1259272336     1.1259259259     1.1259259259   (0.57244, -0.00000, -0.00000)
  1.00   1.000000  1.1851865617     1.1851851852     1.1851851852   (0.57735, 0.00000, -0.00000)
  (B) ALL PASSED  (0.2 s)
(C) NUMERICAL: components by voxel labelling (spacing 0.0125 on [-1.35, 1.35]^3)
  E = {P <= 1}, a^6 = 0.9    face   components=7  component of 0: max|x| = 0.3511 < a/sqrt(3) = 0.5673; labels of the x_i [7, 6, 5, 1, 3, 4]; 0's label 2 -> OK
  E = {P <= 1}, a^6 = 0.9    vertex components=7  component of 0: max|x| = 0.3511 < a/sqrt(3) = 0.5673; labels of the x_i [7, 6, 5, 1, 3, 4]; 0's label 2 -> OK
  {P < 1.1}, a = 1           face   components=7  component of 0: max|x| = 0.3377 < a/sqrt(3) = 0.5774; labels of the x_i [7, 6, 5, 1, 3, 4]; 0's label 2 -> OK
  {P < 1.1}, a = 1           vertex components=7  component of 0: max|x| = 0.3377 < a/sqrt(3) = 0.5774; labels of the x_i [7, 6, 5, 1, 3, 4]; 0's label 2 -> OK
  (C) ALL PASSED  (2.6 s)
TOTAL: ALL CHECKS PASSED  (2.6 s)
