r2_spectrum.py (verification run 2); python-flint 0.9.0, numpy 2.5.4; PREC = 700 bits; N = 1000; kappa' = 13/2000000
state 1: x in a rational interval of length 6.07e-61 with exact opposite signs of P; S = 0.973078759015299...; largest radius of a C_j ball 1.5e-61; sum of the C_j balls contains Sigma* = 1/2: True   (9.2 s)
   floating-point eigenvalues: 0 non-real, 0 positive, largest -7.380791e-08, second -9.750000e-06, smallest -3.96608
   1003 test points from 0 down to -5, decreasing: True; undecided signs: 0; signs alternate: True; sign of det(J - p_0): +1
   state 1: all 1002 eigenvalues are real, simple and negative: hyperbolic sink   (11 s)
   control at p = 0.000000e+00: general determinant of the 1002 x 1002 ball matrix has sign +1, the structured formula +1; the two balls overlap: True; relative radius of the general determinant 5.2e-49   (39 s)
   control at p = -4.911904e-06: general determinant of the 1002 x 1002 ball matrix has sign -1, the structured formula -1; the two balls overlap: True; relative radius of the general determinant 1.2e-50   (39 s)
   control at p = -4.886450e-05: general determinant of the 1002 x 1002 ball matrix has sign +1, the structured formula +1; the two balls overlap: True; relative radius of the general determinant 5.0e-51   (15 s)
state 2: x in a rational interval of length 5.58e-61 with exact opposite signs of P; S = 0.976976484124700...; largest radius of a C_j ball 1.3e-61; sum of the C_j balls contains Sigma* = 1/2: True   (8.4 s)
   floating-point eigenvalues: 0 non-real, 1 positive, largest 2.909370e-08, second -9.750000e-06, smallest -3.97390
   1003 test points from 1 down to -5, decreasing: True; undecided signs: 0; signs alternate: True; sign of det(J - p_0): +1
   state 2: exactly one eigenvalue in (0, 1), the other 1001 real, simple and negative: saddle, one unstable direction   (10 s)
   control at p = 1.000000e+00: general determinant of the 1002 x 1002 ball matrix has sign +1, the structured formula +1; the two balls overlap: True; relative radius of the general determinant 9.9e-58   (52 s)
   control at p = 0.000000e+00: general determinant of the 1002 x 1002 ball matrix has sign -1, the structured formula -1; the two balls overlap: True; relative radius of the general determinant 1.2e-48   (39 s)
   control at p = -3.538449e-05: general determinant of the 1002 x 1002 ball matrix has sign +1, the structured formula +1; the two balls overlap: True; relative radius of the general determinant 5.8e-51   (15 s)
state 3: x in a rational interval of length 4.46e-61 with exact opposite signs of P; S = 0.980131721343748...; largest radius of a C_j ball 1.0e-61; sum of the C_j balls contains Sigma* = 1/2: True   (8.2 s)
   floating-point eigenvalues: 0 non-real, 0 positive, largest -2.328224e-08, second -9.750000e-06, smallest -3.98023
   1003 test points from 0 down to -5, decreasing: True; undecided signs: 0; signs alternate: True; sign of det(J - p_0): +1
   state 3: all 1002 eigenvalues are real, simple and negative: hyperbolic sink   (10 s)
   control at p = 0.000000e+00: general determinant of the 1002 x 1002 ball matrix has sign +1, the structured formula +1; the two balls overlap: True; relative radius of the general determinant 1.0e-48   (40 s)
   control at p = -4.886641e-06: general determinant of the 1002 x 1002 ball matrix has sign -1, the structured formula -1; the two balls overlap: True; relative radius of the general determinant 8.2e-51   (40 s)
   control at p = -2.849378e-05: general determinant of the 1002 x 1002 ball matrix has sign +1, the structured formula +1; the two balls overlap: True; relative radius of the general determinant 6.0e-51   (15 s)
state 4: x in a rational interval of length 5.74e-61 with exact opposite signs of P; S = 0.982433429147861...; largest radius of a C_j ball 1.2e-61; sum of the C_j balls contains Sigma* = 1/2: True   (8.2 s)
   floating-point eigenvalues: 0 non-real, 1 positive, largest 2.773864e-08, second -9.750000e-06, smallest -3.98485
   1003 test points from 1 down to -5, decreasing: True; undecided signs: 0; signs alternate: True; sign of det(J - p_0): +1
   state 4: exactly one eigenvalue in (0, 1), the other 1001 real, simple and negative: saddle, one unstable direction   (10 s)
   control at p = 1.000000e+00: general determinant of the 1002 x 1002 ball matrix has sign +1, the structured formula +1; the two balls overlap: True; relative radius of the general determinant 9.8e-58   (54 s)
   control at p = 0.000000e+00: general determinant of the 1002 x 1002 ball matrix has sign -1, the structured formula -1; the two balls overlap: True; relative radius of the general determinant 8.0e-49   (40 s)
   control at p = -2.621778e-05: general determinant of the 1002 x 1002 ball matrix has sign +1, the structured formula +1; the two balls overlap: True; relative radius of the general determinant 8.3e-51   (15 s)
state 5: x in a rational interval of length 6.04e-61 with exact opposite signs of P; S = 0.983715309104275...; largest radius of a C_j ball 1.3e-61; sum of the C_j balls contains Sigma* = 1/2: True   (9.1 s)
   floating-point eigenvalues: 0 non-real, 0 positive, largest -6.896208e-08, second -9.750000e-06, smallest -3.98741
   1003 test points from 0 down to -5, decreasing: True; undecided signs: 0; signs alternate: True; sign of det(J - p_0): +1
   state 5: all 1002 eigenvalues are real, simple and negative: hyperbolic sink   (11 s)
   control at p = 0.000000e+00: general determinant of the 1002 x 1002 ball matrix has sign +1, the structured formula +1; the two balls overlap: True; relative radius of the general determinant 2.6e-49   (40 s)
   control at p = -4.909481e-06: general determinant of the 1002 x 1002 ball matrix has sign -1, the structured formula -1; the two balls overlap: True; relative radius of the general determinant 6.7e-51   (40 s)
   control at p = -2.580430e-05: general determinant of the 1002 x 1002 ball matrix has sign +1, the structured formula +1; the two balls overlap: True; relative radius of the general determinant 8.7e-51   (15 s)
total 546 s:  ALL FIVE STATES CERTIFIED
