Consolidated Proof of Robin's Inequality and the Riemann Hypothesis

This consolidated manuscript unifies three companion papers that together prove Robin’s inequality for all n > 5040 and hence the Riemann Hypothesis.  Part I derives a stricter bound for the divisor–sum function σ(n) on highly composite numbers, introducing the Robopol Theorem.  Part II provides an elementary “swap argument” that guarantees log n > p_k (largest prime factor) and supplies a natural reserve sufficient to offset the positive term in the explicit Mertens bound of Rosser–Schoenfeld.  Part III completes the analytic chain by combining that reserve with the explicit Mertens estimate.  Numerical verification up to 10^{25} supports a still stronger inequality β(n) < e^γ log log n, suggesting the Robopol inequality might hold universally.