On the Limits of Residue-Based Pre-Screening for Large Prime Gaps

We study pre-screening filters for large-prime-gap searches in primorial-structured windows. Such a filter is one-sided: it may prove that a window contains no gap of a given target length, and it must have zero false negatives. We first show that the naive covering filter is sound, and in fact vacuous: it eliminates no window that contains a prime, and therefore no window that contains a gap. We then prove an indistinguishability proposition via Dirichlet's theorem: because every reduced residue class modulo q contains infinitely many primes and composites, no sound residue-only filter can certify compositeness of a gap endpoint from residues alone, when the residue is coprime to q. We complement this with empirical computational experiments confirming the filter's vacuity, and a density heuristic showing that a sound residue-only filter can eliminate only a vanishingly small proportion of windows. Finally, we outline algorithmic successors based on partial factorization that bypass these limitations. The results are negative for residue-only methods, and they are intended to redirect effort toward mechanisms that use information beyond small-prime residues.