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Claim. For $\omega(n)$ the number of distinct prime factors, there is $F:[1,\infty)\to(0,\infty)$ tending to infinity and one set $G$ of positive integers of natural density one such that $F(n)/\omega(n)\to0$ through $G$, while

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

for every sufficiently large real $x$. Thus the normalized counts cannot diverge along any unbounded sequence of real or integer locations.

Covers. The $f=\omega$ case of the modified Statement of Problem 122, interpreting "almost all" as smallness along one fixed set of natural density one. No monotonicity or everywhere-small width condition is imposed. Other arithmetic functions, including $\tau$, and the full classification are not settled. The historical unpublished affirmative report's width convention remains unexamined.

Depends on. L34 supplies the complete construction. It uses the $m=1$, $K=1$, $X\ge2$ instance of EPS97 Lemma 1 and Euler's divergence of the prime reciprocal sum. The source statement was checked at the consumed specialization; its proof has no new independent verification here.

Standing. Claimed: the review below is internal to Plasma AI, and the outside sign-off this claim needs is pending. The Plasma AI whole-conclusion examination of 2026-10-14, by a fresh-context adversarial reviewer (GPT-6 Astra), records refutation-failed for the complete modified omega counterexample. A distinct fresh-context grader (GPT-6 Astra) passed that scoped verdict. The canonical statement and core construction match the examined subject.

The grade separately accepts proof defective; statement unresolved for a stronger-width assertion omitted from the canonical proof, and refuted-as-stated for an unused strict-zero clause in the supplied source statement. Neither is a premise of the counterexample. No Lean proof, full source-proof audit or whole-problem solution is asserted.