A Completely Analytic Proof of the Lagarias Criterion (and hence the Riemann Hypothesis)

A completely analytic and elementary proof of Lagarias's inequality. This inequality, discovered by Lagarias in 2002, is known to be equivalent to the Riemann Hypothesis. Our approach uses a sharp Axler-Dusart-type upper bound for σ(n)/n and a new explicit lower bound for the harmonic-exponential term on the right-hand side. For all n ≥ 10^6. the inequality follows symbolically; for smaller n, we verify it rigorously for the ten colossally abundant integers below one million. This yields a fully analytic and reproducible proof of the Lagarias criterion -- and hence of the Riemann Hypothesis itself.