[cert] rates: t3=4053/5120 tau3=9/40 t4=609/1024 j4=1/40 ; rho*(level 4) = 13833/5120 ; equals claimed rho? True
[cert] level 2: q>=0 True, sum q=1 True, failing partitions: none
[cert] level 3: q>=0 True, sum q=1 True, failing partitions: none
[cert] level 4: q>=0 True, sum q=1 True, failing partitions: none
[cert] level 5: q>=0 True, sum q=1 True, failing partitions: none
[cert] level 6: q>=0 True, sum q=1 False, failing partitions: [(4, 2), (3, 3), (3, 2, 1), (2, 2, 2), (2, 2, 1, 1)]
[cert] level 7: q>=0 True, sum q=1 False, failing partitions: [(5, 2), (4, 3), (4, 2, 1), (3, 3, 1), (3, 2, 2), (3, 2, 1, 1), (2, 2, 2, 1), (2, 2, 1, 1, 1)]
[cert] closed-form (Z-distribution) sums vs brute-force type sums, m=2..8: mismatches = 0
[cert] sign of Delta_n, n=4..40: 00++++++++++-------------------------
[cert] Delta_n < 0 for n in [16, 17, 18] ... [40] ; Delta_n > 0 for n in [6, 7, 8, 9, 10, 11, 12, 13, 14, 15]
[cert] n=40: Bin(n)W(n-1)/C(n,2) = 0.212499 (aM = 0.212500), Phi(n-1)X(n)/C(n,2) = 5.141e-05
[ThmC] delta_(1/3,1/3,1/3), rho=1: Bin(5)=0 X(5)=10/9 ; level-4 failing: [(2, 2)] ; level-5 failing: [(3, 2), (2, 2, 1)]
[ThmC] delta_(1/3,1/3,1/3), rho=5/2: Bin(5)=0 X(5)=10/9 ; level-4 failing: [(2, 2)] ; level-5 failing: [(3, 2), (2, 2, 1)]
[ThmD] atoms=[('2', ('5/7', '1/7')), ('3', ('5/9', '2/9', '1/9'))] rho=7/4 predicted eventual sign + ; n=4..60: +++++++++++++++++++++++++++++++++++++++++++++++++++++++++
[ThmD] atoms=[('2', ('1/6',))] rho=1 predicted eventual sign - ; n=4..60: ---------------------------------------------------------
[ThmD] atoms=[('1', ('2/9', '1/9', '1/9')), ('2', ('5/11', '1/11')), ('3', ('5/8', '1/4', '1/8'))] rho=4/3 predicted eventual sign + ; n=4..60: -------++++++++++++++++++++++++++++++++++++++++++++++++++
[ThmD] atoms=[('1', ('1/3', '4/15', '2/15'))] rho=3/2 predicted eventual sign + ; n=4..60: +++++++++++++++++++++++++++++++++++++++++++++++++++++++++
[ThmD] atoms=[('4', ('2/5', '2/5')), ('3', ('4/15', '1/5', '1/5')), ('1', ('2/5',))] rho=3/4 predicted eventual sign - ; n=4..60: +---+++--------------------------------------------------
[ThmD] atoms=[('1', ('5/16', '5/16', '3/16')), ('3', ('4/9', '1/9', '1/9'))] rho=8 predicted eventual sign + ; n=4..60: +++++++++++++++++++++++++++++++++++++++++++++++++++++++++
[ThmD] atoms=[('3', ('2/7', '2/7', '3/14'))] rho=1/4 predicted eventual sign + ; n=4..60: +++++++++++++++++++++++++++++++++++++++++++++++++++++++++
[ThmD] atoms=[('2', ('3/8', '1/4', '1/8')), ('2', ('4/7', '1/7'))] rho=2 predicted eventual sign + ; n=4..60: +++++++++++++++++++++++++++++++++++++++++++++++++++++++++
[ThmD] atoms=[('3', ('1',)), ('4', ('1/2', '1/4')), ('1', ('2/3',))] rho=4 predicted eventual sign - ; n=4..60: -+++++++++++++++++++++++++++++++-------------------------
[ThmD] atoms=[('3', ('5/8',)), ('3', ('1/2',))] rho=1/4 predicted eventual sign - ; n=4..60: ---------------------------------------------------------
[ThmD] prediction agrees on 10/10 random examples
[ThmB] a=1/11 rho=1 atoms=2 : n=4..60 ++-------------------------------------------------------
[ThmB] a=1/10 rho=7/2 atoms=1 : n=4..60 +++------------------------------------------------------
[ThmB] a=2/3 rho=4 atoms=2 : n=4..60 ---------------------------------------------------------
[ThmB] a=3/14 rho=2/3 atoms=1 : n=4..60 ++-------------------------------------------------------
[ThmB] a=2/17 rho=8/3 atoms=2 : n=4..60 +++------------------------------------------------------
[ThmB] a=2/13 rho=2 atoms=1 : n=4..60 ++-------------------------------------------------------
[ThmB] a=1/8 rho=1/4 atoms=1 : n=4..60 ---------------------------------------------------------
[ThmB] a=1 rho=3/2 atoms=2 : n=4..60 ++-------------------------------------------------------
[ThmB] eventually negative on 8/8
