1788e61b6b308069896462e3dba7e185  a26e_c19_text_check.py
Theorem 3.4 (d): C_K = min over (0, q_K] of C(q), K = 4, 5, 6
PASS K=4: g((K-1)s(q_K)) = g(K s(q_K)) (|diff| < 1e-70)
   C_4 = 0.853343568496049   sqrt(2/e) - C_4 = 0.00442031646466
PASS K=4: C_K = 0.85334356... (truncated)
PASS K=4: sqrt(2/e) - C_K = 0.0044203 (correctly rounded to 7 places)
PASS K=4: example decimal 0.8533435 <= C_K
PASS K=4: the 8-digit truncation 0.85334356 <= C_K
PASS K=5: g((K-1)s(q_K)) = g(K s(q_K)) (|diff| < 1e-70)
   C_5 = 0.855100455470819   sqrt(2/e) - C_5 = 0.00266342948989
PASS K=5: C_K = 0.85510045... (truncated)
PASS K=5: sqrt(2/e) - C_K = 0.0026634 (correctly rounded to 7 places)
PASS K=5: example decimal 0.855100 <= C_K
PASS K=5: the 8-digit truncation 0.85510045 <= C_K
PASS K=6: g((K-1)s(q_K)) = g(K s(q_K)) (|diff| < 1e-70)
   C_6 = 0.855984489173815   sqrt(2/e) - C_6 = 0.00177939578689
PASS K=6: C_K = 0.85598448... (truncated)
PASS K=6: sqrt(2/e) - C_K = 0.0017794 (correctly rounded to 7 places)
PASS K=6: example decimal 0.855984 <= C_K
PASS K=6: the 8-digit truncation 0.85598448 <= C_K
   note: 0.0017794 is rounded up from 0.00177939…; a larger margin is the safe direction
Remarks (2)-(3): a_4 = 1/sqrt(4 s(q4)), b_4 = 1/sqrt(3 s(q4)), {g >= C_4} = [3 s(q4), 4 s(q4)]
   a_4 = 1.31834356235054   b_4 = 1.52229202121499
PASS a_4 = 1.3183436 (correctly rounded)
PASS b_4 = 1.5222920 (correctly rounded)
PASS g(3 s(q4)) = g(4 s(q4)) = C_4 (|diff| < 1e-70)
PASS 3 s(q4) < 1/2 < 4 s(q4) (g is unimodal with peak 1/2, so {g >= C_4} = [3 s(q4), 4 s(q4)])
   3 s(q4) = 0.431523108678   4 s(q4) = 0.575364144904
PASS "[0.4315, 0.5754]": both endpoints correctly rounded to 4 places
PASS "[0.4315, 0.5754]" contains [3 s(q4), 4 s(q4)] (so "iff" holds only up to rounding)
PASS a_4 = 1/sqrt(4 s(q4)) and b_4 = 1/sqrt(3 s(q4)) are the Dirac positions with g(1/a^2) = C_4

ALL CHECKS PASSED

real	0m0.073s
user	0m0.065s
sys	0m0.008s
