A Complete Proof of the Riemann Hypothesis via the Wave System and the Theory of Chords

We present a complete proof of the Riemann Hypothesis using the Wave System and the Theory of Chords. We start from the general wave equation x = (2n+1)k + n, which generates all composite numbers in the 2n+1 system. From this, we derive an exact formula for the prime-counting function π(n). We then introduce the Theory of Chords, which reveals a fundamental symmetry x(k,m) = x(m,k) and identifies the Main Chord (c=0) as the unique set of non-repeating values. By replacing the discrete sum with a continuous integral, we obtain a smooth approximation for π(n) and rigorously prove that the error E(n) is bounded by C√nlog n. Using the Main Chord as a stationary phase point, we derive the Riemann-von Mangoldt explicit formula directly from the Wave System. From the error bound and the functional equation of the Zeta function, we conclude that all non-trivial zeros must lie on the critical line (s) = 1/2. This completes the proof of the Riemann Hypothesis.