{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:JOTKK75XFOWKSPR55UYIAEALYT","short_pith_number":"pith:JOTKK75X","canonical_record":{"source":{"id":"1501.06165","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-01-25T14:21:28Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"c2fbe390bb28d766859c68caa8b725edc6f916dc5ae0acfc549ab38fd8c10d39","abstract_canon_sha256":"9cb6d4dee467ab4dc789df1c95f47402d519f29e58f97f01767fd4618c4cc081"},"schema_version":"1.0"},"canonical_sha256":"4ba6a57fb72baca93e3ded3080100bc4d7ab3f4e174a709817c2e27e9241edb8","source":{"kind":"arxiv","id":"1501.06165","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.06165","created_at":"2026-05-18T02:28:42Z"},{"alias_kind":"arxiv_version","alias_value":"1501.06165v1","created_at":"2026-05-18T02:28:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.06165","created_at":"2026-05-18T02:28:42Z"},{"alias_kind":"pith_short_12","alias_value":"JOTKK75XFOWK","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JOTKK75XFOWKSPR5","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JOTKK75X","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:JOTKK75XFOWKSPR55UYIAEALYT","target":"record","payload":{"canonical_record":{"source":{"id":"1501.06165","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-01-25T14:21:28Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"c2fbe390bb28d766859c68caa8b725edc6f916dc5ae0acfc549ab38fd8c10d39","abstract_canon_sha256":"9cb6d4dee467ab4dc789df1c95f47402d519f29e58f97f01767fd4618c4cc081"},"schema_version":"1.0"},"canonical_sha256":"4ba6a57fb72baca93e3ded3080100bc4d7ab3f4e174a709817c2e27e9241edb8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:42.172106Z","signature_b64":"3qIYVazpV3QDC7c3N7CyXdmcmWDC+zWxm3Bv4KlaI0C+K+uhaybrPBzDt1mWF0JtScVqk1OfXmNlHhkxi97TDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4ba6a57fb72baca93e3ded3080100bc4d7ab3f4e174a709817c2e27e9241edb8","last_reissued_at":"2026-05-18T02:28:42.171475Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:42.171475Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.06165","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":"2026-05-18T02:28:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z6IsnfKJdJ9HodkDkxXj960cs0aEBdgeZdG+XOAecYZMw8C9loVWO1WADGWV+ZhkuuvQwbT0/2MHlwxzcYiBDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T20:56:39.180707Z"},"content_sha256":"f667cedbffcc95fd5c7a20b5e023b37947fb21c529c59672fc4dcb8da52a5060","schema_version":"1.0","event_id":"sha256:f667cedbffcc95fd5c7a20b5e023b37947fb21c529c59672fc4dcb8da52a5060"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:JOTKK75XFOWKSPR55UYIAEALYT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The multiplicity of eigenvalues of the Hodge Laplacian on 5-dimensional compact manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.SP","authors_text":"Megan E. Gier, Peter D. Hislop","submitted_at":"2015-01-25T14:21:28Z","abstract_excerpt":"We study multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact two-forms, has nonzero eigenvalues of multiplicity two. The proof is based on the fact that Hodge Laplacian restricted to the subspace of co-exact two-forms is minus the square of the Beltrami operator, a first-order operator. We prove that for generic metrics the spectrum of the Beltrami operator is simple. Because the Beltrami operator in this setting is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.06165","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":"2026-05-18T02:28:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v4cTXfEPThrZ/EnWiFHWtSXpFWGMvvNRzNhveXvMBIaFOjQAdiqR7Fvk+Y4mAN7Za8XWYf+FoJ2DJw5HkGdSBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T20:56:39.181171Z"},"content_sha256":"046f478d247b614f55a633837e5f88fc15b73f7833cf7c7756522c6a18bc1cc1","schema_version":"1.0","event_id":"sha256:046f478d247b614f55a633837e5f88fc15b73f7833cf7c7756522c6a18bc1cc1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JOTKK75XFOWKSPR55UYIAEALYT/bundle.json","state_url":"https://pith.science/pith/JOTKK75XFOWKSPR55UYIAEALYT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JOTKK75XFOWKSPR55UYIAEALYT/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":"2026-05-27T20:56:39Z","links":{"resolver":"https://pith.science/pith/JOTKK75XFOWKSPR55UYIAEALYT","bundle":"https://pith.science/pith/JOTKK75XFOWKSPR55UYIAEALYT/bundle.json","state":"https://pith.science/pith/JOTKK75XFOWKSPR55UYIAEALYT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JOTKK75XFOWKSPR55UYIAEALYT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JOTKK75XFOWKSPR55UYIAEALYT","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":"9cb6d4dee467ab4dc789df1c95f47402d519f29e58f97f01767fd4618c4cc081","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-01-25T14:21:28Z","title_canon_sha256":"c2fbe390bb28d766859c68caa8b725edc6f916dc5ae0acfc549ab38fd8c10d39"},"schema_version":"1.0","source":{"id":"1501.06165","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.06165","created_at":"2026-05-18T02:28:42Z"},{"alias_kind":"arxiv_version","alias_value":"1501.06165v1","created_at":"2026-05-18T02:28:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.06165","created_at":"2026-05-18T02:28:42Z"},{"alias_kind":"pith_short_12","alias_value":"JOTKK75XFOWK","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JOTKK75XFOWKSPR5","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JOTKK75X","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:046f478d247b614f55a633837e5f88fc15b73f7833cf7c7756522c6a18bc1cc1","target":"graph","created_at":"2026-05-18T02:28:42Z","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"},"paper":{"abstract_excerpt":"We study multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact two-forms, has nonzero eigenvalues of multiplicity two. The proof is based on the fact that Hodge Laplacian restricted to the subspace of co-exact two-forms is minus the square of the Beltrami operator, a first-order operator. We prove that for generic metrics the spectrum of the Beltrami operator is simple. Because the Beltrami operator in this setting is ","authors_text":"Megan E. Gier, Peter D. Hislop","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-01-25T14:21:28Z","title":"The multiplicity of eigenvalues of the Hodge Laplacian on 5-dimensional compact manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.06165","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:f667cedbffcc95fd5c7a20b5e023b37947fb21c529c59672fc4dcb8da52a5060","target":"record","created_at":"2026-05-18T02:28:42Z","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":"9cb6d4dee467ab4dc789df1c95f47402d519f29e58f97f01767fd4618c4cc081","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2015-01-25T14:21:28Z","title_canon_sha256":"c2fbe390bb28d766859c68caa8b725edc6f916dc5ae0acfc549ab38fd8c10d39"},"schema_version":"1.0","source":{"id":"1501.06165","kind":"arxiv","version":1}},"canonical_sha256":"4ba6a57fb72baca93e3ded3080100bc4d7ab3f4e174a709817c2e27e9241edb8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4ba6a57fb72baca93e3ded3080100bc4d7ab3f4e174a709817c2e27e9241edb8","first_computed_at":"2026-05-18T02:28:42.171475Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:28:42.171475Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3qIYVazpV3QDC7c3N7CyXdmcmWDC+zWxm3Bv4KlaI0C+K+uhaybrPBzDt1mWF0JtScVqk1OfXmNlHhkxi97TDA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:28:42.172106Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.06165","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f667cedbffcc95fd5c7a20b5e023b37947fb21c529c59672fc4dcb8da52a5060","sha256:046f478d247b614f55a633837e5f88fc15b73f7833cf7c7756522c6a18bc1cc1"],"state_sha256":"45a8a0b387e646e7b0b4fd0d94f282430f698ccee64cbf3aa20f13b5e9e5ce54"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hJmDZT6ZpJB2mIH8ta92kYA2hH0FPNxCmGGLWltpgUamxBc7iyjyOekQyQT1YiG8GUMIJVwcv7dPxhSC0pjYBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T20:56:39.184510Z","bundle_sha256":"792f9051be15ed6f2149834c6fe80d1ce701225209b7537795e4cb61ab40a422"}}