A Proof of the Twin Prime Conjecture Based on the Multiple-Sieve Compensation Method

The twin prime conjecture asserts that there are infinitely many pairs of primes differing by $2$. This paper presents a deterministic proof. We work directly on the observation interval $A=[1,P_t^2]$ and apply the multiple-sieve compensation method: by introducing a third bad residue, utilizing the magnitude comparison between the compensation term $D = |A_0|$ and the deviation $E_3$ — $D$ is approximately $N/P_i$, which is much larger than $|E_3|$ (since $E_3$ is a linear combination of three $\delta$'s with an absolute constant upper bound, while $D$ grows with $N$) — we rigorously prove that at each sieve level the number of survivors satisfies $N^{(2)} > N \cdot (P_i-3)/P_i$. Iteration yields $N(A) > \frac{P_t^2}{6} \prod_{i=3}^t (1-3/P_i)$. By Mertens' theorem, the right-hand side tends to infinity, proving the existence of infinitely many twin primes.