r2_check.py (verification run 2); Python 3.13.5; parts 012345; back end both; PB = 320
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N655-A:  N = 655;  Sigma* = 1/2 (computed);  nu = 8346141/1000000000000, x0 = 1/1000, X = 5006255330114563/5000000000000000, rho = 8873328232221/1000000000000000000
  [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 26054 to 34170 bits)
      time: 0.4 s
  [3] Lemma 4.1 (PB = 320): 18002 closed intervals cover [x0, X]: 17976 without a root, 26 monotone; deepest bisection 33, 35974 point evaluations, smallest interval 1.16e-10
      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 (0.9844512889764762, 0.9844512894422544),  S in (0.9834512889764762, 0.9834512894422544)
        root 2:  x in (0.9848046891023192, 0.9848046893352083),  S in (0.9838046891023192, 0.9838046893352083)
        root 3:  x in (0.9853245485751553, 0.9853245486915998),  S in (0.9843245485751553, 0.9843245486915998)
        root 4:  x in (0.9858275715634307, 0.9858275717963199),  S in (0.9848275715634307, 0.9848275717963199)
        root 5:  x in (0.9861309252191220, 0.9861309256849002),  S in (0.9851309252191220, 0.9851309256849002)
      time: 40.1 s
  [4] coefficients of a positive multiple of P: degree 656, up to 26181 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: 14 intervals between x0 and X, sign changes: 0 0 0 0 0 0 0 1 1 1 1 1 0 0
      cut points: 1/2, 4/5, 9/10, 19/20, 49/50, 491/500, 123/125, 1969/2000, 197/200, 1971/2000, 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: 5.5 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 0 1 1 1 1 1 0 0
      cut points: 1/2, 4/5, 9/10, 19/20, 49/50, 491/500, 123/125, 1969/2000, 197/200, 1971/2000, 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: 13.0 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)
  N655-A: ALL EXECUTED PARTS PASS   (total 61.0 s)
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1 parameter points: ALL PASS
