Bypassing the Parity Obstacle: A Proof of Goldbach's Conjecture via Period Block Decomposition

This paper proposes a fixed-window tower sieve based on same-period block deviation control and single-block deviation control, and proves the Goldbach conjecture. We take the window
\[
A=[0,L],\qquad \frac{P_t^2}{2}\le L=N-3\le \frac{P_{t+1}^2}{2}-4.
\]
The sieve covers up to \(P_t\). We cut \(|A|\) by \(Q_k\). The core lemma proves that \textbf{for any two \(Q_k\)-blocks of the same length and the same periodic structure, the difference of the number of bad points at the \(i\)-th layer is \(\le 16\)}. This lemma depends only on left-right endpoint residuals and Cartesian-product phase differences; the deviations from subsequent-layer conditions cancel in sign over \(\mathcal R_k\) and do not produce \(O(i-k)\) accumulation. This yields the deletion-count bound \(D_i\le 8N_{i-1}/P_i\). Iterating gives \(N_t\gg N/(\ln N)^{10}\to\infty\), and hence each survivor \(x\in A\) corresponds to a Goldbach representation \((N-x)+(N+x)=2N\). The proof uses only elementary number theory, the Chinese remainder theorem, Chebyshev's theorem, and Mertens' theorem.