{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:X7WVU3TM4PBNDN45GWEPU3WWB6","short_pith_number":"pith:X7WVU3TM","canonical_record":{"source":{"id":"2508.07591","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SP","submitted_at":"2025-08-11T03:37:28Z","cross_cats_sorted":[],"title_canon_sha256":"8af778b8036d2d36f48bfb0230368fe955fc3042de0bfc40d34e3fe0f027878d","abstract_canon_sha256":"2e699bd4170b36a9d99f1ea8d5e6c313305ba874e00d31fa6a5305e9cfc7b8a3"},"schema_version":"1.0"},"canonical_sha256":"bfed5a6e6ce3c2d1b79d3588fa6ed60fbd5cf302e80f7e8158f2dedbc899b7e5","source":{"kind":"arxiv","id":"2508.07591","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2508.07591","created_at":"2026-07-05T12:00:07Z"},{"alias_kind":"arxiv_version","alias_value":"2508.07591v2","created_at":"2026-07-05T12:00:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.07591","created_at":"2026-07-05T12:00:07Z"},{"alias_kind":"pith_short_12","alias_value":"X7WVU3TM4PBN","created_at":"2026-07-05T12:00:07Z"},{"alias_kind":"pith_short_16","alias_value":"X7WVU3TM4PBNDN45","created_at":"2026-07-05T12:00:07Z"},{"alias_kind":"pith_short_8","alias_value":"X7WVU3TM","created_at":"2026-07-05T12:00:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:X7WVU3TM4PBNDN45GWEPU3WWB6","target":"record","payload":{"canonical_record":{"source":{"id":"2508.07591","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SP","submitted_at":"2025-08-11T03:37:28Z","cross_cats_sorted":[],"title_canon_sha256":"8af778b8036d2d36f48bfb0230368fe955fc3042de0bfc40d34e3fe0f027878d","abstract_canon_sha256":"2e699bd4170b36a9d99f1ea8d5e6c313305ba874e00d31fa6a5305e9cfc7b8a3"},"schema_version":"1.0"},"canonical_sha256":"bfed5a6e6ce3c2d1b79d3588fa6ed60fbd5cf302e80f7e8158f2dedbc899b7e5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:00:07.837024Z","signature_b64":"Diryx2Q5lNbVOJ3kRmTBiY2mwhqmGdfrMldqtqA2PIEJIKiJmuP10/uoW0POquN/hxokxabfXjjAeCSggfqbCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bfed5a6e6ce3c2d1b79d3588fa6ed60fbd5cf302e80f7e8158f2dedbc899b7e5","last_reissued_at":"2026-07-05T12:00:07.836573Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:00:07.836573Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2508.07591","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T12:00:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OGKj7pV3+MiJAvMJqmMNgeQw0FX1fZBS4GhvWVsYGzke88MIyAKGlypL8Saxyo4b7jR37M7dNb4e5NTsModFDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T08:17:44.417618Z"},"content_sha256":"82ffe03e6a014f1961e2308a2ee6a0cd8030a5ec11b76182e3c1d2356f49197e","schema_version":"1.0","event_id":"sha256:82ffe03e6a014f1961e2308a2ee6a0cd8030a5ec11b76182e3c1d2356f49197e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:X7WVU3TM4PBNDN45GWEPU3WWB6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weighted eigenvalues of Dirac operators: complete continuity and comparison","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Ruijun Wu, Zixuan Qiu","submitted_at":"2025-08-11T03:37:28Z","abstract_excerpt":"We give a min-max characterization of the weighted Dirac eigenvalues, and show that the weighted eigenvalues and eigenspaces of Dirac operators are continuous with respect to weak $L^p$ convergence of the inverse weight, for any $p>n$.\n  Moreover, we establish a comparison result for such weighted eigenvalue problems when there are no harmonic spinors."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.07591","kind":"arxiv","version":2},"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/2508.07591/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":"2026-07-05T12:00:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nxdRe0LSqOy/qQ1XXX9iIoadQDF+SCFO1izV7neCgGo79gG974CvSObgE8a27ZIHyVyQe7i0CZGXoTITSiMpBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T08:17:44.418140Z"},"content_sha256":"c71abf84c3e7cfe0ef1179c7b139e57c13bac5d71bc6d19ada1c6308425c336c","schema_version":"1.0","event_id":"sha256:c71abf84c3e7cfe0ef1179c7b139e57c13bac5d71bc6d19ada1c6308425c336c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/X7WVU3TM4PBNDN45GWEPU3WWB6/bundle.json","state_url":"https://pith.science/pith/X7WVU3TM4PBNDN45GWEPU3WWB6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/X7WVU3TM4PBNDN45GWEPU3WWB6/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":"2026-08-21T08:17:44Z","links":{"resolver":"https://pith.science/pith/X7WVU3TM4PBNDN45GWEPU3WWB6","bundle":"https://pith.science/pith/X7WVU3TM4PBNDN45GWEPU3WWB6/bundle.json","state":"https://pith.science/pith/X7WVU3TM4PBNDN45GWEPU3WWB6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/X7WVU3TM4PBNDN45GWEPU3WWB6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:X7WVU3TM4PBNDN45GWEPU3WWB6","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":"2e699bd4170b36a9d99f1ea8d5e6c313305ba874e00d31fa6a5305e9cfc7b8a3","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SP","submitted_at":"2025-08-11T03:37:28Z","title_canon_sha256":"8af778b8036d2d36f48bfb0230368fe955fc3042de0bfc40d34e3fe0f027878d"},"schema_version":"1.0","source":{"id":"2508.07591","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2508.07591","created_at":"2026-07-05T12:00:07Z"},{"alias_kind":"arxiv_version","alias_value":"2508.07591v2","created_at":"2026-07-05T12:00:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.07591","created_at":"2026-07-05T12:00:07Z"},{"alias_kind":"pith_short_12","alias_value":"X7WVU3TM4PBN","created_at":"2026-07-05T12:00:07Z"},{"alias_kind":"pith_short_16","alias_value":"X7WVU3TM4PBNDN45","created_at":"2026-07-05T12:00:07Z"},{"alias_kind":"pith_short_8","alias_value":"X7WVU3TM","created_at":"2026-07-05T12:00:07Z"}],"graph_snapshots":[{"event_id":"sha256:c71abf84c3e7cfe0ef1179c7b139e57c13bac5d71bc6d19ada1c6308425c336c","target":"graph","created_at":"2026-07-05T12:00:07Z","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/2508.07591/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We give a min-max characterization of the weighted Dirac eigenvalues, and show that the weighted eigenvalues and eigenspaces of Dirac operators are continuous with respect to weak $L^p$ convergence of the inverse weight, for any $p>n$.\n  Moreover, we establish a comparison result for such weighted eigenvalue problems when there are no harmonic spinors.","authors_text":"Ruijun Wu, Zixuan Qiu","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SP","submitted_at":"2025-08-11T03:37:28Z","title":"Weighted eigenvalues of Dirac operators: complete continuity and comparison"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.07591","kind":"arxiv","version":2},"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":"sha256:82ffe03e6a014f1961e2308a2ee6a0cd8030a5ec11b76182e3c1d2356f49197e","target":"record","created_at":"2026-07-05T12:00:07Z","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":"2e699bd4170b36a9d99f1ea8d5e6c313305ba874e00d31fa6a5305e9cfc7b8a3","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SP","submitted_at":"2025-08-11T03:37:28Z","title_canon_sha256":"8af778b8036d2d36f48bfb0230368fe955fc3042de0bfc40d34e3fe0f027878d"},"schema_version":"1.0","source":{"id":"2508.07591","kind":"arxiv","version":2}},"canonical_sha256":"bfed5a6e6ce3c2d1b79d3588fa6ed60fbd5cf302e80f7e8158f2dedbc899b7e5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bfed5a6e6ce3c2d1b79d3588fa6ed60fbd5cf302e80f7e8158f2dedbc899b7e5","first_computed_at":"2026-07-05T12:00:07.836573Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T12:00:07.836573Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Diryx2Q5lNbVOJ3kRmTBiY2mwhqmGdfrMldqtqA2PIEJIKiJmuP10/uoW0POquN/hxokxabfXjjAeCSggfqbCw==","signature_status":"signed_v1","signed_at":"2026-07-05T12:00:07.837024Z","signed_message":"canonical_sha256_bytes"},"source_id":"2508.07591","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:82ffe03e6a014f1961e2308a2ee6a0cd8030a5ec11b76182e3c1d2356f49197e","sha256:c71abf84c3e7cfe0ef1179c7b139e57c13bac5d71bc6d19ada1c6308425c336c"],"state_sha256":"ad1ea4baa5d334aaf57b199d16d752fc1003f6ad5752c56dec94065660da2717"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c1LWg/QzyB57mlbB/kxQYRbs/coZ/i63eKlXXr+VN9zBhNzKrafFRx5uEw7ZpwCtBHIi5+5Tz6376lNszRtJDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T08:17:44.423217Z","bundle_sha256":"7824519125f122194d42c74d612673d5a7d1761f19cc0b0feb809db48a0964ff"}}