Sieving primes up to 1000000000...
  50847533 primes, 69s
  power sums built, 681s
logG[1..8] (real, s=1):
  logG[1] = [-0.8157184520 +/- 5.50e-11]
  logG[2] = [0.1011237100 +/- 2.67e-11]
  logG[3] = [0.01658754643 +/- 3.15e-12]
  logG[4] = [0.003623284941 +/- 6.18e-14]
  logG[5] = [0.0009010034968 +/- 1.52e-14]
  logG[6] = [0.0002408478084 +/- 3.95e-14]
  logG[7] = [6.733326516e-5 +/- 1.95e-15]
  logG[8] = [1.939442081e-5 +/- 4.73e-15]
  Cauchy extraction done, 0s

Certified |c_i| * 2^i (i=1..20), P=1000000000, prec=400 bits, N=2048 samples:
  i= 1: |c_i|=[0.8157184521 +/- 1.87e-11]   |c_i|*2^i=[1.6314369 +/- 4.24e-9]   rad=8.778e-11
  i= 2: |c_i|=[0.4338220066 +/- 2.33e-11]   |c_i|*2^i=[1.7352880 +/- 2.64e-8]   rad=8.745e-11
  i= 3: |c_i|=[0.1563636430 +/- 2.78e-11]   |c_i|*2^i=[1.2509091 +/- 4.38e-8]   rad=8.682e-11
  i= 4: |c_i|=[0.04729723182 +/- 2.80e-12]   |c_i|*2^i=[0.75675571 +/- 8.36e-10]   rad=8.683e-11
  i= 5: |c_i|=[0.01118690889 +/- 3.15e-12]   |c_i|*2^i=[0.35798108 +/- 4.59e-9]   rad=8.671e-11
  i= 6: |c_i|=[0.002494624654 +/- 4.59e-13]   |c_i|*2^i=[0.15965598 +/- 2.18e-9]   rad=8.675e-11
  i= 7: |c_i|=[0.0004232965453 +/- 2.07e-14]   |c_i|*2^i=[0.054181958 +/- 2.05e-10]   rad=8.670e-11
  i= 8: |c_i|=[8.397351014e-5 +/- 2.40e-16]   |c_i|*2^i=[0.021497219 +/- 4.05e-10]   rad=8.672e-11
  i= 9: |c_i|=[8.414949067e-6 +/- 2.55e-16]   |c_i|*2^i=[0.0043084539 +/- 2.25e-11]   rad=8.669e-11
  i=10: |c_i|=[2.566089391e-6 +/- 3.05e-16]   |c_i|*2^i=[0.0026276755 +/- 3.67e-11]   rad=8.671e-11
  i=11: |c_i|=[1.266336993e-7 +/- 3.40e-17]   |c_i|*2^i=[0.00025934582 +/- 3.77e-12]   rad=8.669e-11
  i=12: |c_i|=[1.339480400e-7 +/- 4.83e-17]   |c_i|*2^i=[0.00054865117 +/- 2.04e-12]   rad=8.670e-11
  i=13: |c_i|=[3.396072742e-8 +/- 3.42e-18]   |c_i|*2^i=[0.00027820628 +/- 1.01e-12]   rad=8.669e-11
  i=14: |c_i|=[1.255997131e-8 +/- 4.68e-18]   |c_i|*2^i=[0.00020578257 +/- 1.34e-13]   rad=8.670e-11
  i=15: |c_i|=[4.122315539e-9 +/- 5.19e-20]   |c_i|*2^i=[0.00013508004 +/- 4.42e-12]   rad=8.669e-11
  i=16: |c_i|=[1.443859329e-9 +/- 5.28e-21]   |c_i|*2^i=[9.4624765e-5 +/- 1.53e-14]   rad=8.667e-11
  i=17: |c_i|=[5.379774301e-10 +/- 7.34e-21]   |c_i|*2^i=[7.0513778e-5 +/- 2.82e-13]   rad=8.669e-11
  i=18: |c_i|=[2.372202496e-10 +/- 3.08e-20]   |c_i|*2^i=[6.2185865e-5 +/- 1.04e-13]   rad=8.669e-11
  i=19: |c_i|=[1.368556667e-10 +/- 4.62e-20]   |c_i|*2^i=[7.1751784e-5 +/- 2.42e-13]   rad=8.669e-11
  i=20: |c_i|=[1.034133581e-10 +/- 3.54e-20]   |c_i|*2^i=[0.00010843677 +/- 4.66e-12]   rad=8.669e-11

Head sum (i=1..20) of |c_i|*2^i: [5.976476731 +/- 4.26e-10]
Compare to the session's certified S* = 6.98 (head to i=14 + geometric tail).
If this head sum is noticeably BELOW 6.98 minus the old tail (~1e-8), the
extension genuinely tightened S*. If coefficients plateau or grow with i
well before i=IMAX, the precision floor has been hit again -- increase P further.

Total time: 750s