A bound of 240 for gaps between primes

GPT 5.6 Sol made the discovery of the proof presented in this paper. We prove that there are infinitely many pairs of consecutive primes whose gap is at most 240, improving the previously published unconditional bound of 246. The analytic input is obtained by directly combining three distribution estimates of Maynard for primes in arithmetic progressions to large moduli. These estimates yield a nested distribution theorem in which the modulus is decomposed into three ordered factors, with the residue class modulo the first two factors fixed before summation over the third. We then construct a single non-negative Selberg sieve weight whose prime cross terms are compatible with this three-factor structure. The construction is based on an explicit symmetric source in forty-nine variables together with a finite packing scheme that assigns the relevant divisor contributions to three prescribed modulus ranges. The required variational inequality is certified using an explicit rational polynomial and exact rational arithmetic, including rigorous bounds for the energy lost outside the packable region. After a retreat and tensor approximation, the resulting smooth sieve sources satisfy the necessary bilinear asymptotics and produce infinitely many translates of an explicit admissible forty-nine-element set containing at least two primes. Since this set has diameter 240, the claimed bound for gaps between consecutive primes follows.