{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:MOE34XKTFWIA262KD5N6R5BHUZ","short_pith_number":"pith:MOE34XKT","schema_version":"1.0","canonical_sha256":"6389be5d532d900d7b4a1f5be8f427a649fea0301574b6a8d759fd7c4dfb49f3","source":{"kind":"arxiv","id":"2511.20264","version":4},"attestation_state":"computed","paper":{"title":"Symmetry and uniqueness of the positive solution for the critical Hartree equation on the Heisenberg group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jialin Wang, Jijie Xu, Shuijin Zhang, Xiang Li, Yu Zheng","submitted_at":"2025-11-25T12:49:28Z","abstract_excerpt":"We apply the moving plane method in integral forms to classify the positive solutions of the critical Hartree equation on Heisenberg group \\begin{equation}\\label{0.1}\n  -\\Delta_{\\mathbb{H}}u=\\left(\\int_{\\mathbb{H}^{n}}\\frac{|u(\\xi)|^{Q^{\\ast}_{\\mu}}}{|\\zeta^{-1}\\xi|^{\\mu}}\\mathrm{d}\\xi\\right)|u|^{Q^{\\ast}_{\\mu}-2}u,~~~\\zeta,\\xi\\in\\mathbb{H}^{n}, \\end{equation} where $\\Delta_{\\mathbb{H}}$ denotes the Kohn Laplacian, $u(\\xi)$ is a real-valued function, $Q=2n+2$ is the homogeneous dimension of $\\mathbb{H}^{n}$, $\\mu\\in (0,Q)$ is a real parameter and $Q^{\\ast}_{\\mu}=\\frac{2Q-\\mu}{Q-2}$ is the uppe"},"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":"2511.20264","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-11-25T12:49:28Z","cross_cats_sorted":[],"title_canon_sha256":"f81b76d31623558935e1bf59415effcd15df731f1a19ec21590cf9fb5804d2aa","abstract_canon_sha256":"ece561f4545cb61e8b75430bf8164fbb220bc44537346f9dde990dab93b6304e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-19T00:26:16.825256Z","signature_b64":"H/tbV+MZ07I0cT04h32cXg2A2Vak1KT4mxTa472P4k3vv6ujW74xWqUWPmEkX+DDmujjpSUkqhtHGVgJyciaCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6389be5d532d900d7b4a1f5be8f427a649fea0301574b6a8d759fd7c4dfb49f3","last_reissued_at":"2026-08-19T00:26:16.821997Z","signature_status":"signed_v1","first_computed_at":"2026-08-19T00:26:16.821997Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Symmetry and uniqueness of the positive solution for the critical Hartree equation on the Heisenberg group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jialin Wang, Jijie Xu, Shuijin Zhang, Xiang Li, Yu Zheng","submitted_at":"2025-11-25T12:49:28Z","abstract_excerpt":"We apply the moving plane method in integral forms to classify the positive solutions of the critical Hartree equation on Heisenberg group \\begin{equation}\\label{0.1}\n  -\\Delta_{\\mathbb{H}}u=\\left(\\int_{\\mathbb{H}^{n}}\\frac{|u(\\xi)|^{Q^{\\ast}_{\\mu}}}{|\\zeta^{-1}\\xi|^{\\mu}}\\mathrm{d}\\xi\\right)|u|^{Q^{\\ast}_{\\mu}-2}u,~~~\\zeta,\\xi\\in\\mathbb{H}^{n}, \\end{equation} where $\\Delta_{\\mathbb{H}}$ denotes the Kohn Laplacian, $u(\\xi)$ is a real-valued function, $Q=2n+2$ is the homogeneous dimension of $\\mathbb{H}^{n}$, $\\mu\\in (0,Q)$ is a real parameter and $Q^{\\ast}_{\\mu}=\\frac{2Q-\\mu}{Q-2}$ is the uppe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2511.20264","kind":"arxiv","version":4},"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/2511.20264/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":"2511.20264","created_at":"2026-08-19T00:26:16.822961+00:00"},{"alias_kind":"arxiv_version","alias_value":"2511.20264v4","created_at":"2026-08-19T00:26:16.822961+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2511.20264","created_at":"2026-08-19T00:26:16.822961+00:00"},{"alias_kind":"pith_short_12","alias_value":"MOE34XKTFWIA","created_at":"2026-08-19T00:26:16.822961+00:00"},{"alias_kind":"pith_short_16","alias_value":"MOE34XKTFWIA262K","created_at":"2026-08-19T00:26:16.822961+00:00"},{"alias_kind":"pith_short_8","alias_value":"MOE34XKT","created_at":"2026-08-19T00:26:16.822961+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2608.09034","citing_title":"Optimal Stability Bounds, Minimizers, and Critical Points for a Critical Nonlocal Sobolev Inequality on the Heisenberg Group","ref_index":48,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MOE34XKTFWIA262KD5N6R5BHUZ","json":"https://pith.science/pith/MOE34XKTFWIA262KD5N6R5BHUZ.json","graph_json":"https://pith.science/api/pith-number/MOE34XKTFWIA262KD5N6R5BHUZ/graph.json","events_json":"https://pith.science/api/pith-number/MOE34XKTFWIA262KD5N6R5BHUZ/events.json","paper":"https://pith.science/paper/MOE34XKT"},"agent_actions":{"view_html":"https://pith.science/pith/MOE34XKTFWIA262KD5N6R5BHUZ","download_json":"https://pith.science/pith/MOE34XKTFWIA262KD5N6R5BHUZ.json","view_paper":"https://pith.science/paper/MOE34XKT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2511.20264&json=true","fetch_graph":"https://pith.science/api/pith-number/MOE34XKTFWIA262KD5N6R5BHUZ/graph.json","fetch_events":"https://pith.science/api/pith-number/MOE34XKTFWIA262KD5N6R5BHUZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MOE34XKTFWIA262KD5N6R5BHUZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MOE34XKTFWIA262KD5N6R5BHUZ/action/storage_attestation","attest_author":"https://pith.science/pith/MOE34XKTFWIA262KD5N6R5BHUZ/action/author_attestation","sign_citation":"https://pith.science/pith/MOE34XKTFWIA262KD5N6R5BHUZ/action/citation_signature","submit_replication":"https://pith.science/pith/MOE34XKTFWIA262KD5N6R5BHUZ/action/replication_record"}},"created_at":"2026-08-19T00:26:16.822961+00:00","updated_at":"2026-08-19T00:26:16.822961+00:00"}