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Claim. P. Chojecki, Erdős Problem #783: sharp asymptotic value and a stability program (text dated February 25, 2026), collects the asymptotic value $\rho(e^C)+o(1)$ from Tao's theorem and formulates a structural program for which sets minimize up to $o(N)$, sharpening the answer to the corrected Statement for $C>\log2$: every near-minimizer is close in reciprocal sum to a prime tail. Its Theorem 30 states, conditionally on three hypotheses (a stability form of the Granville-Soundararajan equality mechanism for the prime case, a discretization step, and a near-extremal reduction from coprime sets to prime sets), that for fixed $C>\log2$ every near-minimizer $A=A(N)$ is close in reciprocal sum to a prime tail: there is $y=N^{1/e^C+o(1)}$ with $\sum_{a\in A\triangle\mathcal P(y,N]}1/a=o(1)$, where $\mathcal P(y,N]$ is the set of primes in $(y,N]$. Section 9, added on 25 February 2026, claims proofs of the three hypotheses (Proposition 33, Theorem 37 and Proposition 41) and states the unconditional consequence as Theorem 42, coprime tail rigidity. The write-up's Remark 31 says that this does not determine the exact minimizer for a given $N$, since changes at the threshold alter the budget and the unsifted count by $o(1)$ and improve lower-order terms.

Submission note. Posted to the site's forum by Przemyslaw Chojecki on 23 February 2026:

I've worked a bit with GPT-5.2 to see whether Terence's argument can be pushed to the full resolution of the problem and here's the write-up. Basically it shows you that minimizing set $A$ are $\mu$-close to prime tails conditional on some expected results. Asymptotically the result seems to be within reach (but still not easy), finite $N$ seems to be much harder.

Posted to the site's forum by Przemyslaw Chojecki on 25 February 2026:

With some more back and forth with GPT-5.2 I've managed to complete an argument for the asymptotic version of the problem modifying Terence's approach to account for stability conditions. I've added it at the end of the last write-up here.

There are 3 main things proven in Section 9:

• stability of Granville–Soundararajan argument
• prime-tail rigidity
• re-working of Terence's argument with these ingredients.

All 3 are fairly independent and also Section 9 is independent from previous Sections. I'm saying that because this turned out to be pretty lengthy and there might be minor things to correct here and there, but I believe the overall argument should be fine.

Before trying to polish it, I want to have a go at the finite $N$ (i.e. the full problem), though it seems preliminarily that there are some hard things we would have to prove to finish this line of thought.

Covers. For $C>\log2$, the structure of all sets that minimize up to $o(N)$: they are prime tails up to $o(1)$ in reciprocal sum, a refinement of the answer the corrected Statement receives on Tao's page. Not covered: $C\le\log2$, the exact minimizer at finite $N$ (the site's wording; the write-up's Remark 31 says so), and any error term.

Postings. The write-up was first posted to the thread on 23 February 2026, when the author described its conclusion as conditional on some expected results; on 25 February 2026 the author posted the version with Section 9, saying the argument for the asymptotic version was complete, that it is lengthy and may need minor corrections, and that the finite-$N$ problem looks much harder. The author's posts present the work as done with GPT-5.2. A further comment of 28 February 2026 proposes using Saias's theorem on smooth numbers to sharpen the prime-tail competitor's count at finite $N$; it claims no result and has no page.

Standing. Claimed: the site's commentary (page last edited 28 May 2026, accessed 2026-09-05) credits Tao's asymptotic value and Chojecki's $C\le\log2$ case but does not mention this write-up, no thread comment reviews it, and nothing was read here beyond its abstract, the statements of Theorem 30, Hypothesis 29 and the opening of Section 9.

Depends on. Tao 2026, whose theorem supplies the asymptotic value and whose reduction to primes the near-extremal bookkeeping refines, and Hildebrand 1987 for the prime case; the Granville-Soundararajan paper the stability step refines is not held here.