bibkey: vega2026explicitwindow authors: Frank Vega year: 2026 title: An Explicit Window for Hypothetical Colossally Abundant Counterexamples to Robin’s Criterion doi: 10.20944/preprints202408.0348.v13 url: https://www.preprints.org/manuscript/202408.0348/v13 claim: The preprint gives an explicit primorial window for hypothetical colossally abundant Robin counterexamples and isolates an uncontrolled fractional-part sum; it does not provide a FIB source map or an unconditional RH proof. strata_touched: [] license: CC BY 4.0 (source statement) triage: anchor

An explicit CA window and its residual obstruction

The inspected source is Frank Vega, version 13 of the preprint, posted in May 2026. The manuscript is marked as not peer reviewed. This card records its stated results and their interface with the FIB Robin program; it is not a Lean verification or an independent audit of every cited estimate.

The explicit window

Let (n>5040) be a colossally abundant number violating Robin’s inequality, and let (p_k) be its largest prime factor. For (p_k>7{,}232{,}121{,}212), Theorem 1 states that, with

$$N_k=\prod _{i=1}^k{p}_i,$$

one has

$$N_k\le n<N_k^{Y_k}.$$

The source gives (Y_k\to1^+) and records (Y_k-1<0.00021) once (p_k>e^{31.018189471}). Its Theorem 2 turns this into an equivalent RH test restricted to CA integers in these windows, using Robin’s theorem that a false RH would yield infinitely many CA counterexamples.

The explicit constant is

$$Y_k=\frac{\exp (0.2/(\log p_k{)}^2)}{1-0.01/(\log p_k{)}^3}.$$

Consequently,

$$(Y_k-1)\vartheta (p_k)=\left(0.2+o(1)\right)\frac{p_k}{(\log p_k{)}^2}.$$

The remaining obstruction in the source

The Alaoglu–Erdős exponent formula writes the exponent of a prime (q\le p_k) in a CA integer with parameter (\varepsilon) as a floor of a smooth quantity (x_q(\varepsilon)). Consequently the logarithmic size has the decomposition

$$\log n=S(\epsilon )-F(\epsilon )-\vartheta (p_k),F(\epsilon )=\sum _{q\le p_k}\{x_q(\epsilon )\}\log q.$$

The source states that explicit Chebyshev and Mertens estimates control (S(\varepsilon)) at the required scale, while the universal bound (0\le F(\varepsilon)<\vartheta(p_k)) is too weak by a factor of order ((\log p_k)^2). A uniform estimate for this joint fractional-part sum is left open. Thus the window theorem narrows the possible source but does not determine the Robin sign inside it.

Interface with the FIB inventory

For the strict inventory already defined in the project, put

$$P_{-}(t)=\prod _{p<t}p,N^{-}(t)=P_{-}(t)H_t.$$

The project’s repeated-layer estimate gives

$$\log H_t=\sqrt{2}\sqrt{t}+o(\sqrt{t}),\log P_{-}(t)=t+o(t),$$

and therefore

$$\frac{\log N^{-}(t)}{\log P_{-}(t)}=1+\frac{\sqrt{2}+o(1)}{\sqrt{t}}.$$

Since (\sqrt t=o(t/(\log t)^2)), the repeated-layer excess of the strict inventory is asymptotically smaller than Vega’s window width. Thus the window does not exclude the strict-inventory scale; it places it inside the same primorial neighborhood at the level of these known asymptotics.

When a strict-inventory point is also a CA point, this is the project’s primorial-window coordinate. Consuming Vega’s theorem would require comparing this exponent with the explicit (Y_k), matching (p_k) with the actual largest support prime, and transporting the residual (F(\varepsilon)) to the same FIB source. The preprint supplies none of those FIB identifications. Its fractional-part obstruction is therefore a useful external description of the same signed-tail gap, not a proof of Robin or RH for the FIB family.