Conformal Spectral Positivity and Arithmetic Decomposition in Li's Criterion for the Riemann Hypothesis

We present a self-contained, rigorous mathematical study of the Riemann Hypothesis via Li's positivity criterion and the arithmetic-Archimedean decomposition of the completed Riemann Xi function. We give complete, step-by-step proofs of: (1) the exact conformal equivalence between the critical-line condition Re(rho) = 1/2 and the unit-circle modulus bound |1 - 1/rho| = 1; (2) the Hadamard-product proof that any off-line zero induces exponential divergence in the Li coefficients; and (3) the explicit decomposition into an unconditionally positive Archimedean Stirling term and an oscillatory prime-power von Mangoldt sum, isolating the exact conditional prime-cancellation inequality (Open Hypothesis 4.1) required to close the full Millennium Prize Problem.