r2_check.py (verification run 2); Python 3.13.5; parts 012345; back end both; PB = 320
============================================================================================================
N690-A:  N = 690;  Sigma* = 1/2 (computed);  nu = 3709/500000000, x0 = 1/1000, X = 5005788396031/5000000000000, rho = 7869205719/1000000000000000
  [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 19894 to 29119 bits)
      time: 0.3 s
  [3] Lemma 4.1 (PB = 320): 3350 closed intervals cover [x0, X]: 3325 without a root, 25 monotone; deepest bisection 26, 6672 point evaluations, smallest interval 1.49e-08
      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 (0.9832776094256622, 0.9832776392326839),  S in (0.9822776094256622, 0.9822776392326839)
        root 2:  x in (0.9843705881969903, 0.9843706031005011),  S in (0.9833705881969903, 0.9833706031005011)
        root 3:  x in (0.9859717617829893, 0.9859717766865000),  S in (0.9849717617829893, 0.9849717766865000)
        root 4:  x in (0.9873839737555916, 0.9873839886591024),  S in (0.9863839737555916, 0.9863839886591024)
        root 5:  x in (0.9883350562006203, 0.9883351158146636),  S in (0.9873350562006203, 0.9873351158146636)
      time: 8.1 s
  [4] coefficients of a positive multiple of P: degree 691, up to 19999 bits   (time: 0.8 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 0 1 1 1 1 1 0
      cut points: 1/2, 4/5, 9/10, 19/20, 49/50, 491/500, 123/125, 197/200, 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: 4.0 s)
      each root interval of part [3] lies in one of the five intervals with one sign change: True
      back end plain Python integers: 12 intervals between x0 and X, sign changes: 0 0 0 0 0 0 1 1 1 1 1 0
      cut points: 1/2, 4/5, 9/10, 19/20, 49/50, 491/500, 123/125, 197/200, 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: 9.2 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)
  N690-A: ALL EXECUTED PARTS PASS   (total 22.9 s)
============================================================================================================
1 parameter points: ALL PASS
