Translational Tower Sieve and Precise Period Cutting: An Elementary Proof of the Twin Prime Conjecture and Bypassing the Parity Obstacle

The Twin Prime Conjecture asserts that there are infinitely many pairs of primes differing by 2. This paper presents a new method—the Translational Tower Sieve with Precise Period Cutting. Using the square interval property, we transform the problem into finding integers $x$ in the interval $A=[1,P_t^2-2]$ satisfying $x\not\equiv1\pmod2$ and $x\not\equiv\pm1\pmod{P_i}$ ($i\ge2$). We construct a base interval $B=[1,Q_t]$ (a complete residue system) and a translated interval $C=Q_t+A$, and define the total interval $U=B\cup C$. Using the complete residue system property of $B$, we prove that the number of survivors on $B$ is exactly $Q_tA_t$. Using precise period cutting of $C$, we prove that at each sieving layer, the deviation in complete periods is zero, while the deviation in incomplete periods is $O(\ln t)$. From this we establish the recurrence relation $N_i\ge N_{i-1}(1-2/P_i)-C_1\ln t$, and iteration yields the lower bound $N_t\ge c P_t^2/(\ln P_t)^2-O(t\ln t)$. Applying Mertens' theorem gives $N_t\to\infty$, thus proving the Twin Prime Conjecture. This method uses only elementary number theory and successfully overcomes the parity obstacle in classical sieve methods, relying on no unproven conjectures.