r2_check.py (verification run 2); Python 3.13.5; parts 012345; back end both; PB = 320
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N680-A:  N = 680;  Sigma* = 1/2 (computed);  nu = 7667/1000000000, x0 = 1/1000, X = 1001183091843/1000000000000, rho = 4069150047/500000000000000
  [1] (2), (3), (4) vanish exactly at the six states of Lemma 3.2, all C_j > 0, Sigma = Sigma*, feasible: True;  signs of the right-hand side of (1): + - + - + -   PASS
  [2] signs of P at the six test points: + - + - + -   PASS   (integers chat_1, that_0 of 20284 to 27118 bits)
      time: 0.3 s
  [3] Lemma 4.1 (PB = 320): 4262 closed intervals cover [x0, X]: 4238 without a root, 24 monotone; deepest bisection 27, 8496 point evaluations, smallest interval 7.45e-09
      roots of P in [x0, X]: 5, all simple;  signs of P': - + - + -   PASS   (one in each interval between consecutive test points: True)
        root 1:  x in (0.9834621884106541, 0.9834622182184332),  S in (0.9824621884106541, 0.9824622182184332)
        root 2:  x in (0.9843984507474682, 0.9843984581994130),  S in (0.9833984507474682, 0.9833984581994130)
        root 3:  x in (0.9857869567065091, 0.9857869641584538),  S in (0.9847869567065091, 0.9847869641584538)
        root 4:  x in (0.9870181446601974, 0.9870181521121422),  S in (0.9860181446601974, 0.9860181521121422)
        root 5:  x in (0.9878994936162412, 0.9878995234240202),  S in (0.9868994936162412, 0.9868995234240202)
      time: 10.4 s
  [4] coefficients of a positive multiple of P: degree 681, up to 20389 bits   (time: 0.7 s)
      Horner values of the coefficient list at the six test points equal the values of the recursion: True
      back end python-flint 0.9.0: 14 intervals between x0 and X, sign changes: 0 0 0 0 0 0 1 1 1 0 1 1 0 0
      cut points: 1/2, 4/5, 9/10, 19/20, 49/50, 491/500, 123/125, 197/200, 493/500, 987/1000, 79/80, 247/250, 99/100
      every interval has 0 or 1 sign changes: True;  number of roots of P in (x0, X): 5   PASS   (time: 4.4 s)
      each root interval of part [3] lies in one of the five intervals with one sign change: True
      back end plain Python integers: 14 intervals between x0 and X, sign changes: 0 0 0 0 0 0 1 1 1 0 1 1 0 0
      cut points: 1/2, 4/5, 9/10, 19/20, 49/50, 491/500, 123/125, 197/200, 493/500, 987/1000, 79/80, 247/250, 99/100
      every interval has 0 or 1 sign changes: True;  number of roots of P in (x0, X): 5   PASS   (time: 10.9 s)
      each root interval of part [3] lies in one of the five intervals with one sign change: True
  [5] gcd(P, P') is constant (all complex roots of P are simple): True   PASS   (time: 0.0 s)
  N680-A: ALL EXECUTED PARTS PASS   (total 27.3 s)
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1 parameter points: ALL PASS
