The Geometry of Prime Conjectures - Unified Resolution via Torsion Singularity Theory

Abstract As an addendum to The Infinite Polyrhymic Dance of Primes, we demonstrate that classical prime conjectures — including Goldbach, Twin Primes, Sophie Germain, Mersenne, Polignac, Legendre, and Brocard — are not independent problems but geometric consequences of a unified framework: the quasi-periodic placement of prime numbers as torsion singularities arising from constructive interference among three incommensurate transcendental constants (π, φ, e). This framework, previously validated through prime counting predictions achieving 0.011 parts per million accuracy across the range 10⁶ ≤ x ≤ 10¹⁴, implies specific structural properties of the prime distribution. We show that each conjecture reduces to one of three question types—coverage, spacing, or intersection—all of which are resolved by the quasi-periodicity and incommensurability inherent in the torsion singularity model. Keywords: Prime distribution, Goldbach conjecture, Twin prime conjecture, Torsion singularities, Quasi-periodic systems, Harmonic analysis