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Claim. For $k\ge5$ there are irreducible covering sets of size $k$ and none of size at most $4$; writing $I(k)$ for their number, $M(k)$ and $m(k)$ for the largest and smallest possible largest modulus, and $R(k)$ for the largest reciprocal sum, the manuscript's Theorem 1.1 asserts

$$I(k)=\exp\Bigl(\bigl(\tfrac{4\sqrt\tau}{3}+o(1)\bigr) k^{3/2}(\log k)^{-1/2}\Bigr),\qquad M(k)=3\cdot2^{k-3},$$

$k+1\le m(k)\le Ck(\log k)^2$ for an absolute $C$, so $m(k)=k^{1+o(1)}$, and $R(k)=\Theta(\log k)$, where $\tau=\sum_{t\ge1}\log^2(1+1/t)\approx0.977$ is the constant of Balister, Bollobás, Morris, Sahasrabudhe and Tiba for the number of minimal covering systems with $k$ classes; the fifth part restates Sun's theorem that the divisors of $2^{p-1}p$ above one form an irreducible covering set. These answer the site's first three questions of Problem 1189; the fifth part restates Sun's answer to the fourth, the divisor question. The engine, as the manuscript presents it, is an irreducibility criterion on moduli alone: with $F(N)=\sum_p\alpha_p(p-1)$ for $N=\prod p^{\alpha_p}$, a coverable set $S$ every proper subset $T$ of which satisfies $|T|\le F(\operatorname{lcm}T)$ is irreducible, because a cover on a proper subset could be thinned to a minimal cover, which by Simpson's theorem has more than $F(\operatorname{lcm})$ classes. From this criterion the manuscript derives $\operatorname{lcm}S\le3\cdot2^{k-3}$, matched by the set $\{2,4,\ldots,2^{k-3},3,3\cdot2^{k-4},3\cdot2^{k-3}\}$; it credits the lower bound $M(k)\ge3\cdot2^{k-3}$ to van Doorn's construction in a comment of 2026-04-16 on the problem's discussion thread; an edit of 2026-06-21 to that comment announced without proof that the value is optimal. The count transfers the leading constant of Balister, Bollobás, Morris, Sahasrabudhe and Tiba's minimal-system asymptotic to irreducible sets, adding a verification, which the manuscript says the published source does not give, that their frame construction survives the restriction to distinct moduli at a cost of $O(k\log k)$ in the exponent; digit frames built from the prime-power coordinates of a modulus give the bounds on $m(k)$ and $R(k)$. The manuscript (dated 22 July 2026, 19 pages, by "Paratelligent Research Agent and Jeff Pickhardt") was posted on the Omniscience Project site, later renamed Paratelligent, whose page linked above gives the online date 2026-08-27; the site's proof-claims tab carries it as a full proof claim submitted on 2026-07-28 by Pickhardt and credited to them and the Omniscience Research Agent, the AI system named on the claim; its notes say that the thread's comments and Star Fleet Math's proof were good and that the manuscript now supplies the full proof. The claim's one comment, of the same day, reported that the original link showed a sign-in screen and gave the public address, which a moderator added.

Submission note. Posted to erdosproblems.com as a proof claim by Jeff Pickhardt and Omniscience Research Agent (account JPickhardt) on 28 July 2026, giving "Omniscience Research Agent" as the AI used:

Here's a proof that finds the largest modulus can be exactly 32^{k-3}, the smallest is k^{1+o(1)}, the largest reciprocal sum is Θ(log k), and the number of irreducible covering sets of size k is exp((4√τ/3+o(1))k^{3/2}/√log k), τ=Σ_t log²(1+1/t) being the constant of Balister, Bollobás, Morris, Sahasrabudhe and Tiba. The sharp constant is new. It uses the following: let F(N)=Σα_p(p-1). If S covers and every proper subset T has |T|≤F(lcm T), then S is irreducible: if some T covered, delete classes (not moduli) down to a minimal cover, which by Simpson's theorem needs more than F(lcm) of them. So one never has to quantify over residue assignments and irreducibility becomes a condition on the moduli alone. From this, lcm S≤32^{k-3}, matched by {2, 4, ... ,2^{k-3}, 3, 32^{k-4}, 32^{k-3}} (van Doorn's comment had the bound). And their extremal frames are irreducible too, with distinct moduli bought too cheaply to move the exponent, so the constant carries over. Notes: The comments and Starfleet's proof were good, but the full proof is now provided by this paper.

Depends on. Sun's theorem supplies the fifth part and the divisor question; Simpson's inequality drives the irreducibility criterion; and the counting theorem of Balister, Bollobás, Morris, Sahasrabudhe and Tiba, with their frame construction, gives the count.

Standing. Claimed: the manuscript is a web-posted preprint with no journal publication, referee report, curator acceptance or outside review located through 2026-10-06; the site labels the problem OPEN (page last edited 8 April 2026). The manuscript describes as concurrent work the Lean 4 development released by Star Fleet Math, which has its own page (Snyder's claim page): a proposed solution reaching the same $M(k)$, $m(k)=O(k(\log k)^6)$, the same order for $R(k)$ and the divisor property, but, in the form the manuscript cites, only $\log I(k)\ge(\log2/4096)\,k^{3/2}(\log k)^{-1/2}$ along a sequence of sizes, so without the constant; the manuscript claims the sharp constant as its new content, while the release's own page reduces the count in Lean to two hypotheses, a distinct-moduli frame datum and an upper count of displayed minimal systems, and reaches the asymptotic with the constant by hand from Theorem 1.1 of Balister, Bollobás, Morris, Sahasrabudhe and Tiba, a distinct-moduli bridging step the release's referee checked by hand, and elementary estimates.