C1: frame-sum identity in the original coordinates
  a=1.0 trial 0: frame sum (original coords) = 1.557479870877395e-01, fibre/Bloch side = 1.557479870877392e-01, rel diff 1.60e-15; ratio to ||f||^2 = 1.188661
  a=1.0 trial 1: frame sum (original coords) = 1.798309815903034e-01, fibre/Bloch side = 1.798309815903037e-01, rel diff 1.39e-15; ratio to ||f||^2 = 1.241167
  a=0.5 trial 0: frame sum (original coords) = 5.116620593577816e-01, fibre/Bloch side = 5.116620593577834e-01, rel diff 3.47e-15; ratio to ||f||^2 = 0.887329
  a=0.5 trial 1: frame sum (original coords) = 4.477767424721601e-01, fibre/Bloch side = 4.477767424721600e-01, rel diff 2.48e-16; ratio to ||f||^2 = 1.114932
C2: Gram symbol of Remark 2.6 against P*P
  a=1.0: max |P*P - Gram symbol| = 6.68e-16
  a=0.5: max |P*P - Gram symbol| = 1.33e-15
  a=1.0: max |P*P - Gram symbol| = 1.89e-15
C3: numerical minimum of sigma_min(P)
  a=1.0: sigma_min at t=(1/14,1/14), theta=(1/2,1/2): 0.2370731445
  a=1.0: smallest local minimum from 14 starts: 0.2370731445 at t=(0.071429,0.071429), theta=(0.500000,0.500000); A_opt = (10a/sqrt21) min^2 = 0.1226464757
  a=0.5: smallest local minimum from 14 starts: 0.1939896348 at t=(-0.071429,-0.071429), theta=(0.500000,0.500000); A_opt = (10a/sqrt21) min^2 = 0.0410598547
C3b: upper frame bound (10/sqrt21) max sigma_max(P)^2 for a = 1 (Remark 1.4: about 2.33)
  largest local maximum of sigma_max from 10 starts: 1.03415089; B = (10/sqrt21) max^2 = 2.333771
C4: ranges for the rounding budget (Section 5.3)
  cert_s023.npz: max over D of |2 pi w_p.M_a y| = 19.1, of 2 pi |nu_j| = 10.2
  cert_s023.npz: max over D of pi y^T M_a y = 509.1; max over D of |2 pi nu.theta| = 14.1; max |nu_j| = 1.627
  cert_halfwidth_s018.npz: max over D of |2 pi w_p.M_a y| = 9.6, of 2 pi |nu_j| = 9.7
  cert_halfwidth_s018.npz: max over D of pi y^T M_a y = 254.5; max over D of |2 pi nu.theta| = 14.1; max |nu_j| = 1.537
C5: Lambda_1 control
  25 x 24 matrix at t=(5/6,1/3), theta=(1/2,1/2): sigma_min = 6.68e-16, next = 0.2855
nice -n 10 python -I r2_checks.py    132.48s user 0.58s system 99% cpu 2:13.17 total
