{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:37ODXGA3UNMUPCR5DTTSOIWYDL","short_pith_number":"pith:37ODXGA3","canonical_record":{"source":{"id":"1311.2418","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-11-11T11:19:45Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"b992bab5206b9e9e1a9f4b62b01013dc19a3ca5a7c076643f87adb8c64e9e35e","abstract_canon_sha256":"efbab12df07df4121997492eb8013e670c3abd21bbc181d9f1cb66f3b0af5f5f"},"schema_version":"1.0"},"canonical_sha256":"dfdc3b981ba359478a3d1ce72722d81acff704c5b05c82e5e1ac8af8a1b36483","source":{"kind":"arxiv","id":"1311.2418","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.2418","created_at":"2026-05-18T03:07:32Z"},{"alias_kind":"arxiv_version","alias_value":"1311.2418v1","created_at":"2026-05-18T03:07:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.2418","created_at":"2026-05-18T03:07:32Z"},{"alias_kind":"pith_short_12","alias_value":"37ODXGA3UNMU","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"37ODXGA3UNMUPCR5","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"37ODXGA3","created_at":"2026-05-18T12:27:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:37ODXGA3UNMUPCR5DTTSOIWYDL","target":"record","payload":{"canonical_record":{"source":{"id":"1311.2418","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-11-11T11:19:45Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"b992bab5206b9e9e1a9f4b62b01013dc19a3ca5a7c076643f87adb8c64e9e35e","abstract_canon_sha256":"efbab12df07df4121997492eb8013e670c3abd21bbc181d9f1cb66f3b0af5f5f"},"schema_version":"1.0"},"canonical_sha256":"dfdc3b981ba359478a3d1ce72722d81acff704c5b05c82e5e1ac8af8a1b36483","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:07:32.708833Z","signature_b64":"yX0CDd1aRRsO01pOS2G4GGUznZu3nD7wxecI6nfNJIvzZr6+Cw+YwNUMhArYTki23ExO7O0l/IpeMtoscGpzCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dfdc3b981ba359478a3d1ce72722d81acff704c5b05c82e5e1ac8af8a1b36483","last_reissued_at":"2026-05-18T03:07:32.708257Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:07:32.708257Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1311.2418","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":"2026-05-18T03:07:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fhcXMneZJX5IOiiGLfgasdFFhzlZQeuKGDSmZAiGyfmTD52TZy8kGGORnEZYIDfIPeWwKwRJiKvsFTXm+axlAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T14:12:21.689056Z"},"content_sha256":"222191cacdef5038017667f5a38dfd16a20cdb6c8b3faea40d655839e6cf9da8","schema_version":"1.0","event_id":"sha256:222191cacdef5038017667f5a38dfd16a20cdb6c8b3faea40d655839e6cf9da8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:37ODXGA3UNMUPCR5DTTSOIWYDL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spectra of sub-Dirac operators on certain nilmanifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.SP","authors_text":"Ines Kath, Oliver Ungermann","submitted_at":"2013-11-11T11:19:45Z","abstract_excerpt":"We study sub-Dirac operators that are associated with left-invariant bracket-generating sub-Riemannian structures on compact quotients of nilpotent semi-direct products $G=\\mathbb{R}^n\\rtimes_A\\mathbb{R}$. We will prove that these operators admit an $L^2$-basis of eigenfunctions. Explicit examples show that the spectrum of these operators can be non-discrete and that eigenvalues may have infinite multiplicity."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.2418","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":""},"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":"2026-05-18T03:07:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yuOwKngQU8qJsI3knVlRh08QxNvH64XnrUgulpPzQC1y1Ggeks+YMF2varxU3HtOL/5jPvqHQWArWz8Ah8YvDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T14:12:21.689409Z"},"content_sha256":"db20eacc4140271dfa5b5cccf46b4f95b62ccb747ccf28d3556eb86af5bb2532","schema_version":"1.0","event_id":"sha256:db20eacc4140271dfa5b5cccf46b4f95b62ccb747ccf28d3556eb86af5bb2532"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/37ODXGA3UNMUPCR5DTTSOIWYDL/bundle.json","state_url":"https://pith.science/pith/37ODXGA3UNMUPCR5DTTSOIWYDL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/37ODXGA3UNMUPCR5DTTSOIWYDL/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":"2026-06-23T14:12:21Z","links":{"resolver":"https://pith.science/pith/37ODXGA3UNMUPCR5DTTSOIWYDL","bundle":"https://pith.science/pith/37ODXGA3UNMUPCR5DTTSOIWYDL/bundle.json","state":"https://pith.science/pith/37ODXGA3UNMUPCR5DTTSOIWYDL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/37ODXGA3UNMUPCR5DTTSOIWYDL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:37ODXGA3UNMUPCR5DTTSOIWYDL","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":"efbab12df07df4121997492eb8013e670c3abd21bbc181d9f1cb66f3b0af5f5f","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-11-11T11:19:45Z","title_canon_sha256":"b992bab5206b9e9e1a9f4b62b01013dc19a3ca5a7c076643f87adb8c64e9e35e"},"schema_version":"1.0","source":{"id":"1311.2418","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.2418","created_at":"2026-05-18T03:07:32Z"},{"alias_kind":"arxiv_version","alias_value":"1311.2418v1","created_at":"2026-05-18T03:07:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.2418","created_at":"2026-05-18T03:07:32Z"},{"alias_kind":"pith_short_12","alias_value":"37ODXGA3UNMU","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"37ODXGA3UNMUPCR5","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"37ODXGA3","created_at":"2026-05-18T12:27:32Z"}],"graph_snapshots":[{"event_id":"sha256:db20eacc4140271dfa5b5cccf46b4f95b62ccb747ccf28d3556eb86af5bb2532","target":"graph","created_at":"2026-05-18T03:07:32Z","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"},"paper":{"abstract_excerpt":"We study sub-Dirac operators that are associated with left-invariant bracket-generating sub-Riemannian structures on compact quotients of nilpotent semi-direct products $G=\\mathbb{R}^n\\rtimes_A\\mathbb{R}$. We will prove that these operators admit an $L^2$-basis of eigenfunctions. Explicit examples show that the spectrum of these operators can be non-discrete and that eigenvalues may have infinite multiplicity.","authors_text":"Ines Kath, Oliver Ungermann","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-11-11T11:19:45Z","title":"Spectra of sub-Dirac operators on certain nilmanifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.2418","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":"sha256:222191cacdef5038017667f5a38dfd16a20cdb6c8b3faea40d655839e6cf9da8","target":"record","created_at":"2026-05-18T03:07:32Z","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":"efbab12df07df4121997492eb8013e670c3abd21bbc181d9f1cb66f3b0af5f5f","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-11-11T11:19:45Z","title_canon_sha256":"b992bab5206b9e9e1a9f4b62b01013dc19a3ca5a7c076643f87adb8c64e9e35e"},"schema_version":"1.0","source":{"id":"1311.2418","kind":"arxiv","version":1}},"canonical_sha256":"dfdc3b981ba359478a3d1ce72722d81acff704c5b05c82e5e1ac8af8a1b36483","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dfdc3b981ba359478a3d1ce72722d81acff704c5b05c82e5e1ac8af8a1b36483","first_computed_at":"2026-05-18T03:07:32.708257Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:07:32.708257Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yX0CDd1aRRsO01pOS2G4GGUznZu3nD7wxecI6nfNJIvzZr6+Cw+YwNUMhArYTki23ExO7O0l/IpeMtoscGpzCA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:07:32.708833Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.2418","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:222191cacdef5038017667f5a38dfd16a20cdb6c8b3faea40d655839e6cf9da8","sha256:db20eacc4140271dfa5b5cccf46b4f95b62ccb747ccf28d3556eb86af5bb2532"],"state_sha256":"f6eef5f91a700dce1b74033d7066baac3dd0e945f1e131728f90f551ee39b546"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ioiSkDQfjUNJYahLP8V0q0F4IeC57CTkbw4NjNPkrXtrU22YR3c5tSPxoy4PV+b0DtMQLmjyaE4ZdJlRPqImDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T14:12:21.691429Z","bundle_sha256":"e9fb3827c3b8ea49354f353a50749b0927f38dc2c860a6f8f2102eb0af204ccc"}}