r2_check.py (verification run 2); Python 3.13.5; parts 012345; back end both; PB = 320
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N653-A:  N = 653;  Sigma* = 1/2 (computed);  nu = 84019273/10000000000000, x0 = 1/100000, X = 3128925568443009/3125000000000000, rho = 446229587927767/50000000000000000000
  [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 28140 to 34276 bits)
      time: 0.4 s
  [3] Lemma 4.1 (PB = 320): 24448 closed intervals cover [x0, X]: 24426 without a root, 22 monotone; deepest bisection 34, 48870 point evaluations, smallest interval 5.83e-11
      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 (0.9845826253777927, 0.9845826256109136),  S in (0.9845726253777928, 0.9845726256109135)
        root 2:  x in (0.9848558910283385, 0.9848558910866186),  S in (0.9848458910283384, 0.9848458910866187)
        root 3:  x in (0.9852794914993059, 0.9852794915575861),  S in (0.9852694914993060, 0.9852694915575861)
        root 4:  x in (0.9856921647827833, 0.9856921648410635),  S in (0.9856821647827834, 0.9856821648410635)
        root 5:  x in (0.9859424898320138, 0.9859424900651346),  S in (0.9859324898320138, 0.9859324900651346)
      time: 51.6 s
  [4] coefficients of a positive multiple of P: degree 654, up to 28282 bits   (time: 1.3 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: 16 intervals between x0 and X, sign changes: 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0
      cut points: 1/2, 4/5, 9/10, 19/20, 49/50, 491/500, 123/125, 1969/2000, 1231/1250, 197/200, 1971/2000, 4929/5000, 493/500, 247/250, 99/100
      every interval has 0 or 1 sign changes: True;  number of roots of P in (x0, X): 5   PASS   (time: 6.2 s)
      each root interval of part [3] lies in one of the five intervals with one sign change: True
      back end plain Python integers: 16 intervals between x0 and X, sign changes: 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0
      cut points: 1/2, 4/5, 9/10, 19/20, 49/50, 491/500, 123/125, 1969/2000, 1231/1250, 197/200, 1971/2000, 4929/5000, 493/500, 247/250, 99/100
      every interval has 0 or 1 sign changes: True;  number of roots of P in (x0, X): 5   PASS   (time: 14.5 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)
  N653-A: ALL EXECUTED PARTS PASS   (total 74.7 s)
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1 parameter points: ALL PASS
