A Geometric-Spectral Framework for the Twin Prime Conjecture: Symmetry Invariance and Prime Reflection Waves

This paper proposes a novel resolution to the Twin Prime Conjecture by shifting the analytical paradigm from linear sieving to modular symmetry invariance within primorial product spaces $\mathbb{Z}/M_{n}\mathbb{Z}$. We introduce the Prime Reflection Wave (PRW), a dynamic counting function that interprets prime distributions as harmonic oscillations. The core of the framework addresses the classical "parity problem" of sieve theory through a Spectral Parity Filtration mechanism. By establishing a formal orthogonality between the structural symmetry of the primorial ring and the oscillatory noise of the Riemann Zeta non-trivial zeros, we demonstrate that semiprime interference is destructively suppressed. Consequently, twin prime pairs emerge as necessary, invariant resonance nodes of the integer distribution. Through numerical simulation and spectral analysis, we show that the expansion of free twin channels ($K_{n}$) possesses a strictly positive lower bound, ensuring the infinitude of twin primes. This work bridges the gap between analytic number theory and wave mechanics, suggesting that the distribution of primes is a direct consequence of geometric resonance in the vacuum of natural numbers.