s(3) in [[1.05126995479423663626079492997487946409381834557750750736156934023179942983006185386329889297485351562499999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 +/- 1.11e-270], [1.05126995479423663626079492997488735270287055569556162464722216809409616189441294409334659576416015625000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 +/- 2.97e-270]]
P_3(s)   = 2.23756642787
h_1 = c_1 = P_3(s) - 3 = -0.762433572130   (negative: this is the defect)

h_j(s(3)) for j <= 8: 1.00000 -0.762434 0.376884 -0.125985 0.0353111 -0.00770917 0.00159346 -0.000247136 4.61279e-5

   k      A_k*3^k        S_k*3^k      A_k/S_k   (S_k = P_k^odd(s(3)))
    1      11.287301      6.7126993    1.6814846   |S-P|<=2.92e-268
    2      64.477660      23.306247    2.7665398   |S-P|<=8.91e-268
    3      248.06486      55.990303    4.4304969   |S-P|<=1.47e-267
    4      722.13076      105.08787    6.8716849   |S-P|<=1.73e-267
    5      1694.8681      165.01014    10.271297   |S-P|<=1.60e-267
    6      3338.0653      226.70948    14.723978   |S-P|<=1.24e-267
    7      5670.6741      281.45784    20.147508   |S-P|<=8.27e-268
    8      8477.4943      323.55658    26.200964   |S-P|<=4.88e-268
    9      11324.727      351.05403    32.259215   |S-P|<=2.59e-268
   10      13681.543      364.94931    37.488886   |S-P|<=1.26e-268
   11      15093.967      367.84350    41.033664   |S-P|<=5.76e-269
   12      15328.840      362.76158    42.255963   |S-P|<=2.48e-269
   20      1771.5665      248.21857    7.1371232   |S-P|<=1.45e-271
   30      142.41435      141.49812    1.0064752   |S-P|<=6.38e-272
   50      45.868392      45.868392    1.0000000   |S-P|<=3.53e-272
   75      11.219252      11.219252    1.0000000   |S-P|<=2.79e-272
   80      8.4655263      8.4655263    1.0000000   |S-P|<=2.38e-272

largest A_k/S_k over the k sampled: 42.25596339
a single all-k constant would be about 15543.58147 in place of C_N = 367.8435

A_k*3^k <= C_N for all k >= 24  (and A_23*3^23 = 472.26966 exceeds it)
max of A_k*3^k over 59 <= k <= 80: 27.628395, i.e. C_N has 13.3x slack there
=> the sign repair changes no certified constant; it changes the RANGE
   of Proposition 3.5, whose only use downstream is k >= 59.