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Claim. With $p_n$ the least prime $\equiv1\pmod n$ and $m_n$ the least integer with $n\mid\varphi(m_n)$ (so $m_n\le p_n$ always), David Turturean, A positive-density equality set in Erdős Problem 456, and a Dickson-conditional family of uniqueness primes, a 71-page manuscript posted to the problem's thread on 4 May 2026 and digested on its card, proves in its Theorem 1.1 that $\#\{n\le x:m_n=p_n\}\ge cx$ for some $c>0$ and all large $x$. Its Corollary 1.2 deduces that the first two questions of Problem 456 have the answer no: $m_n<p_n$ fails on a positive proportion of $n$, and $p_n/m_n$, equal to $1$ there, does not tend to infinity for almost all $n$. The proof counts base triples $(b,s,P)$ with $n=sP$, $p=bsP+1$ prime and $P^2>p$, so that any smaller totient cover of $n$ would contain a prime $aP+1$ (its Definition 4.1). The author writes that the proof was produced by an automated audit-and-revise scaffold the author designed, querying GPT-5.5-Pro, and that the author verified the final proof.

Covers. The first two questions, both answered no. Not covered: the third question, which the manuscript answers only under Dickson's conjecture for the triple $t,2t+1,8t+1$ (its Theorem 1.4), a conditional result that settles no instance; the unconditional third answer is the claim of 2026-09-23.

Formalization. The repository linked above, at its head commit of 2 September 2026, formalizes this manuscript. By its README, the first two answers (erdos_456_questions_one_two) rest on ten cited literature results stated as axioms, and the third question is proved only under the Dickson-triple hypothesis (erdos_456_question_three_of_dickson). No build, axiom audit or statement audit of the repository was made in this corpus.

Depends on. Nothing in this wiki.

Standing. A manuscript statement, pending: the site's label is OPEN (page last edited 7 October 2025), no referee report or arXiv record exists, and no outside reviewer has accepted the argument. The card's digest is author-recorded and is not an acceptance.