r2_check.py (verification run 2); Python 3.13.5; parts 012345; back end both; PB = 320
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N660-B:  N = 660;  Sigma* = 1/2 (computed);  nu = 7147/871200000, x0 = 1/1000, X = 250309242277367963817/250000000000000000000, rho = 8718964109872981977743/1000000000000000000000000000
  [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 19555 to 54900 bits)
      time: 1.0 s
  [3] Lemma 4.1 (PB = 320): 9805 closed intervals cover [x0, X]: 9777 without a root, 28 monotone; deepest bisection 30, 19578 point evaluations, smallest interval 9.32e-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 (0.9840406915367755, 0.9840406952629487),  S in (0.9830406915367755, 0.9830406952629487)
        root 2:  x in (0.9847016857628271, 0.9847016866943703),  S in (0.9837016857628271, 0.9837016866943703)
        root 3:  x in (0.9854377838018791, 0.9854377847334224),  S in (0.9844377838018791, 0.9844377847334224)
        root 4:  x in (0.9861677774378874, 0.9861677783694307),  S in (0.9851677774378874, 0.9851677783694307)
        root 5:  x in (0.9866064933298223, 0.9866064951929089),  S in (0.9856064933298223, 0.9856064951929089)
      time: 23.4 s
  [4] coefficients of a positive multiple of P: degree 661, up to 19701 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: 15 intervals between x0 and X, sign changes: 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, 197/200, 493/500, 1973/2000, 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: 5.2 s)
      each root interval of part [3] lies in one of the five intervals with one sign change: True
      back end plain Python integers: 15 intervals between x0 and X, sign changes: 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, 197/200, 493/500, 1973/2000, 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: 11.5 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)
  N660-B: ALL EXECUTED PARTS PASS   (total 42.7 s)
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1 parameter points: ALL PASS
