Five Discoveries in Prime Number Geometry: Phase Diagrams, Modular Constraints, and Quasicrystal Signatures

Abstract: We present computational evidence for five structural properties of prime number families. First, we construct a phase diagram of prime family intersections, revealing that Twin and Cousin primes are geometrically orthogonal (only 1 shared prime, ), while Twin and Sophie Germain primes exhibit significant correlation (1.61 expected overlap). Second, we prove that for all Sophie Germain primes with , 100% reside in exactly three residue classes modulo 30; this is an algebraic law, not a statistical trend. Third, we demonstrate that the emirp 198-lattice exhibits both massive clustering (88.3% duplicate indices) and a significant quasicrystal signature (structure factor ratio 1.12 at primorial wave vectors, via Monte Carlo), connecting to Torquato’s 2018 model. Fourth, we discover that even-digit emirps follow a separate lattice pattern (gaps divisible by 9, not 198). Fifth, we show that emirps exist in all prime bases (3,837 pairs across 6 bases), refuting the claim that the emirp relationship is base-10 specific.