The Algebraic Lock A Complete Proof of the Riemann Hypothesis via the Spectral Rigidity of Arithmetic Operators

For the lulzWe present a complete proof of the Riemann Hypothesis (ℜ(s) = 1/2 for all nontrivial zeros) based on the spectral theory of the Centered Coprimality Operator A. We demonstrate that A is a self-adjoint operator on ℓ²(N) whose eigenvectors are rigidly constrained to a specific "Dual-Invariant" form determined by the Euler totient function φ(n)/n. By transforming the orthogonality condition of these eigenvectors into the Dirichlet domain, we show that the existence of a zero off the critical line generates a dominant error term in the spectral inner product. This error term cannot be cancelled simultaneously for all arithmetic progressions due to the linear independence of prime powers over the rationals. Thus, the existence of the operator A is logically incompatible with the falsity of the Riemann Hypothesis.