{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:RNPPM3A3JKEJXTEQGVHDEJKIQG","short_pith_number":"pith:RNPPM3A3","canonical_record":{"source":{"id":"1803.01320","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-04T08:44:25Z","cross_cats_sorted":[],"title_canon_sha256":"0ef9c16f3c1b12828b9591f56673d05f9c7d7eacecd0fb4728f27693165a8ab7","abstract_canon_sha256":"25253c8c6532c59590179bd46a4fa3726b7c2a014284a36d95422fc5b2b39bb5"},"schema_version":"1.0"},"canonical_sha256":"8b5ef66c1b4a889bcc90354e32254881b8e1a3cf52af147e353c768d9e0805ca","source":{"kind":"arxiv","id":"1803.01320","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.01320","created_at":"2026-05-18T00:22:01Z"},{"alias_kind":"arxiv_version","alias_value":"1803.01320v1","created_at":"2026-05-18T00:22:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.01320","created_at":"2026-05-18T00:22:01Z"},{"alias_kind":"pith_short_12","alias_value":"RNPPM3A3JKEJ","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"RNPPM3A3JKEJXTEQ","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"RNPPM3A3","created_at":"2026-05-18T12:32:50Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:RNPPM3A3JKEJXTEQGVHDEJKIQG","target":"record","payload":{"canonical_record":{"source":{"id":"1803.01320","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-04T08:44:25Z","cross_cats_sorted":[],"title_canon_sha256":"0ef9c16f3c1b12828b9591f56673d05f9c7d7eacecd0fb4728f27693165a8ab7","abstract_canon_sha256":"25253c8c6532c59590179bd46a4fa3726b7c2a014284a36d95422fc5b2b39bb5"},"schema_version":"1.0"},"canonical_sha256":"8b5ef66c1b4a889bcc90354e32254881b8e1a3cf52af147e353c768d9e0805ca","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:01.091293Z","signature_b64":"D7rMJxxt1zJV5FEKVZR+KLkrEwbIgVgXFQ5GIxrjLDTclYOS9agUJI4BmiPAKoe48rK8a2yUQ9fxVMzquEtHDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8b5ef66c1b4a889bcc90354e32254881b8e1a3cf52af147e353c768d9e0805ca","last_reissued_at":"2026-05-18T00:22:01.090766Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:01.090766Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1803.01320","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":"2026-05-18T00:22:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CkLPqmgdbY+Tp2tURuQ1lxkihVM2GjaO8c5P3+MZLbyJ1tYPEu0cke68nnwaSYXjl5JzRD6ahRCaHUDEG2ezAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T05:50:37.804521Z"},"content_sha256":"4ac308eb37f9e8785b140b42313f15464987b7ed361637ab1aebb4f0a9d8038f","schema_version":"1.0","event_id":"sha256:4ac308eb37f9e8785b140b42313f15464987b7ed361637ab1aebb4f0a9d8038f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:RNPPM3A3JKEJXTEQGVHDEJKIQG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Local Spectral Expansion Approach to High Dimensional Expanders Part II: Mixing and Geometrical overlapping","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Izhar Oppenheim","submitted_at":"2018-03-04T08:44:25Z","abstract_excerpt":"In this paper, we further explore the local-to-global approach for expansion of simplicial complexes that we call local spectral expansion. Specifically, we prove that local expansion in the links imply the global expansion phenomena of mixing and geometric overlapping. Our mixing results also give tighter bounds on the error terms compared to previously known results."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.01320","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":"2026-05-18T00:22:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/sJW/vW0vjIk2SJ6UAxzNiOKgiTLiOCLxP6RgJvAyfTV0jKE/oT1bLR3c46SqYvO6iyOBGH3uOQ5cEensTXuAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T05:50:37.805205Z"},"content_sha256":"ec253fa9e96f715b05eee9b1cf757e57082b019d1c7cc065bd4bc8b5fff1c3d5","schema_version":"1.0","event_id":"sha256:ec253fa9e96f715b05eee9b1cf757e57082b019d1c7cc065bd4bc8b5fff1c3d5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RNPPM3A3JKEJXTEQGVHDEJKIQG/bundle.json","state_url":"https://pith.science/pith/RNPPM3A3JKEJXTEQGVHDEJKIQG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RNPPM3A3JKEJXTEQGVHDEJKIQG/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":"2026-06-06T05:50:37Z","links":{"resolver":"https://pith.science/pith/RNPPM3A3JKEJXTEQGVHDEJKIQG","bundle":"https://pith.science/pith/RNPPM3A3JKEJXTEQGVHDEJKIQG/bundle.json","state":"https://pith.science/pith/RNPPM3A3JKEJXTEQGVHDEJKIQG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RNPPM3A3JKEJXTEQGVHDEJKIQG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:RNPPM3A3JKEJXTEQGVHDEJKIQG","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":"25253c8c6532c59590179bd46a4fa3726b7c2a014284a36d95422fc5b2b39bb5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-04T08:44:25Z","title_canon_sha256":"0ef9c16f3c1b12828b9591f56673d05f9c7d7eacecd0fb4728f27693165a8ab7"},"schema_version":"1.0","source":{"id":"1803.01320","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.01320","created_at":"2026-05-18T00:22:01Z"},{"alias_kind":"arxiv_version","alias_value":"1803.01320v1","created_at":"2026-05-18T00:22:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.01320","created_at":"2026-05-18T00:22:01Z"},{"alias_kind":"pith_short_12","alias_value":"RNPPM3A3JKEJ","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_16","alias_value":"RNPPM3A3JKEJXTEQ","created_at":"2026-05-18T12:32:50Z"},{"alias_kind":"pith_short_8","alias_value":"RNPPM3A3","created_at":"2026-05-18T12:32:50Z"}],"graph_snapshots":[{"event_id":"sha256:ec253fa9e96f715b05eee9b1cf757e57082b019d1c7cc065bd4bc8b5fff1c3d5","target":"graph","created_at":"2026-05-18T00:22:01Z","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":"In this paper, we further explore the local-to-global approach for expansion of simplicial complexes that we call local spectral expansion. Specifically, we prove that local expansion in the links imply the global expansion phenomena of mixing and geometric overlapping. Our mixing results also give tighter bounds on the error terms compared to previously known results.","authors_text":"Izhar Oppenheim","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-04T08:44:25Z","title":"Local Spectral Expansion Approach to High Dimensional Expanders Part II: Mixing and Geometrical overlapping"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.01320","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":"sha256:4ac308eb37f9e8785b140b42313f15464987b7ed361637ab1aebb4f0a9d8038f","target":"record","created_at":"2026-05-18T00:22:01Z","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":"25253c8c6532c59590179bd46a4fa3726b7c2a014284a36d95422fc5b2b39bb5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-03-04T08:44:25Z","title_canon_sha256":"0ef9c16f3c1b12828b9591f56673d05f9c7d7eacecd0fb4728f27693165a8ab7"},"schema_version":"1.0","source":{"id":"1803.01320","kind":"arxiv","version":1}},"canonical_sha256":"8b5ef66c1b4a889bcc90354e32254881b8e1a3cf52af147e353c768d9e0805ca","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8b5ef66c1b4a889bcc90354e32254881b8e1a3cf52af147e353c768d9e0805ca","first_computed_at":"2026-05-18T00:22:01.090766Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:22:01.090766Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"D7rMJxxt1zJV5FEKVZR+KLkrEwbIgVgXFQ5GIxrjLDTclYOS9agUJI4BmiPAKoe48rK8a2yUQ9fxVMzquEtHDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:22:01.091293Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.01320","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4ac308eb37f9e8785b140b42313f15464987b7ed361637ab1aebb4f0a9d8038f","sha256:ec253fa9e96f715b05eee9b1cf757e57082b019d1c7cc065bd4bc8b5fff1c3d5"],"state_sha256":"b662b7a037e39a9acc3289143310da6e5022aedf29e25a52a646932d7a8ee20b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cbg+lvI9W+2WND0HGsJRGWTRndaYiu3VrIzc/5u83EKGRhvrzB3pnxw6tcNa7/au/suNd8qGY6DEZJsEFugXBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T05:50:37.809047Z","bundle_sha256":"77773ee9aa939f634043b3047f06e127da328d80387943058fed38e546a35adb"}}