{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:TOJKTCTGLPOW77WP4LNRKNCPUW","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0ecc59adf3cfb94a5e72db32ce3d4f86c3c82c045b18b15e7d1131a6b2b291d0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-06-18T19:02:01Z","title_canon_sha256":"625aa393b0ac4ebff46318a1fc6c4ec2de4e8066e97ac1f77444180c360bfbd7"},"schema_version":"1.0","source":{"id":"2606.20870","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.20870","created_at":"2026-06-23T00:12:01Z"},{"alias_kind":"arxiv_version","alias_value":"2606.20870v1","created_at":"2026-06-23T00:12:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.20870","created_at":"2026-06-23T00:12:01Z"},{"alias_kind":"pith_short_12","alias_value":"TOJKTCTGLPOW","created_at":"2026-06-23T00:12:01Z"},{"alias_kind":"pith_short_16","alias_value":"TOJKTCTGLPOW77WP","created_at":"2026-06-23T00:12:01Z"},{"alias_kind":"pith_short_8","alias_value":"TOJKTCTG","created_at":"2026-06-23T00:12:01Z"}],"graph_snapshots":[{"event_id":"sha256:c20db9541a5ce59f593a2ba74216f4305f6082ff4ea62fdec9eec36497487acf","target":"graph","created_at":"2026-06-23T00:12:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.20870/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This note gives a complete classification of the asymptotic behavior of radial solutions to the two-dimensional parabolic-elliptic Keller-Segel system on the whole space, for general initial data in the large. We review previous separate results, and unify them within a single classification framework. Depending on the mass, the flow exhibits three distinct asymptotic regimes. For a subcritical mass, solutions converge toward the unique self-similar expander of same mass. At the critical mass $8\\pi$, solutions concentrate in infinite time around the stationary state with a universal logarithmi","authors_text":"Charles Collot, Federico Buseghin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-06-18T19:02:01Z","title":"Classification of the dynamics of radial solutions to the 2D parabolic-elliptic Keller-Segel System"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.20870","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:fe576a5161a23e3ca0b9f6e421960e3c0126f643ed39a33d93ee1c03dc3c4b21","target":"record","created_at":"2026-06-23T00:12:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0ecc59adf3cfb94a5e72db32ce3d4f86c3c82c045b18b15e7d1131a6b2b291d0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-06-18T19:02:01Z","title_canon_sha256":"625aa393b0ac4ebff46318a1fc6c4ec2de4e8066e97ac1f77444180c360bfbd7"},"schema_version":"1.0","source":{"id":"2606.20870","kind":"arxiv","version":1}},"canonical_sha256":"9b92a98a665bdd6ffecfe2db15344fa5b6b5202d15bb65a11610b3bfd372c4ce","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9b92a98a665bdd6ffecfe2db15344fa5b6b5202d15bb65a11610b3bfd372c4ce","first_computed_at":"2026-06-23T00:12:01.359758Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-23T00:12:01.359758Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/0hgKH1NITW0FE0Rw1ayMbFEHZaaY2mOuXK5l+tbdJXjjDPWospQ5Fe3ExnCSQgV+0iujpC+MmVSAEy4eT3mCQ==","signature_status":"signed_v1","signed_at":"2026-06-23T00:12:01.360131Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.20870","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fe576a5161a23e3ca0b9f6e421960e3c0126f643ed39a33d93ee1c03dc3c4b21","sha256:c20db9541a5ce59f593a2ba74216f4305f6082ff4ea62fdec9eec36497487acf"],"state_sha256":"5e872155a0b5805e1b1bf7189da63cb45e088fdb1c8826de3b5c60bb1bda173e"}