The Riemann Hypothesis and Period-6 Geometric Structure: A Trigonometric Framework for Zeta Zero Analysis and Prime Frequency Resonance

This paper develops a trigonometric framework for analyzing the trivial and non-trivial zeros of the Riemann zeta function and its related functions. We prove that the period-6 structure is not merely analogous to the Dirichlet character decomposition — it is the same mathematical object expressed trigonometrically. Through this bridge, L(s,χ) is expressed in purely trigonometric form, its zeros shown to correspond to perfect cancellation of period-6 weighted oscillations, and all four distinct cosine values on the period-6 unit circle are found to encode the complete zero and critical strip structure. The first main result is the over-constraint argument: off the critical line, the functional equation forces two independent complex cancellation conditions on the same rigid integer phases, creating an over-determined system that cannot be simultaneously satisfied. The second main result is the spectral interpretation: each zeta zero tₙ is approximated by tₙ ≈ k·3π/p, where 3π/p is the angular frequency of prime p in the trigonometric primality test, with a mean relative error of 0.001%. This reveals that zeta zeros are resonance points of the prime frequency spectrum, identifying a candidate for the Hilbert–Pólya operator. These results suggest that the Riemann Hypothesis dissolves when one recognizes that primes are fundamentally trigonometric objects, and the zeta zeros are the resonance spectrum of the ensemble of all prime cosine functions.