A Proof of the Star-Moon Conjecture by Fixed-Window Tower Sieve

The Star-Moon Conjecture is a number-theoretic conjecture inspired by astronomical observations. In a distant galaxy, $t$ planets orbit a star with prime periods $P_1=2, P_2=3, P_3=5, \ldots, P_t$ days. A mysterious satellite obscures the $i$-th planet on observation day $d$ if $d \equiv R_i \pmod{P_i}$ or $d \equiv -R_i \pmod{P_i}$, where $R_i = N \bmod P_i$ and $N$ is a lucky number. In this paper, we prove using a fixed-window tower sieve and translation construction that when the observation window length $L \ge P_t^2/2$, the number of days when all planets shine simultaneously satisfies $M_t(N, L) > \frac{L}{6} \prod_{i=3}^t \frac{P_i - d_i - 1}{P_i} - 2t$, where $d_i = |B_i(N)|$. In particular, when $L$ is sufficiently large, $M_t(N, L) > 0$, and as $t \to \infty$, $M_t(N, L) \to \infty$. This conjecture directly implies classical problems such as the Twin Prime Conjecture, Goldbach's Conjecture, and Polignac's Conjecture. The paper also discusses in detail how this method successfully overcomes the parity obstacle in classical sieve methods.