Translational Tower Sieve and Precise Period Cutting: A Proof of the Star-Moon Conjecture and Bypassing the Parity Obstacle

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 the Translational Tower Sieve with Precise Period Cutting that when the observation window length $L \ge P_t^2/2$, the number of days when all planets shine simultaneously satisfies
$$M_t(N, L) \ge \frac{L}{6} \prod_{i=3}^t \frac{P_i-2}{P_i} - C_1 t \ln t,$$
where $C_1$ is an absolute constant. In particular, $\lim_{t \to \infty} M_t(N, L) = \infty$, i.e., there are infinitely many such days. This conjecture directly implies classical problems such as the Twin Prime Conjecture, Goldbach's Conjecture, and Polignac's Conjecture.