r2_check.py (verification run 2); Python 3.13.5; parts 012345; back end flint; PB = 320
============================================================================================================
N4000-A:  N = 4000;  Sigma* = 1/2 (computed);  nu = 811/5000000000, x0 = 1/100, X = 10000998411859/10000000000000, rho = 1668474131/10000000000000000
  [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 128823 to 172754 bits)
      time: 25.5 s
  [3] Lemma 4.1 (PB = 320): 533 closed intervals cover [x0, X]: 502 without a root, 31 monotone; deepest bisection 20, 1033 point evaluations, smallest interval 9.44e-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 (0.9908085905400057, 0.9908104790055842),  S in (0.9808085905400057, 0.9808104790055842)
        root 2:  x in (0.9944552175719614, 0.9944561618047506),  S in (0.9844552175719614, 0.9844561618047506)
        root 3:  x in (0.9971812176344306, 0.9971821618672198),  S in (0.9871812176344306, 0.9871821618672197)
        root 4:  x in (0.9983756721127900, 0.9983775605783684),  S in (0.9883756721127900, 0.9883775605783683)
        root 5:  x in (0.9989025540091729, 0.9989063309403299),  S in (0.9889025540091729, 0.9889063309403299)
      time: 6.1 s
  [4] coefficients of a positive multiple of P: degree 4001, up to 128937 bits   (time: 215.2 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, 99/100, 124/125, 199/200, 499/500, 1997/2000, 999/1000
      every interval has 0 or 1 sign changes: True;  number of roots of P in (x0, X): 5   PASS   (time: 329.3 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: 0.1 s)
  N4000-A: ALL EXECUTED PARTS PASS   (total 623.5 s)
============================================================================================================
1 parameter points: ALL PASS
