Adélic Defect Limit Set and a Twisted Patterson-Sullivan Bridge to the Riemann Hypothesis

We construct a conditional bridge between the Riemann Hypothesis (RH) and the Hausdorff dimension of an adélic defect limit set Λdefect. By leveraging the Langlands dynamical correspondence and Hecke-rescaled Ruelle-Perron-Frobenius operators, we bypass the triviality of Ratner's volume-finite limits. Assuming two topological conjectures, we prove that H(Λdefect) = 1/2 if and only if RH holds. We also quantify the Sato-Tate bias in finite-truncation approximations, demonstrating that numerical box-counting converges at an intolerably slow rate of O(log log N / log N). This forces any viable proof to rely entirely on strictly algebraic dependencies rather than empirical computation.