Claim. For $t\ge1$ let $d_t$ be the natural density of the integers $n$ for which $t$ is a sum of distinct divisors of $n$. The answer to the question is no: $d_t$ exists and is positive for every $t$, and there are no constants $c_1,c_2>0$ with $d_t\sim c_1/(\log t)^{c_2}$. The proof gives two estimates that no such asymptotic can satisfy at once. With $\delta=1-(1+\log\log2)/\log2=0.0860\ldots$, the exponent in Ford's theorem on integers with a divisor in a dyadic interval, the lower limit of $-\log d_t/\log\log t$ equals $\delta$ exactly, while $(\log t)^{\delta}d_t\to0$. The first forces the exponent $c_2$ of any asymptotic to be $\delta$, and the second forces its constant $c_1$ to be $0$. The upper estimate follows Erdős's 1970 split of the integers counted by $d_t$ into those with a divisor in a short interval below $t$ and a remainder of density $O(1/\log t)$, an exact product-measure law for the capped valuations at the primes up to $t$, and a Ford-type bound with a $(\log\log t)^{-5/4}$ saving that pays for a dyadic union; the lower estimate builds practical seeds and a tilted prime-pattern law, bounds a harmonic average of the densities from below through Hölder's inequality, and passes to a single $t$ because a weighted average has a term at least as large. The density exists because membership of $n>0$ depends only on $n$ modulo $t!$, so $d_t$ is the average of the indicator over one period.

The formal statement. The Lean file proves the negation of the formal-conjectures statement Erdos859.erdos_859 (FormalConjectures/ErdosProblems/859.lean at the catalog's commit of 18 September 2026; the record pins a catalog commit that the public repository does not contain, and the statement it displays is the one in this file):

∃ c₁ > 0, ∃ c₂ > 0, ∃ d, (∀ t > 0, (Erdos859.DivisorSumSet t).HasDensity (d t)) ∧
  Asymptotics.IsEquivalent Filter.atTop (fun t => d t) fun t => c₁ / Real.log ↑t ^ c₂

Here DivisorSumSet t is the set of $n$ with a subset of the positive divisors of $n$ summing to $t$, HasDensity is the natural density (the limit of the proportion of members below $N$), the existential over d with the density conjunct packages "let $d_t$ be the density" (the natural density is unique when it exists), and Real.log ↑t ^ c₂ is $(\log t)^{c_2}$ with a real exponent, so the formal statement is the site's wording clause for clause and its negation is the disproof. The file proves the density conjunct rather than exploiting it, identifies any d satisfying it with the density by uniqueness of limits, and reaches the contradiction through an elementary lemma excluding $d\sim c/(\log t)^{p}$ for every $c>0$ and real $p$. The record's solution page (second link) holds the file, as of 2026-09-22, as its single source block: 36,197 lines and 1,721,167 bytes, with no import statement, inlining a Mertens-type prime reciprocal estimate adapted from the Apache-2.0 Lean project PrimeNumberTheoremAnd, whose license it reproduces. This corpus has not built the file.

Authorship and tools. The record credits the submitter's handle, JenW1N. The exposition published by Conjectures.io (third link) names the authors as Liam Kruer and Jensen Kohlmeyer, follows their manuscript "Divisor sums and a counterexample to Erdős problem 859" (dated 18 September 2026, 27 pages, not public), and says it was prepared with Codex assistance from that manuscript and the accepted Lean submission; the Lean file's header names no author and declares no AI system, and the site's forum entry gives the system as unknown, so which AI system, if any, generated the proof is not disclosed.

Acceptance. The reviewed evidence is the certification by the bounty site Conjectures.io: its Lean kernel verified the proof under Lean 4.33.1 with the axioms propext, Quot.sound and Classical.choice permitted, its review approved the record on 21 September 2026 under its policy v3, and the record was certified on 23 September 2026 and shows the bounty as paid. The review decision says the submission "refutes the proposed positive-constant logarithmic power asymptotic for the natural density of integers whose distinct divisors can sum to a prescribed target", that two agent assessments of the Codex/GPT-6 family using shared evidence and selected source checks supported approval and a human reviewer authorized the decision, and that the approval is "not an exhaustive novelty certification or fresh independent kernel replay"; its verification report records a static scan with "no imports, no axiom declarations, no sorry, no native_decide, no unsafe options", "Statement unchanged", "Lean kernel accepted", and a second kernel "Not run", so "The verdict rests on a single kernel implementation". Beyond the bounty site there is no acceptance: no refereed publication, and erdosproblems.com lists the problem as open. The curator, Thomas Bloom, posted a proof-claim entry on the problem's forum on 27 September 2026 (fourth link) to make the Conjectures.io claim known. The entry lists the claimant as conjectures.io with the system given as unknown. In it the curator says they had not verified the proof or looked into it, that the entry is no endorsement of the Conjectures.io program, which in their view uses the problems for its own ends without explaining its proofs or engaging with the mathematical community, and that the program is not transparent about who runs the problems through an AI system, for how long, or with which one. The entry's summary states the opposite direction (that the constants exist) and its link points at an earlier, different Lean contribution of 1 September 2026 that proves only that each DivisorSumSet t has positive density, a theorem of Erdős [Er70], so neither its summary nor its link describes this result. This corpus has not built the proof, so no formalized evidence is listed; what it checked is on the problem page.