[PASS] thm:main: S3 (Gaussian sample at t=1.44): zeros of a in D from roots agree with the argument principle on |w|=1 and real sign changes on (-1,1)  (numerical (floating point))  -- winding number 4, roots in D 4, real 0; min ||w|-1| = 0.072
   S1 n=10: real 8, nonreal 2 (certified: True); predicted nonreal 2; max dist to zeta_i 1.75e-01; bulk overshoot 0.00e+00
   S1 n=20: real 18, nonreal 2 (certified: True); predicted nonreal 2; max dist to zeta_i 7.76e-02; bulk overshoot 0.00e+00
   S1 n=40: real 38, nonreal 2 (certified: True); predicted nonreal 2; max dist to zeta_i 3.93e-02; bulk overshoot 0.00e+00
   S1 n=80: real 78, nonreal 2 (certified: True); predicted nonreal 2; max dist to zeta_i 1.98e-02; bulk overshoot 0.00e+00
   S1 n=160: real 158, nonreal 2 (certified: True); predicted nonreal 2; max dist to zeta_i 9.97e-03; bulk overshoot 0.00e+00
   S1 n=320: real 318, nonreal 2 (certified: True); predicted nonreal 2; max dist to zeta_i 5.00e-03; bulk overshoot 0.00e+00
   S1 n=400: real 398, nonreal 2 (certified: True); predicted nonreal 2; max dist to zeta_i 4.00e-03; bulk overshoot 0.00e+00
[PASS] thm:main: S1 = (1-2w+2w^2)(1+5w/3)/(1-10w/13): the certified number of nonreal zeros of Q_n equals the number (2) of nonreal zeros of a in D for all tested n >= n_obs (observed onset reported; the theorem only asserts some n_0)  (interval arithmetic (Arb))  -- n_obs = 10; n=10: 8R+2C; n=20: 18R+2C; n=40: 38R+2C; n=80: 78R+2C; n=160: 158R+2C; n=320: 318R+2C; n=400: 398R+2C
[INFO] thm:main(a),(b): S1 = (1-2w+2w^2)(1+5w/3)/(1-10w/13): n x (distance from zeta_i to the nearest zero) and bulk overshoot, n=400  -- [1.375, 1.375, 1.6]; overshoot 0.0
   S2 n=10: real 6, nonreal 4 (certified: True); predicted nonreal 4; max dist to zeta_i 3.36e-01; bulk overshoot 2.15e-01
   S2 n=20: real 10, nonreal 10 (certified: True); predicted nonreal 4; max dist to zeta_i 1.50e-01; bulk overshoot 2.23e-01
   S2 n=40: real 24, nonreal 16 (certified: True); predicted nonreal 4; max dist to zeta_i 6.62e-02; bulk overshoot 1.59e-01
   S2 n=80: real 62, nonreal 18 (certified: True); predicted nonreal 4; max dist to zeta_i 5.75e-02; bulk overshoot 5.08e-02
   S2 n=160: real 138, nonreal 22 (certified: True); predicted nonreal 4; max dist to zeta_i 3.93e-02; bulk overshoot 2.44e-02
   S2 n=320: real 312, nonreal 8 (certified: True); predicted nonreal 4; max dist to zeta_i 1.48e-02; bulk overshoot 1.04e-02
   S2 n=400: real 394, nonreal 6 (certified: True); predicted nonreal 4; max dist to zeta_i 1.14e-02; bulk overshoot 7.33e-03
   S2 n=600: real 594, nonreal 6 (certified: True); predicted nonreal 4; max dist to zeta_i 7.26e-03; bulk overshoot 2.14e-03
   S2 n=800: real 796, nonreal 4 (certified: True); predicted nonreal 4; max dist to zeta_i 5.32e-03; bulk overshoot 0.00e+00
   S2 n=1200: real 1196, nonreal 4 (certified: True); predicted nonreal 4; max dist to zeta_i 3.47e-03; bulk overshoot 0.00e+00
[PASS] thm:main: S2 = (1-24w/17+20w^2/17)(1+32w/13+20w^2/13)/(1-20w/21): the certified number of nonreal zeros of Q_n equals the number (4) of nonreal zeros of a in D for all tested n >= n_obs (observed onset reported; the theorem only asserts some n_0)  (interval arithmetic (Arb))  -- n_obs = 800; n=10: 6R+4C; n=20: 10R+10C; n=40: 24R+16C; n=80: 62R+18C; n=160: 138R+22C; n=320: 312R+8C; n=400: 394R+6C; n=600: 594R+6C; n=800: 796R+4C; n=1200: 1196R+4C
[INFO] thm:main: S2 = (1-24w/17+20w^2/17)(1+32w/13+20w^2/13)/(1-20w/21): extra nonreal zeros (upper half plane, z = x/sqrt n) before the onset  -- n=20: [(1.3846+0.2515j), (1.1299+0.2229j), (0.7846+0.1272j), (0.4415+0.0831j), (0.1109+0.033j)]; n=40: [(1.6044+0.1067j), (1.3554+0.1367j), (1.254+0.1586j), (1.0749+0.0827j), (0.8891+0.0575j), (0.7134+0.0422j), (0.5442+0.0302j), (0.3794+0.0176j)]; n=80: [(1.7573+0.0508j), (1.606+0.0455j), (1.4767+0.0453j), (1.3545+0.0491j), (1.2314+0.0415j), (1.1271+0.0285j), (1.0306+0.0192j), (0.9384+0.0113j)]; n=160: [(1.8504+0.0244j), (1.7572+0.0174j), (1.6788+0.0132j), (1.6084+0.0104j), (1.5432+0.0083j), (1.4814+0.007j), (1.4221+0.0061j), (1.3643+0.0052j), (1.3076+0.0019j)]; n=320: [(1.9075+0.0104j), (1.8494+0.0021j)]; n=400: [(1.9207+0.0073j)]; n=600: [(1.9401+0.0021j)]
[INFO] thm:main(a),(b): S2 = (1-24w/17+20w^2/17)(1+32w/13+20w^2/13)/(1-20w/21): n x (distance from zeta_i to the nearest zero) and bulk overshoot, n=1200  -- [1.595, 1.595, 4.169, 4.169]; overshoot 0.0
   S3 n=10: real 10, nonreal 0 (certified: True); predicted nonreal 4; max dist to zeta_i 2.57e-01; bulk overshoot 0.00e+00
   S3 n=20: real 18, nonreal 2 (certified: True); predicted nonreal 4; max dist to zeta_i 1.48e-01; bulk overshoot 0.00e+00
   S3 n=40: real 38, nonreal 2 (certified: True); predicted nonreal 4; max dist to zeta_i 7.33e-02; bulk overshoot 0.00e+00
   S3 n=80: real 76, nonreal 4 (certified: True); predicted nonreal 4; max dist to zeta_i 3.30e-02; bulk overshoot 0.00e+00
   S3 n=160: real 156, nonreal 4 (certified: True); predicted nonreal 4; max dist to zeta_i 1.44e-02; bulk overshoot 0.00e+00
   S3 n=320: real 316, nonreal 4 (certified: True); predicted nonreal 4; max dist to zeta_i 8.29e-03; bulk overshoot 0.00e+00
   S3 n=400: real 396, nonreal 4 (certified: True); predicted nonreal 4; max dist to zeta_i 6.84e-03; bulk overshoot 0.00e+00
   S3 n=600: real 596, nonreal 4 (certified: True); predicted nonreal 4; max dist to zeta_i 4.68e-03; bulk overshoot 0.00e+00
   S3 n=800: real 796, nonreal 4 (certified: True); predicted nonreal 4; max dist to zeta_i 3.54e-03; bulk overshoot 0.00e+00
[PASS] thm:main: S3 = q(5w/6), q Gaussian sample (seed 4): the certified number of nonreal zeros of Q_n equals the number (4) of nonreal zeros of a in D for all tested n >= n_obs (observed onset reported; the theorem only asserts some n_0)  (interval arithmetic (Arb))  -- n_obs = 80; n=10: 10R+0C; n=20: 18R+2C; n=40: 38R+2C; n=80: 76R+4C; n=160: 156R+4C; n=320: 316R+4C; n=400: 396R+4C; n=600: 596R+4C; n=800: 796R+4C
[INFO] thm:main(a),(b): S3 = q(5w/6), q Gaussian sample (seed 4): n x (distance from zeta_i to the nearest zero) and bulk overshoot, n=800  -- [2.832, 2.832, 2.071, 2.071]; overshoot 0.0
[PASS] thm:main: cross-check: Arb isolation of all roots of the monomial-basis Q_n (n=10,20,40, S1 and S2) gives the same nonreal counts  (interval arithmetic (Arb))  -- S1 n=10: 2; S1 n=20: 2; S1 n=40: 2; S2 n=10: 4; S2 n=20: 10; S2 n=40: 16
== part3a_exact_count: 5 passed, 0 failed, 15.2 s