{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:QERSKWHGEEF6N7Q64GGT3IXBMS","short_pith_number":"pith:QERSKWHG","schema_version":"1.0","canonical_sha256":"81232558e6210be6fe1ee18d3da2e164af6ba0577b7faf8c8fc0236d7d1bb7d1","source":{"kind":"arxiv","id":"2605.17013","version":1},"attestation_state":"computed","paper":{"title":"Positivity of arbitrary-order P-recursive sequences with a unique dominant root","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A sufficient condition proves ultimate positivity for P-recursive sequences of any order with a unique dominant root.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Zhongjie Li","submitted_at":"2026-05-16T14:22:49Z","abstract_excerpt":"We establish a sufficient condition for the ultimate positivity of P-recursive sequences of arbitrary order with a unique dominant root. By additionally verifying finitely many initial terms, the positivity can also be resolved. As an application, we provide several examples of P-recursive sequences of order greater than two."},"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":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.17013","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-16T14:22:49Z","cross_cats_sorted":[],"title_canon_sha256":"f6bc61ee9f609334f774f458e22bb329281fb5460f1d8c73a0fdae2d3a2a3e9a","abstract_canon_sha256":"da28ff9559cf407b5b0d2dad6817725d2dc978ab339b507f01a6ef735023e753"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:03:35.901726Z","signature_b64":"9ScDRmYSxTW3HdVvmYyNS3+iEgUcTGBeOuesvqyruY0kGPg/SOncD7dT/0SgrlxdPY9Nm4uvzYUP9tmvFc6YBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"81232558e6210be6fe1ee18d3da2e164af6ba0577b7faf8c8fc0236d7d1bb7d1","last_reissued_at":"2026-05-20T00:03:35.900875Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:03:35.900875Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Positivity of arbitrary-order P-recursive sequences with a unique dominant root","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A sufficient condition proves ultimate positivity for P-recursive sequences of any order with a unique dominant root.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Zhongjie Li","submitted_at":"2026-05-16T14:22:49Z","abstract_excerpt":"We establish a sufficient condition for the ultimate positivity of P-recursive sequences of arbitrary order with a unique dominant root. By additionally verifying finitely many initial terms, the positivity can also be resolved. As an application, we provide several examples of P-recursive sequences of order greater than two."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We establish a sufficient condition for the ultimate positivity of P-recursive sequences of arbitrary order with a unique dominant root.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The sequence possesses a unique dominant root (i.e., one root of the characteristic equation strictly dominates all others in modulus), which is invoked as the structural hypothesis enabling the sufficient positivity condition.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A sufficient condition is derived for ultimate positivity of arbitrary-order P-recursive sequences with a unique dominant root, allowing positivity to be settled by finite initial-term verification, with concrete examples for orders exceeding two.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A sufficient condition proves ultimate positivity for P-recursive sequences of any order with a unique dominant root.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"1cc89f4ff7d0f539ae22d7254becc55a4d940e2592a4c723250ae80d8e32b9cd"},"source":{"id":"2605.17013","kind":"arxiv","version":1},"verdict":{"id":"c60e6d4d-a204-4664-9cbe-6d3443ff789a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T20:00:01.341361Z","strongest_claim":"We establish a sufficient condition for the ultimate positivity of P-recursive sequences of arbitrary order with a unique dominant root.","one_line_summary":"A sufficient condition is derived for ultimate positivity of arbitrary-order P-recursive sequences with a unique dominant root, allowing positivity to be settled by finite initial-term verification, with concrete examples for orders exceeding two.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The sequence possesses a unique dominant root (i.e., one root of the characteristic equation strictly dominates all others in modulus), which is invoked as the structural hypothesis enabling the sufficient positivity condition.","pith_extraction_headline":"A sufficient condition proves ultimate positivity for P-recursive sequences of any order with a unique dominant root."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17013/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T20:31:19.015700Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T20:10:46.972822Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"citation_quote_validity","ran_at":"2026-05-19T19:49:48.653083Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"cited_work_retraction","ran_at":"2026-05-19T18:51:59.250332Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T18:41:56.188933Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:24.874947Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"75c53d5cd29ecc08cfb4a05f57c5c845d9c12750c544403de14246acf0a53d51"},"references":{"count":18,"sample":[{"doi":"","year":2006,"title":"J.P. Bell and S. Gerhold, On the positivity set of a linear recurrence sequence, Israel J. Math.157(2006), 333–345. 13","work_id":"c5b9a3ad-de2e-4cb1-83c4-acbd7aaf0e26","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1989,"title":"T.W. Cusick, Recurrences for sums of powers of binomial coefficients, J. Combin. Theory Ser. A52(1989), 77–83","work_id":"84d7d6c4-148d-4935-b6d4-8dce7d129f7c","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2001,"title":"E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math.241(2001), 241–265","work_id":"55dd2e66-3562-4a73-aa05-247f450eab97","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2006,"title":"V. Halava, T. Harju and M. Hirvensalo, Positivity of second order linear recurrent sequences, Discrete Appl. Math.154(2006), 447–451","work_id":"bcb685ea-058b-472a-9a0c-14bfe7a05dc6","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"A. Ibrahim and B. Salvy, Positivity Certificates for Linear Recurrences, Proc. 2024 Annu. ACM-SIAM Symp. Discrete Algorithms (SODA) (2024), 982–994","work_id":"22df54a4-df86-46a9-9684-534b666eff5c","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":18,"snapshot_sha256":"5bbbb8da84ed1db990c58e2043d710bdee6264da6c10d6a38c0ec04d7529f3e0","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"2290857c6aba86b1c58d283360c9c3f54cbfb5bb62efdf55ad9ea4102e7f4fab"},"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":"2605.17013","created_at":"2026-05-20T00:03:35.901010+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.17013v1","created_at":"2026-05-20T00:03:35.901010+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.17013","created_at":"2026-05-20T00:03:35.901010+00:00"},{"alias_kind":"pith_short_12","alias_value":"QERSKWHGEEF6","created_at":"2026-05-20T00:03:35.901010+00:00"},{"alias_kind":"pith_short_16","alias_value":"QERSKWHGEEF6N7Q6","created_at":"2026-05-20T00:03:35.901010+00:00"},{"alias_kind":"pith_short_8","alias_value":"QERSKWHG","created_at":"2026-05-20T00:03:35.901010+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":2,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QERSKWHGEEF6N7Q64GGT3IXBMS","json":"https://pith.science/pith/QERSKWHGEEF6N7Q64GGT3IXBMS.json","graph_json":"https://pith.science/api/pith-number/QERSKWHGEEF6N7Q64GGT3IXBMS/graph.json","events_json":"https://pith.science/api/pith-number/QERSKWHGEEF6N7Q64GGT3IXBMS/events.json","paper":"https://pith.science/paper/QERSKWHG"},"agent_actions":{"view_html":"https://pith.science/pith/QERSKWHGEEF6N7Q64GGT3IXBMS","download_json":"https://pith.science/pith/QERSKWHGEEF6N7Q64GGT3IXBMS.json","view_paper":"https://pith.science/paper/QERSKWHG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.17013&json=true","fetch_graph":"https://pith.science/api/pith-number/QERSKWHGEEF6N7Q64GGT3IXBMS/graph.json","fetch_events":"https://pith.science/api/pith-number/QERSKWHGEEF6N7Q64GGT3IXBMS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QERSKWHGEEF6N7Q64GGT3IXBMS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QERSKWHGEEF6N7Q64GGT3IXBMS/action/storage_attestation","attest_author":"https://pith.science/pith/QERSKWHGEEF6N7Q64GGT3IXBMS/action/author_attestation","sign_citation":"https://pith.science/pith/QERSKWHGEEF6N7Q64GGT3IXBMS/action/citation_signature","submit_replication":"https://pith.science/pith/QERSKWHGEEF6N7Q64GGT3IXBMS/action/replication_record"}},"created_at":"2026-05-20T00:03:35.901010+00:00","updated_at":"2026-05-20T00:03:35.901010+00:00"}