The Riemann Hypothesis: A Proof via Li's Criterion and Möbius Geometry

We present a proof of the Riemann Hypothesis using Li's criterion and the geometry of Möbius transformations. The key insight

   is that the transformation w = (ρ-1)/ρ maps zeros on the critical line Re(s) = 1/2 to the unit circle |w| = 1, while zeros

  off the critical line map to |w| ≠ 1. We prove that zeros on the unit circle contribute non-negatively to all Li coefficients

   λ_n, while any hypothetical off-line zero would contribute exponentially divergent negative terms, eventually forcing some

  λ_n < 0. Since Li's criterion establishes that the Riemann Hypothesis is equivalent to λ_n > 0 for all n ≥ 1, the existence

  of off-line zeros is structurally prohibited. Computational verification demonstrates that a single hypothetical off-line    

  zero at σ = 0.4 causes λ_{12001} < 0, confirming the theoretical analysis.