Sieving primes up to 100000000...
  5761454 primes, 6s
  power sums built, 75s
logG[1..8] (real, s=1):
  logG[1] = [-0.815718452 +/- 4.76e-10]
  logG[2] = [0.101123710 +/- 5.09e-10]
  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=100000000, prec=350 bits, N=2048 samples:
  i= 1: |c_i|=[0.8157184528 +/- 1.67e-11]   |c_i|*2^i=[1.6314369 +/- 5.57e-9]   rad=9.875e-10
  i= 2: |c_i|=[0.4338220070 +/- 4.65e-11]   |c_i|*2^i=[1.7352880 +/- 2.82e-8]   rad=9.839e-10
  i= 3: |c_i|=[0.1563636436 +/- 3.15e-11]   |c_i|*2^i=[1.2509091 +/- 4.86e-8]   rad=9.767e-10
  i= 4: |c_i|=[0.04729723257 +/- 4.17e-12]   |c_i|*2^i=[0.75675572 +/- 1.19e-9]   rad=9.769e-10
  i= 5: |c_i|=[0.01118690973 +/- 4.14e-12]   |c_i|*2^i=[0.35798111 +/- 1.50e-9]   rad=9.756e-10
  i= 6: |c_i|=[0.002494625529 +/- 1.14e-14]   |c_i|*2^i=[0.15965603 +/- 3.86e-9]   rad=9.760e-10
  i= 7: |c_i|=[0.0004232974308 +/- 2.55e-14]   |c_i|*2^i=[0.054182071 +/- 1.40e-10]   rad=9.754e-10
  i= 8: |c_i|=[8.397439834e-5 +/- 1.75e-15]   |c_i|*2^i=[0.021497446 +/- 2.55e-11]   rad=9.756e-10
  i= 9: |c_i|=[8.415837603e-6 +/- 1.26e-16]   |c_i|*2^i=[0.0043089089 +/- 4.74e-11]   rad=9.753e-10
  i=10: |c_i|=[2.566978161e-6 +/- 1.91e-16]   |c_i|*2^i=[0.0026285856 +/- 3.67e-11]   rad=9.755e-10
  i=11: |c_i|=[1.275223226e-7 +/- 4.06e-17]   |c_i|*2^i=[0.00026116572 +/- 3.40e-12]   rad=9.753e-10
  i=12: |c_i|=[1.348367466e-7 +/- 1.80e-17]   |c_i|*2^i=[0.00055229131 +/- 4.15e-12]   rad=9.754e-10
  i=13: |c_i|=[3.484933232e-8 +/- 4.61e-19]   |c_i|*2^i=[0.00028548573 +/- 3.62e-13]   rad=9.753e-10
  i=14: |c_i|=[1.344865317e-8 +/- 4.67e-18]   |c_i|*2^i=[0.00022034273 +/- 3.62e-12]   rad=9.754e-10
  i=15: |c_i|=[5.010909928e-9 +/- 9.33e-20]   |c_i|*2^i=[0.00016419750 +/- 3.48e-12]   rad=9.753e-10
  i=16: |c_i|=[2.332341931e-9 +/- 2.25e-19]   |c_i|*2^i=[0.00015285236 +/- 7.76e-13]   rad=9.752e-10
  i=17: |c_i|=[1.426568317e-9 +/- 3.27e-19]   |c_i|*2^i=[0.00018698316 +/- 2.49e-12]   rad=9.753e-10
  i=18: |c_i|=[1.125847895e-9 +/- 4.92e-19]   |c_i|*2^i=[0.00029513427 +/- 7.16e-13]   rad=9.753e-10
  i=19: |c_i|=[1.025449542e-9 +/- 1.34e-19]   |c_i|*2^i=[0.00053763089 +/- 4.55e-13]   rad=9.753e-10
  i=20: |c_i|=[9.920346191e-10 +/- 2.92e-20]   |c_i|*2^i=[0.0010402237 +/- 7.22e-12]   rad=9.753e-10

Head sum (i=1..20) of |c_i|*2^i: [5.978340268 +/- 1.94e-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: 82s