{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:3342IFULR4DRMR7XZ7UKXZPDDJ","short_pith_number":"pith:3342IFUL","canonical_record":{"source":{"id":"2606.11141","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-06-09T17:30:26Z","cross_cats_sorted":[],"title_canon_sha256":"4497116b53cdb5502b343d00534d9519f67556461c0f70a684e4650307e4bb12","abstract_canon_sha256":"716de238f272009527078dbbb04555c7c48db77d0028128dd8abc081f69ffd29"},"schema_version":"1.0"},"canonical_sha256":"def9a4168b8f071647f7cfe8abe5e31a7ef80860faa283a3ec3088014da8e630","source":{"kind":"arxiv","id":"2606.11141","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11141","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11141v1","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11141","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"pith_short_12","alias_value":"3342IFULR4DR","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"pith_short_16","alias_value":"3342IFULR4DRMR7X","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"pith_short_8","alias_value":"3342IFUL","created_at":"2026-06-10T01:11:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:3342IFULR4DRMR7XZ7UKXZPDDJ","target":"record","payload":{"canonical_record":{"source":{"id":"2606.11141","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-06-09T17:30:26Z","cross_cats_sorted":[],"title_canon_sha256":"4497116b53cdb5502b343d00534d9519f67556461c0f70a684e4650307e4bb12","abstract_canon_sha256":"716de238f272009527078dbbb04555c7c48db77d0028128dd8abc081f69ffd29"},"schema_version":"1.0"},"canonical_sha256":"def9a4168b8f071647f7cfe8abe5e31a7ef80860faa283a3ec3088014da8e630","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-10T01:11:13.277847Z","signature_b64":"Pp5HrC0FVJolEb2aQ8wZDICpAY52xcHk/o5y4IWqQRli2TVTGStKqi8Pv3iMcAOAQQbKW+WjXRaNSJ6J/+OSBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"def9a4168b8f071647f7cfe8abe5e31a7ef80860faa283a3ec3088014da8e630","last_reissued_at":"2026-06-10T01:11:13.276979Z","signature_status":"signed_v1","first_computed_at":"2026-06-10T01:11:13.276979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.11141","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-10T01:11:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2mSWMQncNOPCxS+r6LLceWgYoIPIaA7FFAe/HKPnGMoT1ebwkY+8bZMKV+KwODbqah+vsH8XzOTHw9mfpvN/Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T14:46:53.933233Z"},"content_sha256":"dc621ba7579cfcef9201f1a949b345336987f04e82decb05a0c8051a87922cae","schema_version":"1.0","event_id":"sha256:dc621ba7579cfcef9201f1a949b345336987f04e82decb05a0c8051a87922cae"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:3342IFULR4DRMR7XZ7UKXZPDDJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Harmonic extension of Weil-Petersson circle homeomorphisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Abderrahim Mesbah, Farid Diaf, Vladimir Markovi\\'c","submitted_at":"2026-06-09T17:30:26Z","abstract_excerpt":"In this paper, we study Weil--Petersson circle homeomorphisms from the viewpoint of harmonic maps. We prove that a homeomorphism $\\varphi:\\mathbb S^1\\to\\mathbb S^1$ is Weil--Petersson if and only if its unique quasiconformal harmonic extension to the hyperbolic disk $\\mathbb D$ has square-integrable Beltrami differential.\n  Our approach is based on the anti-holomorphic $L^2$-energy of harmonic maps. We show that this energy is finite for the quasiconformal harmonic extension of every Weil--Petersson circle homeomorphism, and that, among suitable quasiconformal extensions, the harmonic extensio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11141","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/2606.11141/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-10T01:11:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E6hCOAGMuvTx9zHsWNTxyUdo737GkYpB+3Tt7zDFLvzYT3EqJHEwW7bdlZQdAxZL4kvqkJ+ljujm+HCIDqP9Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T14:46:53.933671Z"},"content_sha256":"898c0a47da256bf14ac99db93a42736d70170ec6cc45be5547c2f71d8237c965","schema_version":"1.0","event_id":"sha256:898c0a47da256bf14ac99db93a42736d70170ec6cc45be5547c2f71d8237c965"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3342IFULR4DRMR7XZ7UKXZPDDJ/bundle.json","state_url":"https://pith.science/pith/3342IFULR4DRMR7XZ7UKXZPDDJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3342IFULR4DRMR7XZ7UKXZPDDJ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-02T14:46:53Z","links":{"resolver":"https://pith.science/pith/3342IFULR4DRMR7XZ7UKXZPDDJ","bundle":"https://pith.science/pith/3342IFULR4DRMR7XZ7UKXZPDDJ/bundle.json","state":"https://pith.science/pith/3342IFULR4DRMR7XZ7UKXZPDDJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3342IFULR4DRMR7XZ7UKXZPDDJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:3342IFULR4DRMR7XZ7UKXZPDDJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"716de238f272009527078dbbb04555c7c48db77d0028128dd8abc081f69ffd29","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-06-09T17:30:26Z","title_canon_sha256":"4497116b53cdb5502b343d00534d9519f67556461c0f70a684e4650307e4bb12"},"schema_version":"1.0","source":{"id":"2606.11141","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11141","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11141v1","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11141","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"pith_short_12","alias_value":"3342IFULR4DR","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"pith_short_16","alias_value":"3342IFULR4DRMR7X","created_at":"2026-06-10T01:11:13Z"},{"alias_kind":"pith_short_8","alias_value":"3342IFUL","created_at":"2026-06-10T01:11:13Z"}],"graph_snapshots":[{"event_id":"sha256:898c0a47da256bf14ac99db93a42736d70170ec6cc45be5547c2f71d8237c965","target":"graph","created_at":"2026-06-10T01:11:13Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.11141/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we study Weil--Petersson circle homeomorphisms from the viewpoint of harmonic maps. We prove that a homeomorphism $\\varphi:\\mathbb S^1\\to\\mathbb S^1$ is Weil--Petersson if and only if its unique quasiconformal harmonic extension to the hyperbolic disk $\\mathbb D$ has square-integrable Beltrami differential.\n  Our approach is based on the anti-holomorphic $L^2$-energy of harmonic maps. We show that this energy is finite for the quasiconformal harmonic extension of every Weil--Petersson circle homeomorphism, and that, among suitable quasiconformal extensions, the harmonic extensio","authors_text":"Abderrahim Mesbah, Farid Diaf, Vladimir Markovi\\'c","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-06-09T17:30:26Z","title":"Harmonic extension of Weil-Petersson circle homeomorphisms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11141","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:dc621ba7579cfcef9201f1a949b345336987f04e82decb05a0c8051a87922cae","target":"record","created_at":"2026-06-10T01:11:13Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"716de238f272009527078dbbb04555c7c48db77d0028128dd8abc081f69ffd29","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-06-09T17:30:26Z","title_canon_sha256":"4497116b53cdb5502b343d00534d9519f67556461c0f70a684e4650307e4bb12"},"schema_version":"1.0","source":{"id":"2606.11141","kind":"arxiv","version":1}},"canonical_sha256":"def9a4168b8f071647f7cfe8abe5e31a7ef80860faa283a3ec3088014da8e630","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"def9a4168b8f071647f7cfe8abe5e31a7ef80860faa283a3ec3088014da8e630","first_computed_at":"2026-06-10T01:11:13.276979Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-10T01:11:13.276979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Pp5HrC0FVJolEb2aQ8wZDICpAY52xcHk/o5y4IWqQRli2TVTGStKqi8Pv3iMcAOAQQbKW+WjXRaNSJ6J/+OSBg==","signature_status":"signed_v1","signed_at":"2026-06-10T01:11:13.277847Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.11141","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dc621ba7579cfcef9201f1a949b345336987f04e82decb05a0c8051a87922cae","sha256:898c0a47da256bf14ac99db93a42736d70170ec6cc45be5547c2f71d8237c965"],"state_sha256":"94374ea4fc151e0c44ca003fc0550cd6763604d40d5050289c485240b8fc6f0a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"moPoPJgAanWWED+RcMC3+yE9MSPxbnvp4u1agvUZNCAv3UCPfmnLtnVR0R2we9k9YacgVE8MuZLdhty0MBnEBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T14:46:53.935853Z","bundle_sha256":"64c605c9b47abe68e026e0107aa661c9cbe5f1559195dfbd47c7efeed66b52f6"}}