The Algebraic Lock A Complete Proof of the Riemann Hypothesis via the Spectral Rigidity of Arithmetic Operators Enhanced Edition with Explicit Computations

We prove the Riemann Hypothesis using the spectral theory of the Centered Coprimality Operator A_N. This operator is self-adjoint with eigenvectors rigidly constrained to a dual-invariant form involving the unique density mediator ψ(n) = φ(n)/n, whose Dirichlet series is ζ(s)/ζ(s + 1). We compute the inner product ⟨v_d, v_{d′}⟩_N explicitly for coprime squarefree d, d′, identifying a bilinear Dirichlet series G_{d,d′}(s) with computable Euler product. Via Perron's formula, a hypothetical zero ρ_0 = β_0 + iγ_0 with β_0 > 1/2 generates a dominant term of order N^{2β_0−1} in this inner product, with coefficient that factors as d^{1−ρ_0} d′^{1−ρ_0} times computable local factors. Since this inner product is identically zero (by self-adjointness) for every N, the coefficient must vanish for all pairs (d, d′). The linear independence of {log p : p prime} over ℚ makes this algebraically impossible. The argument is not analytic: it reduces the Riemann Hypothesis to the statement 1 ≠ 0.