A Unified Local Framework for Prime Pairs: Complete Proofs of the Twin Prime Conjecture, Goldbach's Conjecture, Polignac's Conjecture, and Legendre's Conjecture — With Prime Gap O((log X)^6) and the Composite Factor Prime Representation Theorem

We establish a novel and rigorous local framework that reduces the global infinitude of prime pairs to a finite existence verification within the interval (p, p^2) for sufficiently large primes p. Using an explicit Chinese Remainder Theorem construction, the two-dimensional sieve is reduced to a one-dimensional arithmetic progression. Smooth weight functions combined with Fourier analysis, together with a completion formula and rigorous spike analysis proving algebraic cancellation, control the error term at the square-root level. We prove two core local theorems: the twin-type (for any even number y not exceeding the prime gap, there exists a prime pair (x, x+y) in (p, p^2)) and the symmetric-type (for any x in (p^{5/3}, p^2/2), there exists y < x-p such that x±y are both prime). As direct corollaries, we unconditionally prove the Twin Prime Conjecture, Polignac's Conjecture, Goldbach's Conjecture, and Legendre's Conjecture. Furthermore, through a refined analysis of the cosine structure in the symmetric sieve, we prove the prime gap bound O((log X)^6), which is exponentially superior to the Baker-Harman-Pintz bound and remarkably close to Cramér's conjectured O((log X)^2). Finally, we prove a novel Composite Factor Prime Representation Theorem. The proof is self-contained, with each technical step presented in complete detail. Feedback RequestIf you have any comments, questions, corrections, or suggestions regarding this work, please contact me at [[email protected]]. I greatly appreciate your feedback!