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Claim. Yuren Tang's full proof claim, submitted to the site's proof-claim tab on 7 September 2026, asserts a stronger form of the restricted statement: for every sufficiently large $k$, the number $1$ is the sum of the reciprocals over $k$ pairwise disjoint and non-adjacent intervals of positive integers, each of two or three elements. The manuscript, Reciprocal Sums over Separated Integer Intervals, was published in the claimant's repository on 7 September 2026 as its version 1.0.0, linked above at that tag's commit; the repository also holds an exact-arithmetic verifier for the manuscript's Appendix C and its display (3.1), and a Lean formalization that the claimant describes as in progress and that, by the first claimant's reading of an earlier snapshot, does not yet prove the final theorem. The claim's own account of its method: the argument tracks, for a configuration of intervals, the fractional part of its reciprocal sum and its number of components; residues are moved along a filtration of the least common multiple by prime torsion, with the Dias da Silva–Hamidoune theorem supplying the fixed-cardinality residue responses; switches that leave the reciprocal sum unchanged adjust the component count, and prime stages widen the range of attainable counts enough to cover every large $k$; Haxell's independent-transversal theorem makes the choices realizable by disjoint, non-adjacent intervals; the sum, with fractional part $0$ and value strictly between $0$ and $2$, is then exactly $1$. The claim's tab names the AI systems GPT-5.6 Sol for the core mathematical development, GPT-5.6 Sol, GPT-6 Astra and Claude Opus 5 for drafting and review, and possibly other OpenAI or Anthropic models; the claimant is the human submitter. The claimant's comment under the first claim states that the manuscript was complete in mid-August 2026 and points to a formalization commit of 18 August 2026 as provenance; the repository's first commit is dated 13 August 2026. No build or review of the repository is recorded.

Submission note. Posted to erdosproblems.com as a proof claim by Yuren Tang (account Yuren_Tang) on 7 September 2026, giving "GPT-5.6 Sol (core mathematical development); GPT-5.6 Sol, GPT-6 Astra and Claude Opus 5 (drafting and review); possibly other OpenAI or Anthropic models" as the AI used:

We prove a stronger form of Erdős Problem 289, with every component interval of cardinality 2 or 3. The proof separates the two observables

$$> \Omega(S)=([W(S)],\nu(S))\in(\mathbb Q/\mathbb Z)\times\mathbb N,$$

while

retaining the actual interval configurations as witnesses. Residues are transferred along the lcm torsion filtration, whose nontrivial successive quotients are $\mathbb Z/p\mathbb Z$; the required fixed-cardinality residue responses ultimately come from the Dias da Silva–Hamidoune theorem. Reciprocal-neutral switches vary the component count without changing $W$, at arbitrarily small mass. Prime stages then create count width of order $q/\log q$, dominating the lower-order composite-prime-power losses and eventually covering every sufficiently large count. Haxell’s independent-transversal theorem makes these choices simultaneously realizable by disjoint, nonadjacent intervals. Finally, $[W]=0$ and $0<W<2$ force $W=1$. Notes: Conceptually, I think of the proof as two machines joined by fibrewise compatibility. The paper uses categorical language throughout because this is how the proof was discovered and organized. The tools involved—fibres, pullbacks, commutative diagrams, universal properties—are elementary; for me they are basic tools for exposing structure and discarding accidental machinery (especially in AI-assisted mathematics, where generating heavy machinery is cheap). As an example, this made the restricted-sum step simple enough to prove directly; only later did I recognize it as the Dias da Silva–Hamidoune theorem. I do not think the reduction has reached its natural endpoint everywhere; Appendix C is the clearest example of numerical witnesses that still seem to conceal a structural statement. An exact-arithmetic verifier for Appendix C and (3.1) is included in Version 1.0.0.

Standing. Claimed. The site's label is OPEN (page last edited 22 September 2025; accessed 2026-10-07), the tab carries its standing notice that listing a claim implies no examination, no comment stands under the claim itself, and no curator, referee or named mathematician has accepted the proof; there is no refereed or arXiv version. The claim was posted three days after Johan Land's claim of the same statement by a different method, and the two authors propose in that claim's thread to compare their arguments.