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Claim. For the function $F(x)$ of Problem 1205, the largest multiplicity that some choice of one residue class $a_n\pmod n$ for every $n\le x$ guarantees for every integer $m\le x$,

$$\log x-O\bigl(\sqrt{\log x\log\log x}\bigr)\le F(x)\le\log x+O(1),$$

so $F(x)\sim\log x$. This answers "estimate $F(x)$" to the first order and is the site's reading of its SOLVED label, defined on the page as a resolution other than a proof or disproof. The argument, in the words of this compilation, has three steps. Upper bound: the class of $n$ holds about $x/n$ of the integers up to $x$, so the average number of congruences an integer $m\le x$ satisfies is $\sum_{n\le x}1/n+O(1)=\log x+O(1)$, and some $m$ sits at or below the average. Random lower bound: choose the classes for $n\le x/2$ uniformly at random; a fixed $m\le x$ lies in the class of $n$ with probability $1/n$, so its expected count is $\log x+O(1)$, and a Chernoff-type concentration bound makes the probability of a count below $\log x-O(\sqrt{\log x\log\log x})$ at most $(\log x)^{-2}$, leaving at most $O(x/(\log x)^2)$ exceptional integers for some fixed choice. Greedy repair: each modulus $n\in(x/2,x]$ can choose its class to contain any one prescribed integer $m\le x$; spreading these about $x/2$ moduli over the $O(x/(\log x)^2)$ exceptions covers each exception $\gg(\log x)^2$ times. The commentary says the argument was suggested by the comments on the prime-modulus analogue, Problem 689, which remains open.

Claimant and date. The argument is the commentary of the site's own problem page, unsigned, whose recommended citation names the site's curator, T. F. Bloom, as the page's author; the page is filed under that name. The revision history of the page shows a revision of 8 April 2026, 06:40 UTC, that already carries the argument, and the current page was last edited the same day; the community database records the status as solved, in a record last updated on 4 April 2026. The page name's date is the earliest dated revision found.

Standing. Pending. No paper, preprint or forum discussion of the argument exists (the discussion thread and the proof-claim tab were empty on 2026-09-18), no formalization exists, and the site's label is the claimant's own, since the curator who labels the problem wrote the argument, so it is not independent acceptance and no reviewed evidence is listed; the concentration step and the greedy count are stated, not written out. Nothing here is this project's own review, and the problem's standing is claimed through this page. A written proof, an independent review or a refereed source would be the evidence for acceptance.

Scope. Full for "estimate $F(x)$" to first order. The second-order term is open: the gap between the two bounds is $O(\sqrt{\log x\log\log x})$, and nothing narrower was found.