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Claim. Johan Land's full proof claim, submitted to the site's proof-claim tab on 4 September 2026, asserts a quantitative form of the restricted statement: for every sufficiently large $k$ there are intervals $I_i=[a_i,b_i]\subseteq[1,20k]$ of two or three consecutive integers, pairwise separated by at least one integer ($b_i+1<a_{i+1}$), whose reciprocals sum to exactly $1$. Since such intervals are distinct, non-overlapping and non-adjacent with $|I_i|\ge2$, this would answer the problem affirmatively. The claim's own account of its method: an absorption argument adapted from one the claimant attributes to Conlon and coauthors, with the pairs $(qm,qm+1)$ supplied by a Bourgain–Garaev estimate and the prime powers of the denominators cancelled by sparse inverse coverage; auxiliary powersmooth pairs fix the number of intervals contributed at each stage; a protected core and a pair-to-triple extension in the manner of Liu and Sawhney absorb the final deficit so that exactly $k$ intervals result. The claim's tab names the AI systems Astra, Fabel-5.1 and Gemini-Flash-3.8, working in a Lean repository built for proof search, as its tools; the claimant is the human submitter. The manuscript is the PDF in the claimant's repository, linked at the repository's head of 13 September 2026; the claim's thread comment of 4 September 2026 adds the repository as the formalization, says that the Bourgain–Garaev input was replaced by a modification of the proof, and says that the development compiles with no sorry and no axioms beyond propext, Classical.choice and Quot.sound. That is the claimant's statement; no build or review of the repository is recorded, so it gives no formalized evidence.

Submission note. Posted to erdosproblems.com as a proof claim by Johan Land (account JohanLand) on 4 September 2026, giving "Astra, Fabel 5.1, Gemini-Flash-3.8" as the AI used:

Astra/Fabel-5.1/Gemini-Flash-3.8 working inside a specialized ~3m lean repo for fast proof search. For all sufficiently large $k$, there exist intervals $I_i=[a_i,b_i]\subseteq[1,20k]$ with $|I_i|\in\{2,3\}$, $b_i+1<a_{i+1}$, and $\sum_i\sum_{n\in I_i}1/n=1$. Adapting Conlon et al.’s absorption method, Bourgain–Garaev yields $q^{1/10-o(1)}$ suitable pairs $(qm,qm+1)$ with $4\mid qm$ and $qm+1$ $(q/2)$-powersmooth. Sparse inverse coverage cancels each denominator prime power $q$ using at most $q^{1/20}$ pairs. Auxiliary $(q/2)$-powersmooth pairs make each stage contribute exactly $s_q=\lfloor q^{1/20}\rfloor$ intervals, at cost $O(q^{-21/20})$. Thus $C_H=\sum_{L<q\le H}s_q$ is predetermined. Starting with $k-R-C_H$ main pairs and $R$ protected pairs, Liu–Sawhney absorbs the final deficit $j/K$ through pair-to-triple extensions, giving exactly $k$ intervals. Spacing ensures nonadjacency throughout. Formalization coming...

Standing. Claimed. The site's label is OPEN (page last edited 22 September 2025; accessed 2026-10-07) and the tab carries its standing notice that listing a claim implies no examination; no curator, referee or named mathematician has accepted the proof, and no refereed or arXiv version exists. The thread under the claim holds a priority statement by the author of the second claim, who writes that their own manuscript was complete in mid-August 2026, and an exchange in which the two authors compare ingredients and propose to write a joint account; a later comment under the third claim says that its author had proposed a shorter solution to both and intends a short paper, a remark without a manuscript, which has no page of its own.