This paper presents a novel geometric and analytical framework aimed at addressing the Twin Prime Conjecture, asserting the existence of infinitely many pairs of prime numbers differing by 2, such as (3, 5) and (11, 13).We project prime numbers onto a unit circle, with angles derived from the imaginary parts of the first 100 non-trivial zeros of the Riemann zeta function, defined as θpi = 2π P100n=1 sin(γn ln(pi)) γn mod 2π. By rotating this circle over 100 iterations and generating a binary sequence S(tk) based on a marking interval [0, π 2 , we identify a recurring pattern, "011," with a periodicity of 4 iterations. Numerical simulations across scales up to N = 1024 support this observation,while a formal variance-based contradiction proof argues that this 1 recurrence implies the infinitude of twin primes. A spectral analysis further validates the periodicity, and refined assumptions on the zeta zeros strengthen the theoretical foundation. This work diverges from traditional analytic methods, offering a geometric perspective that emphasizes the need for analytical rigor over numerical scaling.