r2_check.py (verification run 2); Python 3.13.5; parts 012345; back end both; PB = 320
============================================================================================================
N1000:  N = 1000;  Sigma* = 1/2 (computed);  nu = 13/4000000, x0 = 1/100, X = 10006777/10000000, rho = 343447/100000000000
  [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 21892 to 24265 bits)
      time: 0.4 s
  [3] Lemma 4.1 (PB = 320): 934 closed intervals cover [x0, X]: 910 without a root, 24 monotone; deepest bisection 21, 1841 point evaluations, smallest interval 4.72e-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 (0.9830782659145355, 0.9830801554822922),  S in (0.9730782659145355, 0.9730801554822922)
        root 2:  x in (0.9869764441965103, 0.9869769165884494),  S in (0.9769764441965103, 0.9769769165884494)
        root 3:  x in (0.9901315499581814, 0.9901320223501205),  S in (0.9801315499581814, 0.9801320223501205)
        root 4:  x in (0.9924330434857845, 0.9924335158777237),  S in (0.9824330434857845, 0.9824335158777237)
        root 5:  x in (0.9937146428167343, 0.9937155876006126),  S in (0.9837146428167343, 0.9837155876006126)
      time: 2.9 s
  [4] coefficients of a positive multiple of P: degree 1001, up to 21975 bits   (time: 1.6 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 1 1 0 0
      cut points: 1/2, 4/5, 9/10, 19/20, 49/50, 197/200, 99/100, 124/125, 993/1000, 497/500, 199/200
      every interval has 0 or 1 sign changes: True;  number of roots of P in (x0, X): 5   PASS   (time: 5.9 s)
      each root interval of part [3] lies in one of the five intervals with one sign change: True
      back end plain Python integers: 12 intervals between x0 and X, sign changes: 0 0 0 0 0 1 1 1 1 1 0 0
      cut points: 1/2, 4/5, 9/10, 19/20, 49/50, 197/200, 99/100, 124/125, 993/1000, 497/500, 199/200
      every interval has 0 or 1 sign changes: True;  number of roots of P in (x0, X): 5   PASS   (time: 19.0 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)
  N1000: ALL EXECUTED PARTS PASS   (total 30.4 s)
============================================================================================================
1 parameter points: ALL PASS
