A Sieve-Operational Formulation of Prime Gaps

This work introduces a local, sieve-operational formulation for understanding and determining prime gaps. Rather than approaching the problem through global asymptotics or probabilistic models, the study investigates whether the next prime gap can be derived from a forward, divisibility-based elimination process. The analysis begins by testing a wide range of conventional hypotheses, including regression models, memory-based dependencies, pattern classification, and digit-based features. These approaches are shown to provide limited explanatory power when constrained to strictly causal information—i.e., using only information available at a given prime. Motivated by these limitations, the study reformulates the problem in terms of candidate generation and elimination. For a given prime p_n, successive even offsets are evaluated, and candidates are removed if divisible by primes up to a specified depth. The next prime is then identified as the first candidate that survives this process. Empirical results demonstrate that the critical factor governing predictive accuracy is the depth of the sieve. When the depth is scaled proportionally to sqrt(p_n), with a small correction term, the resulting operational rule achieves near-perfect accuracy across large datasets, reproducing prime gaps with an exact match rate exceeding 99.998% over the first 50,000 primes. The study provides a detailed experimental framework, introduces the concept of a survival landscape, and shows that prime gaps can be interpreted as the first position that survives all relevant divisibility constraints. This formulation does not constitute a formal proof or replace analytic number theory. Instead, it offers a complementary, local perspective on prime gaps, grounded in elementary arithmetic yet capable of producing highly accurate empirical results. Keywords: prime numbers, prime gaps, sieve methods, divisibility, computational number theory, experimental mathematics