r2_check.py (verification run 2); Python 3.13.5; parts 012345; back end flint; PB = 320
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N2000:  N = 2000;  Sigma* = 1/2 (computed);  nu = 7/10000000, x0 = 1/100, X = 20004997/20000000, rho = 363553/500000000000
  [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 46464 to 48519 bits)
      time: 2.5 s
  [3] Lemma 4.1 (PB = 320): 600 closed intervals cover [x0, X]: 573 without a root, 27 monotone; deepest bisection 20, 1170 point evaluations, smallest interval 9.44e-07
      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 (0.9878218638301849, 0.9878237525818825),  S in (0.9778218638301850, 0.9778237525818825)
        root 2:  x in (0.9913169988464833, 0.9913179432223320),  S in (0.9813169988464833, 0.9813179432223320)
        root 3:  x in (0.9945307098598480, 0.9945316542356968),  S in (0.9845307098598480, 0.9845316542356968)
        root 4:  x in (0.9963458002411842, 0.9963476889928817),  S in (0.9863458002411842, 0.9863476889928817)
        root 5:  x in (0.9977113677185059, 0.9977189227252961),  S in (0.9877113677185059, 0.9877189227252960)
      time: 3.4 s
  [4] coefficients of a positive multiple of P: degree 2001, up to 46553 bits   (time: 13.3 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: 12 intervals between x0 and X, sign changes: 0 0 0 0 0 1 1 1 0 1 1 0
      cut points: 1/2, 4/5, 9/10, 19/20, 49/50, 99/100, 124/125, 199/200, 249/250, 997/1000, 499/500
      every interval has 0 or 1 sign changes: True;  number of roots of P in (x0, X): 5   PASS   (time: 36.3 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)
  N2000: ALL EXECUTED PARTS PASS   (total 58.9 s)
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1 parameter points: ALL PASS
