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Claim. The five-page write-up An order-$p_k$ initial interval of gaps in primorial wheels, dated 19 August 2026 and signed by the AI system GPT 5.6 Sol, was posted to the proof-claim tab of Problem 854 on 20 August 2026 as a partial claim by the forum account DottedCalculator, the claimant named here; the write-up gives the AI system as its author. Write $P_k=n_k$ for the $k$th primorial, $1=a_1<\cdots<a_{\phi(P_k)}=P_k-1$ for the integers below $P_k$ coprime to it, and $F(k)$ for the least even integer that is not a difference $a_{i+1}-a_i$; this $F(k)$ is the site's even-integer form of the $f(k)$ of Erdős's 1985 problem list, whose library card is erdos_1985_my_problems_number_theory_i_would. Theorem 1.1 of the write-up: there is an absolute constant $c>0$ such that, for every sufficiently large $k$, every even number $2h$ with $1\le h\le cp_k$ is such a difference; hence

$$F(k)>2\lfloor cp_k\rfloor,\qquad F(k)\gg p_k\sim k\log k,$$

and at least $\lfloor cp_k\rfloor$ distinct even gaps occur. The write-up describes this as improving the initial interval of realized gaps known from the public bounds it cites (Ziller, arXiv:2007.01808, recorded on Ziller's page, and an online working report, recorded on White's claim page) by an unbounded factor, and says that it does not resolve the problem's displayed comparison between the number of realized gaps and the largest gap.

Submission note. Posted to erdosproblems.com as a proof claim by GPT 5.6 Sol (account DottedCalculator) on 20 August 2026, giving "GPT 5.6 Sol" as the AI used:

I couldn't find many results on this problem in the literature. GPT proves that every even integer from $2$ to $O(k\log k)$ is of the form $a_{i+1}-a_i$. The problem is equivalent to covering the interval $[1,h-1]$ with one residue modulo every prime from $3$ to $p_k-1$ while leaving $0$ and $h$ uncovered. The construction picks the residues randomly for each prime up to $cp_k$ and fills in the remaining gaps with large primes. GPT also seems to think that the method in Ford-Green-Konyagin-Maynard-Tao (https://arxiv.org/abs/1412.5029) can be easily modified to keep the endpoints uncovered, which would bring the lower bound close to $\max_i(a_{i+1}-a_i)$.

Method, as the write-up states it. Proposition 2.1 restates $2h$ being a gap as a covering: one residue class $b_q$ for each odd prime $q\le p_k$, with $b_q\not\equiv0,h\pmod q$, whose union contains $\{1,\ldots,h-1\}$; the endpoint condition is what distinguishes this from the unrestricted covering of Problem 687. Lemma 3.1 chooses the classes of the primes in $[5,h]$ at random, avoiding the two endpoint residues, and bounds the expected number of uncovered positions by $O(h/\log h)$ through a Mertens product; the case of a prime dividing $h$, where the two forbidden residues coincide, is handled separately. Proposition 4.1 then assigns each uncovered position its own prime in $(h,p_k]$, which the prime number theorem supplies once $h\le cp_k$ for a small enough $c$. The proof-claim summary adds the suggestion, not proved there, that the method of Ford, Green, Konyagin, Maynard and Tao for long prime gaps could be adapted to keep the endpoints uncovered, which would push the initial interval toward the largest gap.

Covers. The lower bound $F(k)\gg p_k\sim k\log k$ for the smallest even non-gap, and the existence of $\gg p_k$ distinct even gaps, for all large $k$. Not covered: an upper bound for $F(k)$, and the displayed question whether $\gg\max_i(a_{i+1}-a_i)$ even integers occur as gaps. A remark made here: the finite and periodic gap sets agree apart from the boundary gap $2$ (the write-up's (2.1)), so the largest gap is Jacobsthal's function at $P_k$, which is $Y(p_k)+1$ for the covering function $Y$ of Problem 687; the refereed bound (1.2) of Ford, Green, Konyagin, Maynard and Tao on the library's result page makes it $\gg p_k\log p_k\log_3p_k/\log_2p_k$, so a count of order $p_k$ falls short of the displayed target by an unbounded factor.

Read depth. This page rests on Theorem 1.1, Proposition 2.1, Lemma 3.1 and Proposition 4.1 of the write-up at statement level, and on the proofs only at the level of their displays. Nothing here is this project's own review.

Standing. Claimed: an unrefereed write-up in a personal repository (the link is pinned to the repository's revision of 2026-10-07), with no arXiv or journal record found. The site's label is OPEN (page last edited 4 November 2025); the proof-claim tab (accessed 2026-10-07) shows the claim with no comments and the site's standing notice that appearing on the tab means no one at the site has examined the proof. The problem's standing is not changed by a partial claim. The claim value is proved: the result proves a lower bound toward the estimate the problem asks for, and it answers neither side of the displayed question.

Depends on. Ford, Green, Konyagin, Maynard and Tao, (1.2) and the Formulation paragraph of Problem 687: only the remark on the largest gap under Covers rests on them, not the claim.