bibkey: zimov2025leastca authors: Bruce Zimov year: 2025 title: On the Least Colossally Abundant Exception to Robin’s Inequality doi: null url: https://arxiv.org/abs/2510.23889v1 claim: The preprint claims that, if RH is false, the least colossally abundant Robin counterexample lies in a narrow asymptotic excess band; this band is much wider than the explicit FIB §250 gap and does not identify a FIB source. strata_touched: [] license: citation-only triage: anchor

On the Least Colossally Abundant Exception to Robin’s Inequality

The source is arXiv:2510.23889v1, submitted 27 October 2025. This card records the stated result and its hypotheses; it is a preprint and was not independently audited or formalized here.

Write

$$G(n)=\frac{\sigma (n)/n}{\log \log n}.$$

Using the classical reduction to consecutive colossally abundant numbers, their prime or semiprime quotients, and Robin’s oscillation theorem under $\neg \mathrm{RH}$, Theorem 8 claims that the least CA counterexample satisfies

$$e^{\gamma }<G(n)<e^{\gamma }\left(1+\frac{c}{(\log n{)}^b}\right),0<b<\frac{1}{2},$$

for a positive constant $c$ depending on the chosen $b$. The paper distinguishes this least CA counterexample from the least integer counterexample; the former is a sufficient test set only through the classical CA reduction.

Relation to the current FIB estimates

The upper band is of order $(\log n{)}^{-b}$, whereas the explicit §250 same-source gap is of order $1/(\sqrt{n}(\log n{)}^3)$. These scales do not contradict one another, and the band does not control the signed $\Phi$ term in the FIB decomposition. Applying it would additionally require proving that a FIB candidate is CA and that the consecutive-CA quotient is the same source used by the FIB price model. Neither implication follows from a five-window address or its congruence.