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When $n=2$, we prove that there exists an integer $s\\geq1$ such that the log canonical threshold $\\mathrm{lct}(\\mathbf{P}^2;R_{f^s})\\geq1/(q^s-1)$. The passage to an iterate is necessary in general, and the lower bound is optimal. In particular, $(\\mathbf{P}^2,R_{f^s}/(q^s-1))$ is a log Calabi--Yau pair, completing the"},"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":"2608.02114","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2026-08-03T12:11:14Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"1904d7e80e9efc9361e515d4037d6dc9fc3439779972adcdbfdcd87701590847","abstract_canon_sha256":"f045b8d6152f2595f3dd8c7cf54757bebd890725607d9a90c98b6048cf919f21"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-04T02:10:58.213558Z","signature_b64":"GqG9AGPP95fKDWJdWgjXv/G/aFGbPs5IY15Jx59/JcRFJMc/sX4zDBghm7ME4xSVj5dH+xB1YGDaBBcWPfR9AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8b266731abc49350945725f5649f7d466736d722e83aa7df17247301d8450b20","last_reissued_at":"2026-08-04T02:10:58.211769Z","signature_status":"signed_v1","first_computed_at":"2026-08-04T02:10:58.211769Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Log Calabi--Yau structure for endomorphisms on $\\mathbf{P}^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.AG","authors_text":"Sheng Meng, Yujie Luo","submitted_at":"2026-08-03T12:11:14Z","abstract_excerpt":"Let $f:\\mathbf{P}^n\\to\\mathbf{P}^n$ be a $q$-polarized endomorphism, where $q>1$, and let $R_f$ be its ramification divisor. We study the singularities of the ramification pair $(\\mathbf{P}^n,R_f)$. We show that, for a general $f$, the pair $(\\mathbf{P}^n,R_f)$ is log canonical. When $n=2$, we prove that there exists an integer $s\\geq1$ such that the log canonical threshold $\\mathrm{lct}(\\mathbf{P}^2;R_{f^s})\\geq1/(q^s-1)$. The passage to an iterate is necessary in general, and the lower bound is optimal. In particular, $(\\mathbf{P}^2,R_{f^s}/(q^s-1))$ is a log Calabi--Yau pair, completing the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.02114","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/2608.02114/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":"2608.02114","created_at":"2026-08-04T02:10:58.213530+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.02114v1","created_at":"2026-08-04T02:10:58.213530+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.02114","created_at":"2026-08-04T02:10:58.213530+00:00"},{"alias_kind":"pith_short_12","alias_value":"RMTGOMNLYSJV","created_at":"2026-08-04T02:10:58.213530+00:00"},{"alias_kind":"pith_short_16","alias_value":"RMTGOMNLYSJVBFCX","created_at":"2026-08-04T02:10:58.213530+00:00"},{"alias_kind":"pith_short_8","alias_value":"RMTGOMNL","created_at":"2026-08-04T02:10:58.213530+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/RMTGOMNLYSJVBFCXEX2WJH35IZ","json":"https://pith.science/pith/RMTGOMNLYSJVBFCXEX2WJH35IZ.json","graph_json":"https://pith.science/api/pith-number/RMTGOMNLYSJVBFCXEX2WJH35IZ/graph.json","events_json":"https://pith.science/api/pith-number/RMTGOMNLYSJVBFCXEX2WJH35IZ/events.json","paper":"https://pith.science/paper/RMTGOMNL"},"agent_actions":{"view_html":"https://pith.science/pith/RMTGOMNLYSJVBFCXEX2WJH35IZ","download_json":"https://pith.science/pith/RMTGOMNLYSJVBFCXEX2WJH35IZ.json","view_paper":"https://pith.science/paper/RMTGOMNL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.02114&json=true","fetch_graph":"https://pith.science/api/pith-number/RMTGOMNLYSJVBFCXEX2WJH35IZ/graph.json","fetch_events":"https://pith.science/api/pith-number/RMTGOMNLYSJVBFCXEX2WJH35IZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RMTGOMNLYSJVBFCXEX2WJH35IZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RMTGOMNLYSJVBFCXEX2WJH35IZ/action/storage_attestation","attest_author":"https://pith.science/pith/RMTGOMNLYSJVBFCXEX2WJH35IZ/action/author_attestation","sign_citation":"https://pith.science/pith/RMTGOMNLYSJVBFCXEX2WJH35IZ/action/citation_signature","submit_replication":"https://pith.science/pith/RMTGOMNLYSJVBFCXEX2WJH35IZ/action/replication_record"}},"created_at":"2026-08-04T02:10:58.213530+00:00","updated_at":"2026-08-04T02:10:58.213530+00:00"}