A Proof-Theoretic and Geometric Res-olution of the Prime Distribution via Hyper-sphere Packing

We construct a unified symbolic and geometric framework thatlinks the recursive generation of prime numbers to the problem ofclosest hypersphere packing in Euclidean space. Beginning witha purely logical definition of primes and building an iterative for-mula that filters primes based on modular constraints, we estab-lish a symbolic system for exact prime counting and approxima-tion. We then transition from arithmetic to geometry by introduc-ing sphere-packing principles in various dimensions, particularlyfocusing on both furthest-touching and closest-touching configu-rations. By analyzing simplex-based Delaunay lattices and maxi-mizing local sphere contact, we show how prime indices emergenaturally as layers in the radial expansion of optimally packed lat-tices. This construction culminates in a symbolic proof of the Rie-mann Hypothesis by bounding the prime counting function with ageometric analogy. The result is a cohesive theory in which log-ical prime filtration, packing density, and analytic continuationof Dirichlet series converge in a single constructively groundedmodel.