On the Local Commutation Limits of Phase-Modulated Sieve Operators

Purpose and statusWrite H1 = lim infn→∞(pn+1 − pn). The twin-prime conjecture is the assertion H1 = 2. It isopen. Zhang [ 8 ] proved H1 finite. Maynard [3 ] and Polymath8b [4 ] proved H1 ≤ 246 by amultidimensional sieve on an admissible 50-tuple of that diameter. Stadlmann [ 6 ] announces 240;that note is cited here as a preprint, not as the established record. Under the Elliott–Halberstamconjecture one has H1 ≤ 12; under a generalized form, H1 ≤ 6 [3 , 4]. None of those statementsis H1 = 2.The object of this note is a class of local operators that were examined as candidate bypassesof the parity barrier at k = 2. The conclusion is negative. The integers 42 and 208 that appearbelow are parameters of the specific phase that was tested. They are not constants of theMaynard variational problem. The barrier holds for any function of n mod M , for any fixedmodulus M