{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:DPIGREOQ6E2LXJA7DYNWSVOCVZ","short_pith_number":"pith:DPIGREOQ","canonical_record":{"source":{"id":"2509.07921","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SG","submitted_at":"2025-09-09T17:00:58Z","cross_cats_sorted":[],"title_canon_sha256":"2849bf64f44802a0ec8aaf432646df148d87285bf1c67eba02244eede20058b0","abstract_canon_sha256":"9b6b8f3a1a456b2b1d2b93036457d3b16d762e7e6e3056b8fa794fffbdbed550"},"schema_version":"1.0"},"canonical_sha256":"1bd06891d0f134bba41f1e1b6955c2ae58180728107116df1b3be123af834be5","source":{"kind":"arxiv","id":"2509.07921","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.07921","created_at":"2026-07-05T12:07:42Z"},{"alias_kind":"arxiv_version","alias_value":"2509.07921v1","created_at":"2026-07-05T12:07:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.07921","created_at":"2026-07-05T12:07:42Z"},{"alias_kind":"pith_short_12","alias_value":"DPIGREOQ6E2L","created_at":"2026-07-05T12:07:42Z"},{"alias_kind":"pith_short_16","alias_value":"DPIGREOQ6E2LXJA7","created_at":"2026-07-05T12:07:42Z"},{"alias_kind":"pith_short_8","alias_value":"DPIGREOQ","created_at":"2026-07-05T12:07:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:DPIGREOQ6E2LXJA7DYNWSVOCVZ","target":"record","payload":{"canonical_record":{"source":{"id":"2509.07921","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SG","submitted_at":"2025-09-09T17:00:58Z","cross_cats_sorted":[],"title_canon_sha256":"2849bf64f44802a0ec8aaf432646df148d87285bf1c67eba02244eede20058b0","abstract_canon_sha256":"9b6b8f3a1a456b2b1d2b93036457d3b16d762e7e6e3056b8fa794fffbdbed550"},"schema_version":"1.0"},"canonical_sha256":"1bd06891d0f134bba41f1e1b6955c2ae58180728107116df1b3be123af834be5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:07:42.447431Z","signature_b64":"VbHb8yn9Z/EHiLS/BltYNO144nf2Ch+QXK2mwskjWOgT3+CtRtZ5nst8U9BFji2Fea++xZqrV4feMeL9yFXRAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1bd06891d0f134bba41f1e1b6955c2ae58180728107116df1b3be123af834be5","last_reissued_at":"2026-07-05T12:07:42.446939Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:07:42.446939Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2509.07921","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":"2026-07-05T12:07:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cGC2wjGZphoU+tvyuvvgqKTd3ooevlotOrVPcBAfIdDEuRrJR+MPndqfLPdF0mQZ8C2d5MACEXHo/keLaPqEDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T10:15:02.350208Z"},"content_sha256":"57170233cfd9119ad7f1b70ac66f1cc9e8b8e7a763d69c551d0604e753202b51","schema_version":"1.0","event_id":"sha256:57170233cfd9119ad7f1b70ac66f1cc9e8b8e7a763d69c551d0604e753202b51"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:DPIGREOQ6E2LXJA7DYNWSVOCVZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bordered Legendrian Rational Symplectic Field Theory","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.SG","authors_text":"Maciej Wlodek","submitted_at":"2025-09-09T17:00:58Z","abstract_excerpt":"Given a Legendrian knot $\\Lambda \\subset \\mathbb{R}^3$ and a vertical line dividing the front projection of $\\Lambda$ into two halves, we construct a differential graded algebra associated to each half-knot. We then show that one may obtain the commutative algebra from Legendrian Rational Symplectic Field Theory as a pushout of the two bordered algebras. This construction extends the bordered Chekanov-Eliashberg differential graded algebra by incorporating disks with multiple positive punctures into the differential."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.07921","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/2509.07921/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":"2026-07-05T12:07:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"p/0sBbJZTSUgbNvmVpySY0V7wDgTv5hjo3uoxkjcT/DxRroc8CXzHuGb0gd2tYG73MgNJS8KeK0scDu3AvncDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T10:15:02.354173Z"},"content_sha256":"9c98284b21b1b0ccee5d9f0b997d4385f5abd872ca8eb76dc7eda7e9da462a7b","schema_version":"1.0","event_id":"sha256:9c98284b21b1b0ccee5d9f0b997d4385f5abd872ca8eb76dc7eda7e9da462a7b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DPIGREOQ6E2LXJA7DYNWSVOCVZ/bundle.json","state_url":"https://pith.science/pith/DPIGREOQ6E2LXJA7DYNWSVOCVZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DPIGREOQ6E2LXJA7DYNWSVOCVZ/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":"2026-08-17T10:15:02Z","links":{"resolver":"https://pith.science/pith/DPIGREOQ6E2LXJA7DYNWSVOCVZ","bundle":"https://pith.science/pith/DPIGREOQ6E2LXJA7DYNWSVOCVZ/bundle.json","state":"https://pith.science/pith/DPIGREOQ6E2LXJA7DYNWSVOCVZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DPIGREOQ6E2LXJA7DYNWSVOCVZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:DPIGREOQ6E2LXJA7DYNWSVOCVZ","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":"9b6b8f3a1a456b2b1d2b93036457d3b16d762e7e6e3056b8fa794fffbdbed550","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SG","submitted_at":"2025-09-09T17:00:58Z","title_canon_sha256":"2849bf64f44802a0ec8aaf432646df148d87285bf1c67eba02244eede20058b0"},"schema_version":"1.0","source":{"id":"2509.07921","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2509.07921","created_at":"2026-07-05T12:07:42Z"},{"alias_kind":"arxiv_version","alias_value":"2509.07921v1","created_at":"2026-07-05T12:07:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2509.07921","created_at":"2026-07-05T12:07:42Z"},{"alias_kind":"pith_short_12","alias_value":"DPIGREOQ6E2L","created_at":"2026-07-05T12:07:42Z"},{"alias_kind":"pith_short_16","alias_value":"DPIGREOQ6E2LXJA7","created_at":"2026-07-05T12:07:42Z"},{"alias_kind":"pith_short_8","alias_value":"DPIGREOQ","created_at":"2026-07-05T12:07:42Z"}],"graph_snapshots":[{"event_id":"sha256:9c98284b21b1b0ccee5d9f0b997d4385f5abd872ca8eb76dc7eda7e9da462a7b","target":"graph","created_at":"2026-07-05T12:07:42Z","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/2509.07921/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Given a Legendrian knot $\\Lambda \\subset \\mathbb{R}^3$ and a vertical line dividing the front projection of $\\Lambda$ into two halves, we construct a differential graded algebra associated to each half-knot. We then show that one may obtain the commutative algebra from Legendrian Rational Symplectic Field Theory as a pushout of the two bordered algebras. This construction extends the bordered Chekanov-Eliashberg differential graded algebra by incorporating disks with multiple positive punctures into the differential.","authors_text":"Maciej Wlodek","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SG","submitted_at":"2025-09-09T17:00:58Z","title":"Bordered Legendrian Rational Symplectic Field Theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.07921","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":"sha256:57170233cfd9119ad7f1b70ac66f1cc9e8b8e7a763d69c551d0604e753202b51","target":"record","created_at":"2026-07-05T12:07:42Z","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":"9b6b8f3a1a456b2b1d2b93036457d3b16d762e7e6e3056b8fa794fffbdbed550","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SG","submitted_at":"2025-09-09T17:00:58Z","title_canon_sha256":"2849bf64f44802a0ec8aaf432646df148d87285bf1c67eba02244eede20058b0"},"schema_version":"1.0","source":{"id":"2509.07921","kind":"arxiv","version":1}},"canonical_sha256":"1bd06891d0f134bba41f1e1b6955c2ae58180728107116df1b3be123af834be5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1bd06891d0f134bba41f1e1b6955c2ae58180728107116df1b3be123af834be5","first_computed_at":"2026-07-05T12:07:42.446939Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:07:42.446939Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VbHb8yn9Z/EHiLS/BltYNO144nf2Ch+QXK2mwskjWOgT3+CtRtZ5nst8U9BFji2Fea++xZqrV4feMeL9yFXRAA==","signature_status":"signed_v1","signed_at":"2026-07-05T12:07:42.447431Z","signed_message":"canonical_sha256_bytes"},"source_id":"2509.07921","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:57170233cfd9119ad7f1b70ac66f1cc9e8b8e7a763d69c551d0604e753202b51","sha256:9c98284b21b1b0ccee5d9f0b997d4385f5abd872ca8eb76dc7eda7e9da462a7b"],"state_sha256":"1ff2d5e72eef34240c2829ed57bf2850aa8b4cee15d2894e426c1e66d7cf390d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"96U3PUO3GUnD2ysb0a15wZM2TbSJccP4wqycUQoXLHOxxDlb+FmIcWa1alIS19I255Pg1CfHtyyTbZzpRwLmCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-17T10:15:02.370572Z","bundle_sha256":"fe91223724a80a659f31716c4af4301a96e52f8b7e402c27f72ae8a3c7bf9045"}}