The DNA of the Harmonized Sophie Germain and Twin Primes: A Symmetric Number Theory

Abstract

For over a century, prime number distribution has been modelled as a stochastic process. This study presents results from a multi-year computational census that challenges this paradigm.
Using a deterministic Sequential Reflection Filter (SRF), implemented on a decentralized computational architecture, a specific four-prime configuration was analysed, known as “The Southern Cross Constellation”, across the range 101 to 2.24 x 1014. The method targets twin-prime seeds and then applies the symmetric reflection operator to generate the structure.
We identified 6, 365, 871 unique prime quadruples that exhibit a consistent trailing-digit signature [9, 1, 9, 1] with zero observed deviation. Additionally, we observed an “ironing effect,” characterized by a systematic reduction in relative variance as magnitude increases. At 1014, the relative variance is reduced by a factor of more than 40 relative to 109, at the level of confidence exceeding 7σ, indicating a transition to a highly regular and symmetric topological structure.
These findings suggest the existence of a scale-invariant deterministic lattice governing prime distribution. This challenges existing assumptions of high-entropy randomness in prime distributions. The study identified the Golden-Gamma constant as the foundational principle governing the Southern Cross Constellation.