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\\nabla \\cdot(u \\nabla v) \\ \\ \\ \\text{in } \\mathbb{R}^2\\times(0,T),\\\\[5pt] v = (-\\Delta_{\\mathbb{R}^2})^{-1} u := \\displaystyle\\frac {1}{2\\pi} \\displaystyle\\int_{\\mathbb{R}^2} \\log \\frac {1}{|x-z|}u(z,t) dz, \\ \\ \\ \\ \\ \\ \\ \\ \\ (\\star)\\\\[5pt] u(\\cdot ,0) = u_{0}^{\\star} \\ge 0 \\ \\ \\ \\text{in } \\mathbb{R}^2. \\end{cases} \\end{equation}$\n  We show that there exists $\\varepsilon>0$ such that for any $m$ satisfying $8\\pi<m\\le 8\\pi+\\varepsilon$ \n  and any $k$ given po"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2312.01475","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2312.01475/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2312.01475","created_at":"2026-07-05T07:30:06.308032+00:00"},{"alias_kind":"arxiv_version","alias_value":"2312.01475v1","created_at":"2026-07-05T07:30:06.308032+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2312.01475","created_at":"2026-07-05T07:30:06.308032+00:00"},{"alias_kind":"pith_short_12","alias_value":"QRYVHCV3AO3W","created_at":"2026-07-05T07:30:06.308032+00:00"},{"alias_kind":"pith_short_16","alias_value":"QRYVHCV3AO3WNZ3Z","created_at":"2026-07-05T07:30:06.308032+00:00"},{"alias_kind":"pith_short_8","alias_value":"QRYVHCV3","created_at":"2026-07-05T07:30:06.308032+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2606.02070","citing_title":"Critical mass threshold for the 2D Patlak-Keller-Segel-Navier-Stokes system","ref_index":5,"is_internal_anchor":false},{"citing_arxiv_id":"2604.13300","citing_title":"Refined temporal asymptotics near blow-up points in the planar Keller-Segel system","ref_index":6,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QRYVHCV3AO3WNZ3Z6AYNEJW5VT","json":"https://pith.science/pith/QRYVHCV3AO3WNZ3Z6AYNEJW5VT.json","graph_json":"https://pith.science/api/pith-number/QRYVHCV3AO3WNZ3Z6AYNEJW5VT/graph.json","events_json":"https://pith.science/api/pith-number/QRYVHCV3AO3WNZ3Z6AYNEJW5VT/events.json","paper":"https://pith.science/paper/QRYVHCV3"},"agent_actions":{"view_html":"https://pith.science/pith/QRYVHCV3AO3WNZ3Z6AYNEJW5VT","download_json":"https://pith.science/pith/QRYVHCV3AO3WNZ3Z6AYNEJW5VT.json","view_paper":"https://pith.science/paper/QRYVHCV3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2312.01475&json=true","fetch_graph":"https://pith.science/api/pith-number/QRYVHCV3AO3WNZ3Z6AYNEJW5VT/graph.json","fetch_events":"https://pith.science/api/pith-number/QRYVHCV3AO3WNZ3Z6AYNEJW5VT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QRYVHCV3AO3WNZ3Z6AYNEJW5VT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QRYVHCV3AO3WNZ3Z6AYNEJW5VT/action/storage_attestation","attest_author":"https://pith.science/pith/QRYVHCV3AO3WNZ3Z6AYNEJW5VT/action/author_attestation","sign_citation":"https://pith.science/pith/QRYVHCV3AO3WNZ3Z6AYNEJW5VT/action/citation_signature","submit_replication":"https://pith.science/pith/QRYVHCV3AO3WNZ3Z6AYNEJW5VT/action/replication_record"}},"created_at":"2026-07-05T07:30:06.308032+00:00","updated_at":"2026-07-05T07:30:06.308032+00:00"}