[Lemma 1] (pf)^(n)(0) = sum_i c_i n^(i) a_{n-i} verified symbolically with sympy (4 random p, n <= 9): PASS
[A] J = 150, prefix = False; degrees of g_1..g_20: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]; n_j first 12: [2, 8, 19, 36, 58, 86, 119, 158, 202, 251, 306, 366]; n_J = 60830
[Thm A step 4] n*a_(n-1) in {0} u {|w|>=2} for every n <= 60831: PASS  (150 non-zero entries; min |n a_(n-1)|^2 over them = 18.0000)
[Rem A'] p = (1+0i)z^2: values in {0} u {|w| >= 2|c|}: PASS
[Rem A'] p = (1/2+0i)z^1: values in {0} u {|w| >= 2|c|}: PASS
[Rem A'] p = (0+3i)z^3: values in {0} u {|w| >= 2|c|}: PASS
[Rem A'] p = (1/7-1/7i)z^5: values in {0} u {|w| >= 2|c|}: PASS
[Rem A'] p = (-1/50+0i)z^1: values in {0} u {|w| >= 2|c|}: PASS
[Rem A'] p = (1/1000+1/1000i)z^8: values in {0} u {|w| >= 2|c|}: PASS
[Rem B2] f_A, p = 1+z: 150 values n with |s_n - 1| < 1; all at block starts n_j: True; number of block starts that violate: 150 of 150
[B] J = 250, prefix = False; degrees of g_1..g_20: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]; n_j first 12: [5, 15, 32, 56, 87, 124, 169, 221, 281, 348, 424, 508]; n_J = 323434; non-zero entries: 31625
[Thm B] p = (1+0i)z^1 p = z: d = 1, j_0' = 6, n_* = 118; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (1+0i)z^0 + (1+0i)z^1 1+z: d = 1, j_0' = 8, n_* = 213; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (1/5+0i)z^0 + (1+0i)z^1 1/5+z: d = 1, j_0' = 32, n_* = 3982; violations at n >= n_*: 0 -> PASS; violations below n_*: 1 (last at n = 4, zone 1)
[Thm B] p = (1+0i)z^0 + (-1+0i)z^1 + (1+0i)z^2 1-z+z^2: d = 2, j_0' = 10, n_* = 338; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (2+0i)z^0 + (1+0i)z^3 2+z^3: d = 3, j_0' = 10, n_* = 338; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (0+1i)z^0 + (1+1i)z^2 + (-1+0i)z^4 i+(1+i)z^2-z^4: d = 4, j_0' = 12, n_* = 496; violations at n >= n_*: 0 -> PASS; violations below n_*: 1 (last at n = 5, zone 1)
[Thm B] p = (1/3+0i)z^0 + (2+0i)z^1 1/3+2z: d = 1, j_0' = 26, n_* = 2554; violations at n >= n_*: 0 -> PASS; violations below n_*: 1 (last at n = 4, zone 1)
[Thm B] p = (1+0i)z^0 + (0+1/2i)z^1 + (1/4+0i)z^5 1+(i/2)z+z^5/4: d = 5, j_0' = 30, n_* = 3468; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (1/50+0i)z^1 z/50 (tiny monomial): d = 1, j_0' = 202, n_* = 205100; violations at n >= n_*: 0 -> PASS; violations below n_*: 6 (last at n = 57, zone 4)
[Thm B] p = (1/4+0i)z^0 + (-1+0i)z^1 + (1+0i)z^2 (z-1/2)^2: d = 2, j_0' = 34, n_* = 4533; violations at n >= n_*: 0 -> PASS; violations below n_*: 1 (last at n = 4, zone 1)
[Thm B] p = (3+0i)z^0 + (3+0i)z^1 + (3+0i)z^2 + (3+0i)z^3 + (3+0i)z^4 + (3+0i)z^5 + (3+0i)z^6 + (3+0i)z^7 + (3+0i)z^8 3(1+z+...+z^8), d = 8: d = 8, j_0' = 20, n_* = 1462; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (1+0i)z^0 + (1+0i)z^2 1+z^2 (c_1 = 0): d = 2, j_0' = 8, n_* = 213; violations at n >= n_*: 0 -> PASS; violations below n_*: 1 (last at n = 5, zone 1)
[Thm B] p = (1+0i)z^1 + (1+0i)z^2 z+z^2 (c_0 = 0): d = 2, j_0' = 8, n_* = 213; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (7+0i)z^0 + (1/7+0i)z^1 7+z/7 (small leading coefficient): d = 1, j_0' = 128, n_* = 77390; violations at n >= n_*: 0 -> PASS; violations below n_*: 2 (last at n = 15, zone 2)
[Thm B] p = (1/40+0i)z^0 + (1+0i)z^1 1/40+z (small constant): d = 1, j_0' = 242, n_* = 301534; violations at n >= n_*: 0 -> PASS; violations below n_*: 2 (last at n = 13, zone 2)
[Thm B] p = (1+1i)z^0 + (-1+1i)z^1 + (0-1/3i)z^2 + (2/5+3/5i)z^3 complex cubic: d = 3, j_0' = 44, n_* = 7869; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (-1+0i)z^0 + (1+0i)z^1 z-1: d = 1, j_0' = 8, n_* = 213; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (1+0i)z^0 + (-2+0i)z^1 + (1+0i)z^2 (z-1)^2: d = 2, j_0' = 12, n_* = 496; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (1/10+0i)z^0 + (1/10+0i)z^1 + (1/10+0i)z^2 + (1/10+0i)z^3 + (1/10+0i)z^4 + (1/10+0i)z^5 + (1/10+0i)z^6 all 1/10, d = 6: d = 6, j_0' = 54, n_* = 12198; violations at n >= n_*: 0 -> PASS; violations below n_*: 3 (last at n = 13, zone 2)
[Thm B] p = (2/3+3i)z^0 + (0-4i)z^1 + (-3/2+3i)z^2 + (-6+0i)z^3 + (2-4i)z^5 random #0: d = 5, j_0' = 18, n_* = 1168; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (-1/3-5/2i)z^0 + (4+1i)z^1 + (-1/4-2i)z^2 + (2-2i)z^3 + (3/4+5i)z^5 + (6-3/2i)z^6 random #1: d = 6, j_0' = 30, n_* = 3468; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (4/3+3/2i)z^0 + (-1/4+0i)z^3 random #2: d = 3, j_0' = 41, n_* = 6766; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (1+1/4i)z^0 + (1-4/3i)z^1 random #3: d = 1, j_0' = 12, n_* = 496; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (5/2+2i)z^0 + (-4/3+1i)z^2 random #4: d = 2, j_0' = 14, n_* = 686; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (0+6i)z^0 + (2/3-6i)z^1 + (4/3-1i)z^3 + (3/4+1/2i)z^4 + (-5/4+3i)z^5 random #5: d = 5, j_0' = 60, n_* = 15286; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (-5/2-1i)z^1 + (0-2i)z^3 + (5/3+0i)z^4 random #6: d = 4, j_0' = 11, n_* = 413; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (1/3+6i)z^0 + (4-5/2i)z^1 random #7: d = 1, j_0' = 8, n_* = 213; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (3-2i)z^0 + (-3/2+3/4i)z^1 + (6-5i)z^2 random #8: d = 2, j_0' = 27, n_* = 2769; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (-2+2i)z^0 + (-5/2-5i)z^1 + (1/2+1i)z^2 + (4/3+2i)z^3 + (1/2-4i)z^4 random #9: d = 4, j_0' = 46, n_* = 8654; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (-1-5/2i)z^0 + (-5/3+0i)z^2 + (5/2-1i)z^3 + (1+4i)z^4 + (-1-1i)z^6 random #10: d = 6, j_0' = 36, n_* = 5123; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (-5+3i)z^1 random #11: d = 1, j_0' = 4, n_* = 52; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (-3/2+1/2i)z^0 + (-5+4/3i)z^1 + (-5/2+0i)z^2 random #12: d = 2, j_0' = 18, n_* = 1168; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (1-2i)z^0 + (0+1i)z^1 + (-1/2+3/2i)z^2 + (3/2+2i)z^3 random #13: d = 3, j_0' = 23, n_* = 1968; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (5/2+4/3i)z^0 + (-5/2+2i)z^1 + (3/4+0i)z^2 random #14: d = 2, j_0' = 30, n_* = 3468; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (-2/3-3/2i)z^1 + (5/3+3/2i)z^2 + (5/3+2i)z^4 random #15: d = 4, j_0' = 15, n_* = 794; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (-5+2i)z^0 + (-1/4+1i)z^2 + (0-1i)z^4 + (-1/3+1i)z^5 random #16: d = 5, j_0' = 26, n_* = 2554; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (1/3+2/3i)z^2 + (5/3-1/2i)z^3 + (2+1/4i)z^4 random #17: d = 4, j_0' = 23, n_* = 1968; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (0-3/2i)z^0 + (2-3/2i)z^1 + (0+3/2i)z^3 + (5-1i)z^4 + (-2+3/2i)z^5 random #18: d = 5, j_0' = 24, n_* = 2154; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (-2-5/2i)z^0 + (1+1/3i)z^1 random #19: d = 1, j_0' = 16, n_* = 910; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (1/2+1i)z^1 random #20: d = 1, j_0' = 7, n_* = 162; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (5/4+3/4i)z^0 + (-4+5/4i)z^1 + (2+1i)z^3 random #21: d = 3, j_0' = 20, n_* = 1462; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (3+3i)z^0 + (4/3+1i)z^1 random #22: d = 1, j_0' = 16, n_* = 910; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (-5/2-1i)z^0 + (-1-1i)z^1 + (-5+4i)z^3 + (2+1/2i)z^4 + (-3-5/3i)z^5 random #23: d = 5, j_0' = 48, n_* = 9479; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] p = (1-4i)z^0 + (5/2-6i)z^1 + (0-3i)z^2 + (-3-3/2i)z^3 + (2+1/3i)z^4 + (1+0i)z^5 + (0+3i)z^6 random #24: d = 6, j_0' = 59, n_* = 14745; violations at n >= n_*: 0 -> PASS; violations below n_*: 0 (last at n = None, zone None)
[Thm B] random polynomials tested: 25
[tails A] sum_(i>j) T_i <= 2^-j for 1 <= j < 60: PASS (max of 2^j * sum = 2.985e-03)
[tails B] sum_(i>j) T_i <= 2^-j for 1 <= j < 60: PASS (max of 2^j * sum = 3.201e-03)
[actual tail, exact, |a| upper bounds, zones <= 40] j = 1: A: 1.389e-03, B: 4.081e-04  (bound 2^-j = 5.000e-01)
[actual tail, exact, |a| upper bounds, zones <= 40] j = 2: A: 1.026e-04, B: 8.303e-05  (bound 2^-j = 2.500e-01)
[actual tail, exact, |a| upper bounds, zones <= 40] j = 3: A: 7.261e-07, B: 1.195e-05  (bound 2^-j = 1.250e-01)
[actual tail, exact, |a| upper bounds, zones <= 40] j = 4: A: 3.130e-08, B: 2.267e-06  (bound 2^-j = 6.250e-02)
[actual tail, exact, |a| upper bounds, zones <= 40] j = 6: A: 1.100e-11, B: 1.513e-07  (bound 2^-j = 1.562e-02)
[float A] j = 1: max |f^(n_j) - g_j| at 8 points of |z| = j (zones <= 60): 0.001389  (bound 2^-j = 5.00e-01)
[float A] j = 2: max |f^(n_j) - g_j| at 8 points of |z| = j (zones <= 60): 7.256e-5  (bound 2^-j = 2.50e-01)
[float A] j = 3: max |f^(n_j) - g_j| at 8 points of |z| = j (zones <= 60): 5.135e-7  (bound 2^-j = 1.25e-01)
[float A] j = 5: max |f^(n_j) - g_j| at 8 points of |z| = j (zones <= 60): 1.222e-10  (bound 2^-j = 3.12e-02)
[float A] j = 8: max |f^(n_j) - g_j| at 8 points of |z| = j (zones <= 60): 2.897e-15  (bound 2^-j = 3.91e-03)
[float A] j = 12: max |f^(n_j) - g_j| at 8 points of |z| = j (zones <= 60): 4.371e-22  (bound 2^-j = 2.44e-04)
[float B] j = 1: max |f^(n_j) - g_j| at 8 points of |z| = j (zones <= 60): 0.0004074  (bound 2^-j = 5.00e-01)
[float B] j = 2: max |f^(n_j) - g_j| at 8 points of |z| = j (zones <= 60): 8.299e-5  (bound 2^-j = 2.50e-01)
[float B] j = 3: max |f^(n_j) - g_j| at 8 points of |z| = j (zones <= 60): 1.195e-5  (bound 2^-j = 1.25e-01)
[float B] j = 5: max |f^(n_j) - g_j| at 8 points of |z| = j (zones <= 60): 1.713e-6  (bound 2^-j = 3.12e-02)
[float B] j = 8: max |f^(n_j) - g_j| at 8 points of |z| = j (zones <= 60): 2.945e-9  (bound 2^-j = 3.91e-03)
[float B] j = 12: max |f^(n_j) - g_j| at 8 points of |z| = j (zones <= 60): 1.004e-13  (bound 2^-j = 2.44e-04)
elapsed 49.1 s
ALL CHECKS PASSED
nice -n 10 ~/.cache/openmathproblems-audit/venv/bin/python -I vA_check.py 150  48.89s user 0.17s system 99% cpu 49.240 total
