Jacobsthal Function Bound Analysis

**Jacobsthal Function Bound Analysis** This paper presents an original refinement of upper bounds for the Jacobsthal function g(n), which describes the longest possible gap between consecutive integers all divisible by primes from a given set. Using methods from combinatorial number theory, residue classes, and probabilistic modeling, we derive a new general upper bound of g(n) ≤ φ(n) log(n)^2. These improvements lay groundwork for further explorations in prime gaps and modular sieving theory.