Exceptional widths prevent unbounded clustering for shifted omega

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_proof: A dyadic second-moment argument bounds the count at ordinary locations and identifies a density-zero set where large widths are harmless.

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The proof is an independently examined argument by Plasma AI using Erdős–Pomerance–Sárközy Lemma 1 at modulus $1$. Its hypotheses and conclusion are inspected at statement depth; the source proof is not independently verified here. This claim has the independently graded review below and no Lean proof.

L34 gives a negative answer for $f=\omega$ under the literal almost-all width condition in the modified statement of Problem 122. It does not classify the other arithmetic functions. The research account distinguishes density-one smallness from a uniform or monotone width condition and from the historical unpublished affirmative report.

There is no computational premise. Verification consists of specializing the source lemma, bounding the exceptional input and interval sets, checking the real-location extension, and proving density-one smallness of the constructed widths.

Current standing and verification

Tier 1 records the independent whole-conclusion examination and its separate passing grade, both dated 2026-10-14. The reviewer and distinct grader each worked in a fresh, isolated context and disclosed GPT-6 Astra. The accepted verdict is refutation-failed for the complete density-one construction, including the everywhere real width limit and eventual normalized-count bound of seven. The canonical statement has exactly that scope, and its proof preserves the examined core argument.

The canonical proof omits the literal stronger-width assertion at lines 104–106 of the frozen argument. Its separately accepted verdict is proof defective; statement unresolved: that assertion requires infinitude of the exceptional set, which was not established and is not consumed by L34. The grade also accepts refuted-as-stated for the unused universal strict EPS97 wording at $x=2,m=1,h=0,f\equiv0$. This does not affect the consumed $K=1$, $X\ge2$ specialization or certify its proof.

No native L-claim is a premise. The EPS97 statement was checked from the supplied source interfaces; its proof was not independently verified, and Euler's divergence remains an explicit classical premise. No result for other $f$, everywhere-small or monotone widths, historical priority, or the unpublished affirmative report is accepted by this grade.