{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:THR4KB723GV5WQJTWT4T4HUVXP","short_pith_number":"pith:THR4KB72","canonical_record":{"source":{"id":"1709.04431","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-13T17:23:18Z","cross_cats_sorted":[],"title_canon_sha256":"728049d4e076a52de38bef93e77d1c56dccea7e8c6328028b095d789f9a61498","abstract_canon_sha256":"86c191c6b306f08c9ea2030c91e0056eb16841d09d5c218377564bd93dec075a"},"schema_version":"1.0"},"canonical_sha256":"99e3c507fad9abdb4133b4f93e1e95bbfc1dd7e240ebded24960f12efff98d46","source":{"kind":"arxiv","id":"1709.04431","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.04431","created_at":"2026-05-18T00:35:14Z"},{"alias_kind":"arxiv_version","alias_value":"1709.04431v1","created_at":"2026-05-18T00:35:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.04431","created_at":"2026-05-18T00:35:14Z"},{"alias_kind":"pith_short_12","alias_value":"THR4KB723GV5","created_at":"2026-05-18T12:31:46Z"},{"alias_kind":"pith_short_16","alias_value":"THR4KB723GV5WQJT","created_at":"2026-05-18T12:31:46Z"},{"alias_kind":"pith_short_8","alias_value":"THR4KB72","created_at":"2026-05-18T12:31:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:THR4KB723GV5WQJTWT4T4HUVXP","target":"record","payload":{"canonical_record":{"source":{"id":"1709.04431","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-13T17:23:18Z","cross_cats_sorted":[],"title_canon_sha256":"728049d4e076a52de38bef93e77d1c56dccea7e8c6328028b095d789f9a61498","abstract_canon_sha256":"86c191c6b306f08c9ea2030c91e0056eb16841d09d5c218377564bd93dec075a"},"schema_version":"1.0"},"canonical_sha256":"99e3c507fad9abdb4133b4f93e1e95bbfc1dd7e240ebded24960f12efff98d46","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:14.374266Z","signature_b64":"A+8+fzCJFleVToWumtOt+6FUgZkMVXdZd7dI9KjYcwHN4x8Kg/5pRTzcK0J2AkgJndfRcRmTQtDMI1ntDxenDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"99e3c507fad9abdb4133b4f93e1e95bbfc1dd7e240ebded24960f12efff98d46","last_reissued_at":"2026-05-18T00:35:14.373793Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:14.373793Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1709.04431","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":"2026-05-18T00:35:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Yhw6C/FcVeM+Tl3WKRv1I8ei5zjp67q/3QoFv6HZpyZY8GP9MdAHhedhS2EuPqZkyAFO23ZQkiDF9rLzHwlODA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T13:44:35.181237Z"},"content_sha256":"3fe1961a5e322cc010280c161258fe4b65bb26b1942ccb4e076f09ab40008858","schema_version":"1.0","event_id":"sha256:3fe1961a5e322cc010280c161258fe4b65bb26b1942ccb4e076f09ab40008858"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:THR4KB723GV5WQJTWT4T4HUVXP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Local spectral expansion approach to high dimensional expanders part I: Descent of spectral gaps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Izhar Oppenheim","submitted_at":"2017-09-13T17:23:18Z","abstract_excerpt":"This paper introduces the notion of local spectral expansion of a simplicial complex as a possible analogue of spectral expansion defined for graphs. We then show that the condition of local spectral expansion for a complex yields various spectral gaps in both the links of the complex and the global Laplacians of the complex."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.04431","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":"2026-05-18T00:35:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RtxQkWS+1npLOjUgqopydy3m/1TAzRdVbdLEtitMr6/PV/zGEulJnSvlNEUpo1FYOjZX8pHHKPCfh/YI0XlKAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T13:44:35.181906Z"},"content_sha256":"c0ab3be1ba856978a12d89647faf15ecc542a920c81bad4405d005861773ebca","schema_version":"1.0","event_id":"sha256:c0ab3be1ba856978a12d89647faf15ecc542a920c81bad4405d005861773ebca"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/THR4KB723GV5WQJTWT4T4HUVXP/bundle.json","state_url":"https://pith.science/pith/THR4KB723GV5WQJTWT4T4HUVXP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/THR4KB723GV5WQJTWT4T4HUVXP/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":"2026-05-25T13:44:35Z","links":{"resolver":"https://pith.science/pith/THR4KB723GV5WQJTWT4T4HUVXP","bundle":"https://pith.science/pith/THR4KB723GV5WQJTWT4T4HUVXP/bundle.json","state":"https://pith.science/pith/THR4KB723GV5WQJTWT4T4HUVXP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/THR4KB723GV5WQJTWT4T4HUVXP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:THR4KB723GV5WQJTWT4T4HUVXP","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":"86c191c6b306f08c9ea2030c91e0056eb16841d09d5c218377564bd93dec075a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-13T17:23:18Z","title_canon_sha256":"728049d4e076a52de38bef93e77d1c56dccea7e8c6328028b095d789f9a61498"},"schema_version":"1.0","source":{"id":"1709.04431","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1709.04431","created_at":"2026-05-18T00:35:14Z"},{"alias_kind":"arxiv_version","alias_value":"1709.04431v1","created_at":"2026-05-18T00:35:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.04431","created_at":"2026-05-18T00:35:14Z"},{"alias_kind":"pith_short_12","alias_value":"THR4KB723GV5","created_at":"2026-05-18T12:31:46Z"},{"alias_kind":"pith_short_16","alias_value":"THR4KB723GV5WQJT","created_at":"2026-05-18T12:31:46Z"},{"alias_kind":"pith_short_8","alias_value":"THR4KB72","created_at":"2026-05-18T12:31:46Z"}],"graph_snapshots":[{"event_id":"sha256:c0ab3be1ba856978a12d89647faf15ecc542a920c81bad4405d005861773ebca","target":"graph","created_at":"2026-05-18T00:35:14Z","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":"This paper introduces the notion of local spectral expansion of a simplicial complex as a possible analogue of spectral expansion defined for graphs. We then show that the condition of local spectral expansion for a complex yields various spectral gaps in both the links of the complex and the global Laplacians of the complex.","authors_text":"Izhar Oppenheim","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-13T17:23:18Z","title":"Local spectral expansion approach to high dimensional expanders part I: Descent of spectral gaps"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.04431","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":"sha256:3fe1961a5e322cc010280c161258fe4b65bb26b1942ccb4e076f09ab40008858","target":"record","created_at":"2026-05-18T00:35:14Z","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":"86c191c6b306f08c9ea2030c91e0056eb16841d09d5c218377564bd93dec075a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-13T17:23:18Z","title_canon_sha256":"728049d4e076a52de38bef93e77d1c56dccea7e8c6328028b095d789f9a61498"},"schema_version":"1.0","source":{"id":"1709.04431","kind":"arxiv","version":1}},"canonical_sha256":"99e3c507fad9abdb4133b4f93e1e95bbfc1dd7e240ebded24960f12efff98d46","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"99e3c507fad9abdb4133b4f93e1e95bbfc1dd7e240ebded24960f12efff98d46","first_computed_at":"2026-05-18T00:35:14.373793Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:35:14.373793Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"A+8+fzCJFleVToWumtOt+6FUgZkMVXdZd7dI9KjYcwHN4x8Kg/5pRTzcK0J2AkgJndfRcRmTQtDMI1ntDxenDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:35:14.374266Z","signed_message":"canonical_sha256_bytes"},"source_id":"1709.04431","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3fe1961a5e322cc010280c161258fe4b65bb26b1942ccb4e076f09ab40008858","sha256:c0ab3be1ba856978a12d89647faf15ecc542a920c81bad4405d005861773ebca"],"state_sha256":"454774220982263506877029c8a66ada006b99870fbe19a67b5da68049b5aa24"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+AkFKjDEAmAN01KlG/GnRpYXnP6S6tCbmZEmUqCyJA9GNIDroa0GSt3aPPVA7F2BsHVpiI9v7cBpoTHRRekRCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T13:44:35.184598Z","bundle_sha256":"e412bd10ac67c6c17a5ce04f0d71b236000a0f164fb2862a54bfdf2910b390a9"}}