r2_check.py (verification run 2); Python 3.13.5; parts 012345; back end both; PB = 320
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N652-A:  N = 652;  Sigma* = 1/2 (computed);  nu = 843117449/100000000000000, x0 = 1/100000, X = 312893455958161/312500000000000, rho = 8956252108919407/1000000000000000000000
  [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 30260 to 33362 bits)
      time: 0.4 s
  [3] Lemma 4.1 (PB = 320): 35847 closed intervals cover [x0, X]: 35825 without a root, 22 monotone; deepest bisection 35, 71668 point evaluations, smallest interval 2.91e-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 (0.9847198690845262, 0.9847198691428066),  S in (0.9847098690845263, 0.9847098691428067)
        root 2:  x in (0.9849290751701930, 0.9849290751993333),  S in (0.9849190751701931, 0.9849190751993332)
        root 3:  x in (0.9852593339173477, 0.9852593339464878),  S in (0.9852493339173476, 0.9852493339464878)
        root 4:  x in (0.9855819961976530, 0.9855819962267932),  S in (0.9855719961976530, 0.9855719962267931)
        root 5:  x in (0.9857780768747354, 0.9857780769330158),  S in (0.9857680768747354, 0.9857680769330158)
      time: 76.4 s
  [4] coefficients of a positive multiple of P: degree 653, up to 30409 bits   (time: 1.4 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: 17 intervals between x0 and X, sign changes: 0 0 0 0 0 0 0 0 1 1 1 1 1 0 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, 616/625, 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: 7.0 s)
      each root interval of part [3] lies in one of the five intervals with one sign change: True
      back end plain Python integers: 17 intervals between x0 and X, sign changes: 0 0 0 0 0 0 0 0 1 1 1 1 1 0 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, 616/625, 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: 16.6 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)
  N652-A: ALL EXECUTED PARTS PASS   (total 102.6 s)
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1 parameter points: ALL PASS
