r2_check.py (verification run 2); Python 3.13.5; parts 012345; back end both; PB = 320
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N700-A:  N = 700;  Sigma* = 1/2 (computed);  nu = 359/50000000, x0 = 1/1000, X = 2502832726089/2500000000000, rho = 1903047399/250000000000000
  [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 17861 to 28841 bits)
      time: 0.3 s
  [3] Lemma 4.1 (PB = 320): 2900 closed intervals cover [x0, X]: 2874 without a root, 26 monotone; deepest bisection 26, 5771 point evaluations, smallest interval 1.49e-08
      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 (0.9831670518034764, 0.9831671114160541),  S in (0.9821670518034764, 0.9821671114160541)
        root 2:  x in (0.9843794822126590, 0.9843795120189478),  S in (0.9833794822126590, 0.9833795120189478)
        root 3:  x in (0.9861312723206730, 0.9861312872238174),  S in (0.9851312723206730, 0.9851312872238174)
        root 4:  x in (0.9876377566719307, 0.9876377864782195),  S in (0.9866377566719307, 0.9866377864782195)
        root 5:  x in (0.9887798144332759, 0.9887799336584311),  S in (0.9877798144332759, 0.9877799336584311)
      time: 6.8 s
  [4] coefficients of a positive multiple of P: degree 701, up to 17961 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: 13 intervals between x0 and X, sign changes: 0 0 0 0 0 0 1 1 0 1 1 1 0
      cut points: 1/2, 4/5, 9/10, 19/20, 49/50, 491/500, 123/125, 197/200, 493/500, 987/1000, 247/250, 99/100
      every interval has 0 or 1 sign changes: True;  number of roots of P in (x0, X): 5   PASS   (time: 3.7 s)
      each root interval of part [3] lies in one of the five intervals with one sign change: True
      back end plain Python integers: 13 intervals between x0 and X, sign changes: 0 0 0 0 0 0 1 1 0 1 1 1 0
      cut points: 1/2, 4/5, 9/10, 19/20, 49/50, 491/500, 123/125, 197/200, 493/500, 987/1000, 247/250, 99/100
      every interval has 0 or 1 sign changes: True;  number of roots of P in (x0, X): 5   PASS   (time: 8.9 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)
  N700-A: ALL EXECUTED PARTS PASS   (total 20.8 s)
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1 parameter points: ALL PASS
