{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:QCAJG6GOSTLCDAJUKBNTP2JZIY","short_pith_number":"pith:QCAJG6GO","canonical_record":{"source":{"id":"2505.19796","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2025-05-26T10:29:28Z","cross_cats_sorted":[],"title_canon_sha256":"1b5864b71062149844464d9652ab1a0fe36f34c1b3f51919d3fb0912443a8216","abstract_canon_sha256":"b2355a70366d093a74f6d949b78d2b7667fa12fc22f7b0e012c95143950ab32e"},"schema_version":"1.0"},"canonical_sha256":"80809378ce94d6218134505b37e93946037c1dfd6f7bf2bc5a0ae4bc9b8f48b2","source":{"kind":"arxiv","id":"2505.19796","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2505.19796","created_at":"2026-07-05T11:09:36Z"},{"alias_kind":"arxiv_version","alias_value":"2505.19796v1","created_at":"2026-07-05T11:09:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.19796","created_at":"2026-07-05T11:09:36Z"},{"alias_kind":"pith_short_12","alias_value":"QCAJG6GOSTLC","created_at":"2026-07-05T11:09:36Z"},{"alias_kind":"pith_short_16","alias_value":"QCAJG6GOSTLCDAJU","created_at":"2026-07-05T11:09:36Z"},{"alias_kind":"pith_short_8","alias_value":"QCAJG6GO","created_at":"2026-07-05T11:09:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:QCAJG6GOSTLCDAJUKBNTP2JZIY","target":"record","payload":{"canonical_record":{"source":{"id":"2505.19796","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2025-05-26T10:29:28Z","cross_cats_sorted":[],"title_canon_sha256":"1b5864b71062149844464d9652ab1a0fe36f34c1b3f51919d3fb0912443a8216","abstract_canon_sha256":"b2355a70366d093a74f6d949b78d2b7667fa12fc22f7b0e012c95143950ab32e"},"schema_version":"1.0"},"canonical_sha256":"80809378ce94d6218134505b37e93946037c1dfd6f7bf2bc5a0ae4bc9b8f48b2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:09:36.197475Z","signature_b64":"eVRd6MLi7d+27FwLQ0FSpvjKBPsJsUKFdV8OmteUoaSzoZoVROQZlvubVrpd1QCIuiBwXmwrsXxINsLg1N75Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"80809378ce94d6218134505b37e93946037c1dfd6f7bf2bc5a0ae4bc9b8f48b2","last_reissued_at":"2026-07-05T11:09:36.196958Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:09:36.196958Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2505.19796","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":"2026-07-05T11:09:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kwUkHhTOCK73tffTod3ZW9TBLrwS/z2TTPKxfDlTCEmtDTiAaQfAEbRkdR2qMys7YIOHV57wpCwAkHcrL9PMBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T12:40:21.134266Z"},"content_sha256":"cd4d23db747bac25c9e2f0405731aa51cba5fa7bf5359d6994cd5867a7f9dbc7","schema_version":"1.0","event_id":"sha256:cd4d23db747bac25c9e2f0405731aa51cba5fa7bf5359d6994cd5867a7f9dbc7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:QCAJG6GOSTLCDAJUKBNTP2JZIY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Topological Perspective on the Birch and Swinnerton Dyer Conjectures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Maisara Shoeib","submitted_at":"2025-05-26T10:29:28Z","abstract_excerpt":"This paper presents a topological framework for investigating the Birch and Swinnerton Dyer conjecture through four dimensional embeddings of elliptic curves. We propose a correspondence between the algebraic rank of an elliptic curve and the number of topologically independent loops in its embedding, which appears to be related to the order of vanishing of its L function at s=1. Our computational function F new and its generalization F_(m,s) provide methods for examining this relationship through asymptotic analysis. Examples with rank curves from 0 to 8 show patterns supporting this correspo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.19796","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/2505.19796/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":"2026-07-05T11:09:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ocWh7XbH9LG53MJLsytB7TIF5knDG15FGD2/04oVkfmnDiigMZHZpEY12yZFcW245/vF1ZQmCDcHtCWsy9iFDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T12:40:21.135154Z"},"content_sha256":"869cbf288cea250590193f69f4dd16a5cbf82b56c537172ac8b5567ff9d38f27","schema_version":"1.0","event_id":"sha256:869cbf288cea250590193f69f4dd16a5cbf82b56c537172ac8b5567ff9d38f27"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QCAJG6GOSTLCDAJUKBNTP2JZIY/bundle.json","state_url":"https://pith.science/pith/QCAJG6GOSTLCDAJUKBNTP2JZIY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QCAJG6GOSTLCDAJUKBNTP2JZIY/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":"2026-08-12T12:40:21Z","links":{"resolver":"https://pith.science/pith/QCAJG6GOSTLCDAJUKBNTP2JZIY","bundle":"https://pith.science/pith/QCAJG6GOSTLCDAJUKBNTP2JZIY/bundle.json","state":"https://pith.science/pith/QCAJG6GOSTLCDAJUKBNTP2JZIY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QCAJG6GOSTLCDAJUKBNTP2JZIY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:QCAJG6GOSTLCDAJUKBNTP2JZIY","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":"b2355a70366d093a74f6d949b78d2b7667fa12fc22f7b0e012c95143950ab32e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2025-05-26T10:29:28Z","title_canon_sha256":"1b5864b71062149844464d9652ab1a0fe36f34c1b3f51919d3fb0912443a8216"},"schema_version":"1.0","source":{"id":"2505.19796","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2505.19796","created_at":"2026-07-05T11:09:36Z"},{"alias_kind":"arxiv_version","alias_value":"2505.19796v1","created_at":"2026-07-05T11:09:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.19796","created_at":"2026-07-05T11:09:36Z"},{"alias_kind":"pith_short_12","alias_value":"QCAJG6GOSTLC","created_at":"2026-07-05T11:09:36Z"},{"alias_kind":"pith_short_16","alias_value":"QCAJG6GOSTLCDAJU","created_at":"2026-07-05T11:09:36Z"},{"alias_kind":"pith_short_8","alias_value":"QCAJG6GO","created_at":"2026-07-05T11:09:36Z"}],"graph_snapshots":[{"event_id":"sha256:869cbf288cea250590193f69f4dd16a5cbf82b56c537172ac8b5567ff9d38f27","target":"graph","created_at":"2026-07-05T11:09:36Z","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/2505.19796/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This paper presents a topological framework for investigating the Birch and Swinnerton Dyer conjecture through four dimensional embeddings of elliptic curves. We propose a correspondence between the algebraic rank of an elliptic curve and the number of topologically independent loops in its embedding, which appears to be related to the order of vanishing of its L function at s=1. Our computational function F new and its generalization F_(m,s) provide methods for examining this relationship through asymptotic analysis. Examples with rank curves from 0 to 8 show patterns supporting this correspo","authors_text":"Maisara Shoeib","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2025-05-26T10:29:28Z","title":"A Topological Perspective on the Birch and Swinnerton Dyer Conjectures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.19796","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:cd4d23db747bac25c9e2f0405731aa51cba5fa7bf5359d6994cd5867a7f9dbc7","target":"record","created_at":"2026-07-05T11:09:36Z","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":"b2355a70366d093a74f6d949b78d2b7667fa12fc22f7b0e012c95143950ab32e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2025-05-26T10:29:28Z","title_canon_sha256":"1b5864b71062149844464d9652ab1a0fe36f34c1b3f51919d3fb0912443a8216"},"schema_version":"1.0","source":{"id":"2505.19796","kind":"arxiv","version":1}},"canonical_sha256":"80809378ce94d6218134505b37e93946037c1dfd6f7bf2bc5a0ae4bc9b8f48b2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"80809378ce94d6218134505b37e93946037c1dfd6f7bf2bc5a0ae4bc9b8f48b2","first_computed_at":"2026-07-05T11:09:36.196958Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:09:36.196958Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eVRd6MLi7d+27FwLQ0FSpvjKBPsJsUKFdV8OmteUoaSzoZoVROQZlvubVrpd1QCIuiBwXmwrsXxINsLg1N75Bw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:09:36.197475Z","signed_message":"canonical_sha256_bytes"},"source_id":"2505.19796","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cd4d23db747bac25c9e2f0405731aa51cba5fa7bf5359d6994cd5867a7f9dbc7","sha256:869cbf288cea250590193f69f4dd16a5cbf82b56c537172ac8b5567ff9d38f27"],"state_sha256":"52f8b6a75fb3f4d0034cabcab39fdda8d9e1d5bfd84e8edf89e086825780996f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8L6YKvpU2jbtJGy2i0hL8ejb41x3JZQ/nRR7h2BUbI37gLH5A+S5Ggw7gjnB5VZPMDArd/uus7qdWkHEATj0Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T12:40:21.141541Z","bundle_sha256":"0b01ddd0d128555d90d59f676f626d88224e5bfc539cab5fef8b3f25f7ed8ed1"}}