Counting Residue Pairs in a Sieve Cycle, and the Frequency of Prime Gaps

Abstract. Counting residue pairs in the sieve cycle W = 30030, and the frequency of prime gaps. The pair count is an exact theorem with no free parameters; it is the truncated Hardy-Littlewood singular series. The empirical half is a worked example of a fitted parameter that looks real and is not: a stretched exponent that fits better and transfers better out of sample, yet moves seven times more with the arbitrary fitting window than with six decades of height.