Proof Study: The Symmetry Invariance of Twin Primes through Sine-Wave Reflection in Primorial Space

Author: Daniel SpeckmannDate: March 26, 2026 Abstract: This paper presents a geometric-combinatorial approach to the Twin Prime Conjecture, introducing the "Prime Reflection Wave" (PRW) as a fundamental model for prime distribution. Unlike traditional linear counting methods, the PRW treats prime numbers as reflection axes within a symmetric product space defined by primorials (Mₙ). The study establishes that every primorial space possesses an inherent mirror symmetry at its midpoint (Mₙ/2), creating a framework where the distribution of coprimality is strictly invariant. By analyzing the expansion rate of "free twin channels" (Kₙ), defined by the product of (pᵢ - 2), the author demonstrates that the available geometric nodes for twin primes grow multiplicatively and tend toward infinity (Kₙ -> inf). The core of the proof lies in the principle of Symmetry Invariance: the infinite nature of prime reflections, coupled with the mandatory symmetry of the product space, necessitates the existence of twin primes as stable, invariant nodes. A termination of twin primes would represent a collapse of the underlying symmetry of the multiplication-based number system. Consequently, the infinity of twin primes is proven to be a geometric necessity within the expanding primorial architecture.