An Explicit Window for Hypothetical Colossally Abundant Counterexamples to Robin's Criterion

Robin's criterion equates the Riemann hypothesis with the inequality $\sigma(n) < e^{\gamma}\,n\,\log\log n$ for every $n > 5040$. By a theorem of Robin, if the Riemann hypothesis is false then infinitely many colossally abundant numbers fail this inequality, so the existence of a counterexample to Robin's criterion is equivalent to the existence of a colossally abundant counterexample. Combining structural properties of colossally abundant numbers with explicit estimates for the Chebyshev theta function and the Mertens product due to Aoudjit, Berkane, and Dusart, we prove an unconditional and effective upper bound on any such counterexample. Specifically, if $n > 5040$ is colossally abundant and violates Robin's inequality, then $n < N_{k}^{Y_{k}}$, where $p_{k}$ is the largest prime factor of $n$, $N_{k} = \prod_{i=1}^{k} p_{i}$ is the primorial of order $k$, and $Y_{k} \to 1^{+}$ is an explicit constant. Together with the lower bound $n \geq N_{k}$ that holds for any Hardy--Ramanujan integer, this confines every hypothetical colossally abundant counterexample to a narrow explicit window above the corresponding primorial. We deduce a conditional reformulation of the Riemann hypothesis localized to this window. This work refines the approach taken in the author's earlier article ``Robin's criterion on divisibility'', published in The Ramanujan Journal.