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Claim. Theorem 1 of P. Chojecki, A Power Lower Bound for Runs of Distinct Prime Gaps (a six-page note dated 24 April 2026), states that there is an absolute constant $c_0>0$ such that, for all sufficiently large $x$, the function $h(x)$ of Problem 852 satisfies

$$h(x)\ge c_0(\log x)^{1/3},$$

and in particular $h(x)>(\log x)^c$ for every fixed $c<1/3$ and all sufficiently large $x$. The proof uses no hypothesis. It first proves the bound at $M=\pi(N)$ (Proposition 5): among the blocks of $H$ consecutive gaps starting at $n\le M-H$, Markov's inequality applied to the total block length leaves a positive proportion with $p_{n+H}-p_n\le AH\log N$; if such a block repeats a gap $q$ at offsets $a<b$ from $p_n$, the five numbers $p_n$, $p_n+a$, $p_n+a+q$, $p_n+b$, $p_n+b+q$ are prime, and the Selberg upper-bound sieve (Lemma 2, cited from Friedlander and Iwaniec and from Lichtman and Teräväinen) together with an average of singular series over affine families (Lemma 3, a special case of Lichtman and Teräväinen's Corollary 2.6) bounds the number of such configurations with offsets at most $Y$ by $\ll Y^3+NY^3/(\log N)^5$, with the degenerate four- and three-prime cases handled alike. With $Y\asymp H\log N$ this is smaller than the number of short blocks once $H$ is a small multiple of $(\log N)^{1/3}$. Passing to the index variable through $p_M\sim M\log M$ gives the theorem. The note says (Introduction and Remark 6) that the argument uses only that the endpoints of the equal gaps are prime, not that the gaps are consecutive, and that this is the source of the exponent $1/3$. The author's thread post of the same day credits the result to GPT-5.5 Pro, and reports the system's further remark that $h(x)=o(\log x)$ may fail under a form of the uniform Hardy-Littlewood conjecture, with $h(x)\sim1.32323\log x$ as a guess; the note itself contains neither.

Submission note. Posted to the site's forum by Przemek Chojecki on 24 April 2026:

GPT-5.5 Pro got a nice partial result showing that

$$h(x)\gg (\log x)^{1/3}. >$$

The idea is this: we look at blocks of $H$ consecutive prime gaps whose

total length is at most a constant multiple of $H\log N$. A positive proportion of starting positions have this property. If such a block has two equal gaps, then five explicitly related numbers must be prime:

$$m,\quad > m+a,\quad m+a+q,\quad m+b,\quad m+b+q.$$

For fixed $a,b,q$, the Selberg sieve

(in a form proved by Lichtman--Teravainen) gives an upper bound of the expected order $N(\log N)^{-5}$, up to the singular series. Averaging the singular series over the three parameters $a,b,q$ then shows that the number of bad blocks is

$$\ll \frac{NY^3}{(\log N)^5}$$

when all offsets are at most

$Y$. Since in our application $Y\asymp H\log N$, this is smaller than the number of available starting positions as soon as $H$ is a sufficiently small multiple of $(\log N)^{1/3}$.

On the other hand, GPT-5.5 also noted that $h(x)= o(\log x)$ might be wrong if we assume a form of uniform Hardy-Littlewood conjecture and more likely something like $h(x)≤(C+o(1))\log x$ is plausible with a guess at possible asymptotics being

$$> h(x) \sim 1.32323 \log x >$$

Covers. The first of the problem's two particular questions, answered yes: $h(x)>(\log x)^c$ holds for every fixed $c<1/3$. The note gives no upper bound, does not estimate $h(x)$, and does not address whether $h(x)=o(\log x)$. An independent argument posted the same day, Turturean's four-prime count, reaches the same bound.

Depends on. Nothing in this wiki: the inputs are the cited sieve estimates of Friedlander and Iwaniec and of Lichtman and Teräväinen.

Standing. Claimed. The note is posted on the author's site and is not refereed; the site's label is OPEN, its commentary records only that Brun's sieve gives $h(x)\to\infty$, and its proof-claims tab lists nothing for the problem, so the curator records no acceptance. No independent review of the argument is recorded.