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Claim. Theorem 1.1 of the preprint: for every integer $k\ge4$ the sequence $d_k(p)$, the density of integers whose $k$th smallest distinct prime factor is $p$, indexed by the primes in increasing order, is not unimodal. With Cambie's theorem for $k=1,2,3$ this gives the paper's Corollary 1.2, the complete classification: $d_k(p)$ is unimodal exactly for $k\in\{1,2,3\}$. That decides the question of Problem 690 for every fixed $k$, the stronger of the two readings recorded on Cambie's claim page; the answer to the yes-or-no question stays no, so the claim value is disproved. The route is a prime-gap criterion for a strict rise or fall of consecutive values, certified finite computations through $k=8600001$ (including its own check, in Table 1, of Cambie's range $4\le k\le20$), and a uniform Chinese-remainder construction for every larger $k$.

Submission note. Posted to the site's forum by Shouqiao Wang on 8 May 2026:

We would like to share a manuscript proposing a resolution. The manuscript includes a complete proof together with a public numerical verifier.

As part of this project, we developed the Multiscalar Fields system, a multi-agent framework for AI-assisted proof generation and verification. The resulting manuscript was checked and refined by the authors.

PDF: https://github.com/multiscalar/results/blob/main/erdos-690/paper.pdf Numerical verifier: https://github.com/multiscalar/results/blob/main/erdos-690/numerical_verifier.py

Comments and verification feedback would be greatly appreciated.

Depends on. Cambie's claim for the three unimodal cases $k=1,2,3$ of the classification; the paper re-verifies $4\le k\le20$ in its own Table 1 (p. 7).

The corpus's compilation of the route is the library's source card and its Theorem 1.1 page: the symbolic route passed an independent review filed there on 2026-09-07, its forty-two finite prime-enumeration, rational-sum and logarithmic certificates are pending, and the analytic estimates, the constant enclosure and the two large prime records are imported premises. That is this project's own work and warrants no acceptance.

Standing. Claimed. The authors announced the manuscript in the site's thread on 2026-05-08 with a public numerical verifier; Nat Sothanaphan posted in the thread on 2026-05-12 that a standard check found no issues, with a caveat that the presentation overstates what the verifier did, and that the proof could be regarded as correct given the checkers named in the acknowledgments. The site's remarks (page last edited 10 May 2026) cite only Cambie's range; no proof claim on the site and no refereed publication were found on 2026-10-07. The title page credits the proof to an AI system ("Discovered by the Multiscalar Fields System") with the two named authors; the attribution is recorded as the source states it.