r2_check.py (verification run 2); Python 3.13.5; parts 012345; back end both; PB = 320
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N660-A:  N = 660;  Sigma* = 1/2 (computed);  nu = 6409/781250000, x0 = 1/1000, X = 1001236952091663/1000000000000000, rho = 4359422493891/500000000000000000
  [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 19451 to 32898 bits)
      time: 0.3 s
  [3] Lemma 4.1 (PB = 320): 9928 closed intervals cover [x0, X]: 9900 without a root, 28 monotone; deepest bisection 30, 19824 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 (0.9841047593981180, 0.9841047612612045),  S in (0.9831047593981180, 0.9831047612612045)
        root 2:  x in (0.9846310366222397, 0.9846310375537830),  S in (0.9836310366222397, 0.9836310375537830)
        root 3:  x in (0.9854217100418144, 0.9854217109733576),  S in (0.9844217100418144, 0.9844217109733576)
        root 4:  x in (0.9861742376967036, 0.9861742386282469),  S in (0.9851742376967036, 0.9851742386282469)
        root 5:  x in (0.9866230990187477, 0.9866231008818341),  S in (0.9856230990187477, 0.9856231008818341)
      time: 23.6 s
  [4] coefficients of a positive multiple of P: degree 661, up to 19566 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: 4.6 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: 10.5 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-A: ALL EXECUTED PARTS PASS   (total 40.3 s)
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1 parameter points: ALL PASS
