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Claim. For the natural density $\delta_1(n,m)$ of the integers with exactly one divisor in the open interval $(n,m)$, the sequence $(\delta_1(n,m))_{m\ge n+2}$ need not be unimodular. Cambie computes the dip $\delta_1(3,6)=7/20>\delta_1(3,7)=1/3<\delta_1(3,8)=38/105$, reports a computer check that unimodularity fails for every $2\le n\le20$ (the notebooks are in Cambie's repository, linked above at its revision of 27 May 2025), and proves (Theorem 3) that for some $c>0$ the sequence has $\omega(\exp(n^c))$ local maxima for all sufficiently large $n$. Theorem 1 of the same paper proves the positive case $n=1$: $\delta_1(1,m)$ is non-increasing in $m$. The problem's precise Statement is therefore answered in the negative. On the variant recorded under the problem page's Formulation, where $\delta_1(n,m)$ attains its maximum for fixed $n$, Theorem 1 answers only $n=1$: it puts the maximum $1/2$ at $m=3$, tied with $m=4$, since $\delta_1(1,3)=\delta_1(1,4)=1/2$ and $\delta_1(1,5)=1/3$. Every $n\ge2$ is left open.

Source. Stijn Cambie, Resolution of Erdős' problems about unimodularity, arXiv:2501.10333v1 (17 January 2025); Journal of Number Theory 280 (March 2026), 271--277, doi:10.1016/j.jnt.2025.08.014. Its source card carries the finite example, Theorem 1, Claim 4 and Theorem 3; Theorem 3 consumes Ford's Theorem 4 on $H_1(x,y,z)/H(x,y,z)$, which the problem page quotes.

Acceptance. Refereed: the paper appeared in the Journal of Number Theory. Reviewed: the site's curator, Thomas Bloom, marks Problem 692 disproved and credits Cambie's computation and many-local-maxima theorem for it. This corpus has not independently reviewed the paper, and its own reading awards no standing.

Formalization. Pietro Monticone posted on the site's thread on 2 April 2026 a Lean 4 file autoformalized by Aristotle (Harmonic), authored as Monticone and Aristotle, which proves delta1 3 7 < delta1 3 6 and delta1 3 7 < delta1 3 8 for its own rational residue-proportion definition of $\delta_1$. It is a formalization of Cambie's finite example and is linked above at the pinned revision. This corpus has not built it; it does not bridge its definition to natural density or state a general negation of unimodularity, and it is no formalized evidence of this corpus. Boris Alexeev's lean-proofs repository re-hosts the file, added on 6 May 2026 and linked above at a pinned revision, naming Cambie as informal author and Aristotle and Monticone as formal authors; this corpus has not built it either.