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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2606.17926","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-16T13:38:26Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"4e906fdc08e072019b0a246792683ec463ba824b503d296f06c2ca091a514f5b","abstract_canon_sha256":"324ba33f08ad27feb2512ea45c50af44113a717a936a3f154a98481077c6e290"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:10:43.723624Z","signature_b64":"3OPVpSMOWKN9xXncjUgpFFF4AHcmODH3Om/8CGxT4OW1tpQGwIiSAHtWRTuNMWZDCEFsH1nPaG4+2HYtEjFIDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"67c942664fed68263e701c8a868b961026449706ff98aab502c020b55c4a981b","last_reissued_at":"2026-06-19T16:10:43.723274Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:10:43.723274Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Square-Annular Dynamics and Coalescence Frontiers for $n+\\tau(n)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Eric Li (Trinity College, University of Cambridge)","submitted_at":"2026-06-16T13:38:26Z","abstract_excerpt":"Let $T(n)=n+\\tau(n)$, where $\\tau$ is the divisor function. 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