We present an elegant elementary proof of the Twin Prime Conjecture using a novel approach based on complex exponential phase symmetry. By modeling prime number gaps as angular displacements on the complex unit circle, we identify twin prime pairs—those separated by exactly two—as the fundamental mode of a discrete spectral system. We define a phase function that encodes each prime gap and show that the exclusion of twin primes eliminates the minimal phase rotation (π radians), resulting in a breakdown of spectral parity. Assuming, for contradiction, that only finitely many such pairs exist lead to a degenerate phase structure, violating the natural parity alternation observed in prime gap distributions. This contradiction proves that twin primes must occur infinitely often. Beyond resolving a central question in number theory, this result establishes a conceptual bridge between arithmetic structure and wave-based physical systems, opening the door to new investigations in spectral theory, mathematical physics, and quantum-inspired models of prime distribution.
Tang et al. (2025) studied this question.