Unconditional Proof of Legendre's Conjecture

We present a fully rigorous proof of Legendre’s conjecture for all integers using explicit stratified sieve method. By carefully applying a linear sieve at level followed by a second sieve for primes in the range , we obtain explicit lower and upper bounds on surviving numbers and semiprimes in the interval . All constants are fully explicit and derived from classical results in sieve theory, Rosser–Schoenfeld bounds, and Dusart (2010) estimates on . Our method guarantees that at least one prime exists in each interval for , with smaller verified computationally. In addition, this approach yields explicit bounds on prime gaps, providing rigorous support for Andrica’s conjecture, and can be adapted to establish new effective limits for conjectures of Oppermann and Cramér on prime distributions.