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Chen, Procaccia and Zong proved the sharp upper bound $\\mathbb{E}[\\omega_{N,\\lambda}]\\le(1+\\varepsilon)(\\log N)/(2\\lambda)$ and conjectured the matching limit. Here we prove the matching lower bound, and therefore \\[\n  \\lim_{N\\to\\infty}\\frac{\\mathbb{E}[\\omega_{N,\\lambda}]}{\\log N}=\\frac{1}{2\\lambda}\n  \\qquad\\t"},"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":"2608.06925","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-08-07T07:59:36Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"ac51ffd34ee29e486717cc669e4743f9cab0a1b141c03da350014b44132e2311","abstract_canon_sha256":"68fd77078ac8c164a9da7041e89b3ba362f014346b0c9e95b00710c2d488196c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-10T01:12:31.815379Z","signature_b64":"/+b8smubXz3sqrD4fmeI6hMshuTXcyRFXHX2Et9VQDHlqPjekL+nkqklQofsaBorym5bTJk8sgv3HZRzZNa5Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"05bde7e3a8795188dadeeada2245ad5df831da481dc64b522aa219ee987bca71","last_reissued_at":"2026-08-10T01:12:31.812902Z","signature_status":"signed_v1","first_computed_at":"2026-08-10T01:12:31.812902Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp asymptotics for the tree-completion time in cylindrical Hastings--Levitov$(0)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Tianyang Sun, Xiao-Ming Fu, Yuxuan Zong","submitted_at":"2026-08-07T07:59:36Z","abstract_excerpt":"Let $\\mathrm{CHL}_N$ be the cylindrical Hastings--Levitov aggregation process with parameter $0$ on a cylinder of width $N$ with particles of fixed size $\\lambda>0$, and let $\\omega_{N,\\lambda}$ be its tree-completion time --- the last time at which a new tree is born on the base circle. Chen, Procaccia and Zong proved the sharp upper bound $\\mathbb{E}[\\omega_{N,\\lambda}]\\le(1+\\varepsilon)(\\log N)/(2\\lambda)$ and conjectured the matching limit. Here we prove the matching lower bound, and therefore \\[\n  \\lim_{N\\to\\infty}\\frac{\\mathbb{E}[\\omega_{N,\\lambda}]}{\\log N}=\\frac{1}{2\\lambda}\n  \\qquad\\t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.06925","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/2608.06925/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":"2608.06925","created_at":"2026-08-10T01:12:31.814027+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.06925v1","created_at":"2026-08-10T01:12:31.814027+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.06925","created_at":"2026-08-10T01:12:31.814027+00:00"},{"alias_kind":"pith_short_12","alias_value":"AW66PY5IPFIY","created_at":"2026-08-10T01:12:31.814027+00:00"},{"alias_kind":"pith_short_16","alias_value":"AW66PY5IPFIYRWW6","created_at":"2026-08-10T01:12:31.814027+00:00"},{"alias_kind":"pith_short_8","alias_value":"AW66PY5I","created_at":"2026-08-10T01:12:31.814027+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AW66PY5IPFIYRWW65LNCERNNLX","json":"https://pith.science/pith/AW66PY5IPFIYRWW65LNCERNNLX.json","graph_json":"https://pith.science/api/pith-number/AW66PY5IPFIYRWW65LNCERNNLX/graph.json","events_json":"https://pith.science/api/pith-number/AW66PY5IPFIYRWW65LNCERNNLX/events.json","paper":"https://pith.science/paper/AW66PY5I"},"agent_actions":{"view_html":"https://pith.science/pith/AW66PY5IPFIYRWW65LNCERNNLX","download_json":"https://pith.science/pith/AW66PY5IPFIYRWW65LNCERNNLX.json","view_paper":"https://pith.science/paper/AW66PY5I","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.06925&json=true","fetch_graph":"https://pith.science/api/pith-number/AW66PY5IPFIYRWW65LNCERNNLX/graph.json","fetch_events":"https://pith.science/api/pith-number/AW66PY5IPFIYRWW65LNCERNNLX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AW66PY5IPFIYRWW65LNCERNNLX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AW66PY5IPFIYRWW65LNCERNNLX/action/storage_attestation","attest_author":"https://pith.science/pith/AW66PY5IPFIYRWW65LNCERNNLX/action/author_attestation","sign_citation":"https://pith.science/pith/AW66PY5IPFIYRWW65LNCERNNLX/action/citation_signature","submit_replication":"https://pith.science/pith/AW66PY5IPFIYRWW65LNCERNNLX/action/replication_record"}},"created_at":"2026-08-10T01:12:31.814027+00:00","updated_at":"2026-08-10T01:12:31.814027+00:00"}