r2_check.py (verification run 2); Python 3.13.5; parts 012345; back end both; PB = 320
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N655-B:  N = 655;  Sigma* = 1/2 (computed);  nu = 35807/4290250000, x0 = 1/1000, X = 1001251064899138279703/1000000000000000000000, rho = 8873320104916924370947/1000000000000000000000000000
  [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 20911 to 57090 bits)
      time: 1.1 s
  [3] Lemma 4.1 (PB = 320): 17982 closed intervals cover [x0, X]: 17956 without a root, 26 monotone; deepest bisection 33, 35934 point evaluations, smallest interval 1.16e-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 (0.9844585661217062, 0.9844585665874844),  S in (0.9834585661217062, 0.9834585665874844)
        root 2:  x in (0.9847987136929590, 0.9847987139258482),  S in (0.9837987136929590, 0.9837987139258482)
        root 3:  x in (0.9853208608348331, 0.9853208609512776),  S in (0.9843208608348331, 0.9843208609512776)
        root 4:  x in (0.9858252257295333, 0.9858252259624224),  S in (0.9848252257295333, 0.9848252259624224)
        root 5:  x in (0.9861356829681893, 0.9861356834339675),  S in (0.9851356829681893, 0.9851356834339675)
      time: 39.7 s
  [4] coefficients of a positive multiple of P: degree 656, up to 21059 bits   (time: 1.0 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: 14 intervals between x0 and X, sign changes: 0 0 0 0 0 0 0 1 1 1 1 1 0 0
      cut points: 1/2, 4/5, 9/10, 19/20, 49/50, 491/500, 123/125, 1969/2000, 197/200, 1971/2000, 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: 5.0 s)
      each root interval of part [3] lies in one of the five intervals with one sign change: True
      back end plain Python integers: 14 intervals between x0 and X, sign changes: 0 0 0 0 0 0 0 1 1 1 1 1 0 0
      cut points: 1/2, 4/5, 9/10, 19/20, 49/50, 491/500, 123/125, 1969/2000, 197/200, 1971/2000, 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: 11.1 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)
  N655-B: ALL EXECUTED PARTS PASS   (total 58.7 s)
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1 parameter points: ALL PASS
