{"assessments":[],"deployments":[],"fuzz":[],"identity":{"adapter":"0xde152afb7db5373f34876e1499fbd893a82dd336","chainId":1,"collection":"0x0000ec93127baa929e58e97dd0095a2bfb38ec1d","registry":"0x8004a169fb4a3325136eb29fa0ceb6d2e539a432"},"interpretation":"Records acceptance and evidence. Neither completion nor an AI assessment establishes correctness, safety, or independent review.","jobId":"d2f6b891-e1aa-4b1f-b724-4324eab9df1b","kind":"skill:research-report","nodes":[{"acceptedSubmissionHash":"17d255f83c84858d69f77494d12015d7123f740a5966c8da39edf41e43c16a1f","dependsOn":[],"execution":{"network":true,"profile":"none","requires":["network"],"skillHash":"3ddca93330036359dd721585e58e67820336a0398b7927b3c89369d6134f30f6","skillId":"research-report","tools":[]},"key":"research_report","kind":"code","role":"implement","skillHash":"3ddca93330036359dd721585e58e67820336a0398b7927b3c89369d6134f30f6","skillId":"research-report","state":"accepted"}],"objective":"TASK: attempt open problems from the Justin Sun Prize bank (github.com/TheJustinSunPrize/awards, problems/). Work on ONE problem per agent.\n\nProblems (JSP id — catalog paraphrase):\n1. JSP-000246 — Can separated integer intervals be chosen so that the reciprocals of all their integers sum to one?\n2. JSP-000294 — Can distinct-element sums from the union of two floored exponential sequences represent every sufficiently large integer?\n3. JSP-000340 — Are there infinitely many n such that m + ω(m) ≤ n for all m < n? Same with c·ω(m) for any fixed c > 0?\n4. JSP-000372 — Is lcm(1..p_{k+1}−1) always smaller than p_k · lcm(1..p_k)?\n5. JSP-000550 — Must two disjoint equal-length intervals of consecutive integers have different lcm?\n6. JSP-000617 — Can an integer set have uniformly bounded two-term representation counts while its sumset density is arbitrarily close to 1?\n7. JSP-000682 — Is there an integer n such that d(n+k) is bounded linearly in k for all k?\n8. JSP-000738 — Is there a uniform constant bound on the number of divisors of n in an interval of length n^{1/4} near √n?\n9. JSP-000773 — Which disjoint intervals of consecutive integers have products with exactly the same set of prime factors?\n10. JSP-000947 — Is there an integer n such that n − 2^k is prime for every permitted 2^k < n?\n11. JSP-000951 — Does adding the set {2^a·3^b} always strictly increase the Schnirelmann density of another integer set?\n12. JSP-001004 — Does every 2-coloring of ℕ contain an infinite set whose specified pairwise sums all have one color?\n\nSTEP 0 (mandatory, before any math):\n- Find the exact original statement: the problem's page on erdosproblems.com or the original Erdős paper. The catalog text is only a paraphrase.\n- Check current status on erdosproblems.com, arXiv, and the open PRs/issues of the repo. If it is solved, or someone has an active complete-solution claim, STOP and report \"taken\".\n\nSTEP 1: try to solve the original statement completely (proof, disproof, or counterexample). Partial progress is not eligible, so do not claim it as a solution.\n\nSTEP 2: if solved, write a Lean 4 + Mathlib formalization of the EXACT original statement, compiling with no sorry and no extra axioms (check with #print axioms).\n\nRULES:\n- Never claim a solution without a compiling Lean proof.\n- Do not post anything publicly, do not open PRs, do not submit claims.\n- Report back only: JSP id, exact statement + source link, status check result, outcome (solved / counterexample / no result), the proof sketch, and the Lean file. If no result, say so in one line.","parentJobId":null,"planHash":"669a6ba28a6bf78b7be1c3f5c4b63a9c85f49f42fa7563ba3f1b6a6f0a4904de","previousHash":"0000000000000000000000000000000000000000000000000000000000000000","projectId":"d2f6b891-e1aa-4b1f-b724-4324eab9df1b","publication":{"commit":null,"deliveredAt":null,"repoUrl":null},"receiptIdentity":{"adapter":"0xde152afb7db5373f34876e1499fbd893a82dd336","chainId":1,"collection":"0x0000ec93127baa929e58e97dd0095a2bfb38ec1d","registry":"0x8004a169fb4a3325136eb29fa0ceb6d2e539a432"},"registry":"0xb6d0a187b050fa5bb0b87033a203f37becf4a775","research":[],"schema":"identitymd-work-v1","signals":[{"agentId":"51080","feedbackHash":"32ab16622d7a81cf41286a02f519495ff06478bf93303984ed4ca794145d1f4b","nodeKey":"research_report","submissionHash":"17d255f83c84858d69f77494d12015d7123f740a5966c8da39edf41e43c16a1f","tag1":"verification:structural","tag2":"acceptance-v2","value":1}],"site":null,"snapshotHash":"8297006bc3b76ba26217829696ba909eaa59eba63ddb3f5073445f94964b8bf9","state":"completed","submissions":[{"artifacts":[{"bytes":2335,"hash":"c134d86ff1243764b2a7636776f7d6f950144787534a87176ee94505412b5e26","mediaType":"text/markdown","name":"report","path":"artifacts/report.md"}],"attempt":1,"bundleHash":"e3fb42892fcc4c823e4a2679189ed63a07cd22703dafe00aba5c6d6322494c80","device":"19e9c602f9934c65","findings":[],"hash":"17d255f83c84858d69f77494d12015d7123f740a5966c8da39edf41e43c16a1f","nodeId":"756e08b0-bae5-4823-99f1-de5b4cc47217","outcome":"completed","summary":"JSP-000340 — no result; no complete proof, counterexample, or Lean file. Exact statement and status evidence: [report](artifacts/report.md).","treeHash":"801b584d81506ca5c74463c010fbf7dc6a2a775d","usage":{"cachedInputTokens":574976,"inputTokens":71473,"model":null,"outputTokens":4333,"runtime":"codex","turns":4,"wallClockMs":137069}}],"verification":[{"checks":[],"detail":"paths and tree verified; no suite was run for this kind of work","evaluation":"structural","profile":"none","status":"accepted","submissionHash":"17d255f83c84858d69f77494d12015d7123f740a5966c8da39edf41e43c16a1f","verifiedTreeHash":"801b584d81506ca5c74463c010fbf7dc6a2a775d","verifierVersion":"0.1.0+18543036"}]}