Density-one widths and shifted prime-divisor counts

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evidence/: Exact mathematical snapshots and independently graded scrutiny of the density-one width construction for omega.

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L34 constructs widths $F(x)\to\infty$ with $F(n)/\omega(n)\to0$ on a set of natural density one, while

$$\frac{\#\{n\ge1:x<n+\omega(n)<x+F(x)\}}{F(x)}\le7$$

for every sufficiently large real $x$. The independent examination and separate passing grade of 2026-10-14 accept this complete argument for the $\omega$ subquestion under the literal modified statement of Problem 122. They do not classify all $f$; their source and formulation limits remain explicit.

The source input is the modulus-one specialization of the published second-moment inequality. It makes atypical inputs sparse. Counting how often their images enter short intervals makes atypical interval locations sparse as well. The proof permits short widths elsewhere and widths of order $x$ at exceptional locations. Density-one smallness allows this definition. The argument establishes neither infinitude of the exceptional set nor failure of an everywhere-small condition for this particular $F$. No conclusion is claimed about the different formulation requiring $F(n)=o(\log\log n)$ everywhere or about monotone widths.

The construction is compatible with large point multiplicities in the published paper: the widths may be enlarged at those exceptional images. The unpublished affirmative report is retained at its recorded standing. Its original passages and any unpublished proof have not been inspected for their width convention. Reconciling that historical formulation is a separate source obligation; it is not an extra premise of L34. The divisor-function case and the whole classification remain outside this result.