A deterministic arithmetic reformulation of the mod 6 lattice revealing geometric symmetry in the distribution of primes. Rotational symmetry when applied to these residuals visits all of the non-prime numbers stepwise, which we refer to as "hops", that generates a deterministic arithmetic framework that reproduces the sequence of primes and their gaps. We present a framework for understanding the distribution of primes using modular arithmetic and iterative hop sequences. Visual patterns such as the Ulam spiral are shownto arise naturally from rotational symmetries within this framework. We provide both anintuitive explanation and a formal arithmetic treatment that reproduces the sequence ofprimes and their gaps. As first noted by Ulam and popularized by Gardner, the arrangement of integersin a spiral lattice reveals that prime numbers tend to cluster along diagonal lines. However,this observation alone does not explain the *mechanism* of the clustering. The hop-based interpretation offers a natural explanation: primes occupy loci defined by arithmetic propagation rather than arbitrary geometric coincidence.