{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:M7EUEZSP5VUCMPTQDSFINC4WCA","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":"324ba33f08ad27feb2512ea45c50af44113a717a936a3f154a98481077c6e290","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-16T13:38:26Z","title_canon_sha256":"4e906fdc08e072019b0a246792683ec463ba824b503d296f06c2ca091a514f5b"},"schema_version":"1.0","source":{"id":"2606.17926","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.17926","created_at":"2026-06-19T16:10:43Z"},{"alias_kind":"arxiv_version","alias_value":"2606.17926v1","created_at":"2026-06-19T16:10:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.17926","created_at":"2026-06-19T16:10:43Z"},{"alias_kind":"pith_short_12","alias_value":"M7EUEZSP5VUC","created_at":"2026-06-19T16:10:43Z"},{"alias_kind":"pith_short_16","alias_value":"M7EUEZSP5VUCMPTQ","created_at":"2026-06-19T16:10:43Z"},{"alias_kind":"pith_short_8","alias_value":"M7EUEZSP","created_at":"2026-06-19T16:10:43Z"}],"graph_snapshots":[{"event_id":"sha256:232310aa3ecb066c6042564c1e8195f39d9c8c70d96b9cb8f26139fb700fa17a","target":"graph","created_at":"2026-06-19T16:10:43Z","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/2606.17926/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $T(n)=n+\\tau(n)$, where $\\tau$ is the divisor function. We study the Erdos-Graham coalescence problem by encoding finite-level obstructions in the divisor-successor graph and in square-annular transfer maps. Coalescence is equivalent both to connectedness of this graph and to synchronization along an infinite non-autonomous sequence of finite annular systems. The basic identities are \\[\n  \\operatorname{im}(\\mathcal A_k)=E_{k+1},\n  \\qquad\n  \\mathcal F_{k^2}=k^2+E_k, \\] where $E_k$ is the set of square-crossing overshoots from below $k^2$. We prove a transfer parity law, dynamic frontier bou","authors_text":"Eric Li (Trinity College, University of Cambridge)","cross_cats":["math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-16T13:38:26Z","title":"Square-Annular Dynamics and Coalescence Frontiers for $n+\\tau(n)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.17926","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:baf2dd061932dc9c587b4c5dfca327486a3d45aa340a97c80e46351843734719","target":"record","created_at":"2026-06-19T16:10:43Z","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":"324ba33f08ad27feb2512ea45c50af44113a717a936a3f154a98481077c6e290","cross_cats_sorted":["math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-16T13:38:26Z","title_canon_sha256":"4e906fdc08e072019b0a246792683ec463ba824b503d296f06c2ca091a514f5b"},"schema_version":"1.0","source":{"id":"2606.17926","kind":"arxiv","version":1}},"canonical_sha256":"67c942664fed68263e701c8a868b961026449706ff98aab502c020b55c4a981b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"67c942664fed68263e701c8a868b961026449706ff98aab502c020b55c4a981b","first_computed_at":"2026-06-19T16:10:43.723274Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-19T16:10:43.723274Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3OPVpSMOWKN9xXncjUgpFFF4AHcmODH3Om/8CGxT4OW1tpQGwIiSAHtWRTuNMWZDCEFsH1nPaG4+2HYtEjFIDA==","signature_status":"signed_v1","signed_at":"2026-06-19T16:10:43.723624Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.17926","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:baf2dd061932dc9c587b4c5dfca327486a3d45aa340a97c80e46351843734719","sha256:232310aa3ecb066c6042564c1e8195f39d9c8c70d96b9cb8f26139fb700fa17a"],"state_sha256":"f319db1cd32efef0ad268e8a1086c3d1f36fbe74eca1bd42d3324bcbb569a57b"}