{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:JAAW3URS4YNXZ5STCF6RSBD27V","short_pith_number":"pith:JAAW3URS","canonical_record":{"source":{"id":"1207.4705","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-19T15:24:05Z","cross_cats_sorted":["math.CO","math.FA"],"title_canon_sha256":"d8fca0a315cb0b806f2aefa13bfad3c55eaee284eac16403b5d897aa53df99c5","abstract_canon_sha256":"7d0f7f109fa7d65cd035ebb3b2efc00ce9f5114e61622c4f3ecdcc189f7410c5"},"schema_version":"1.0"},"canonical_sha256":"48016dd232e61b7cf653117d19047afd41e30b4093f0a1af6aa665680d4b715c","source":{"kind":"arxiv","id":"1207.4705","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.4705","created_at":"2026-05-18T02:41:43Z"},{"alias_kind":"arxiv_version","alias_value":"1207.4705v3","created_at":"2026-05-18T02:41:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.4705","created_at":"2026-05-18T02:41:43Z"},{"alias_kind":"pith_short_12","alias_value":"JAAW3URS4YNX","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JAAW3URS4YNXZ5ST","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JAAW3URS","created_at":"2026-05-18T12:27:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:JAAW3URS4YNXZ5STCF6RSBD27V","target":"record","payload":{"canonical_record":{"source":{"id":"1207.4705","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-19T15:24:05Z","cross_cats_sorted":["math.CO","math.FA"],"title_canon_sha256":"d8fca0a315cb0b806f2aefa13bfad3c55eaee284eac16403b5d897aa53df99c5","abstract_canon_sha256":"7d0f7f109fa7d65cd035ebb3b2efc00ce9f5114e61622c4f3ecdcc189f7410c5"},"schema_version":"1.0"},"canonical_sha256":"48016dd232e61b7cf653117d19047afd41e30b4093f0a1af6aa665680d4b715c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:41:43.033348Z","signature_b64":"DpLTgBlo1cUn9SzWU3XXfbqqXk7N3VGXZzR4q3hD8qV4xHobygDYnJGc2018Mmh8VuI13cS9VDIdUruglrSJBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"48016dd232e61b7cf653117d19047afd41e30b4093f0a1af6aa665680d4b715c","last_reissued_at":"2026-05-18T02:41:43.032728Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:41:43.032728Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.4705","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:41:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zHAoJrZ77hKgei/yIGfZT4OvcqELjbT+wf77hurfR92JqglbLeRB3bJ3mkZSHfdix2rRAGrG4Xytpx3GGJhfAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T16:21:35.157985Z"},"content_sha256":"26f35bb1aa6c14faa4af67fbb332d76989f36b16327314ba0fe4d5b592804d3c","schema_version":"1.0","event_id":"sha256:26f35bb1aa6c14faa4af67fbb332d76989f36b16327314ba0fe4d5b592804d3c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:JAAW3URS4YNXZ5STCF6RSBD27V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Nonlinear spectral calculus and super-expanders","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.FA"],"primary_cat":"math.MG","authors_text":"Assaf Naor, Manor Mendel","submitted_at":"2012-07-19T15:24:05Z","abstract_excerpt":"Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a combinatorial construction of super-expanders, i.e., a sequence of 3-regular graphs that does not admit a coarse embedding into any uniformly convex normed space."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.4705","kind":"arxiv","version":3},"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":"2026-05-18T02:41:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FlEGVsmnMAE4YAhPpvWdPBW/E3Fjy8KAfYn7F9KqYt5QB34dxM0+znikq4TCN8FrjONhiLn7gA5lFuv4vs47Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T16:21:35.158337Z"},"content_sha256":"e06d3941708fa3e8f393840d1a1062400e2e253f514810f9e2a3e37a67c6c740","schema_version":"1.0","event_id":"sha256:e06d3941708fa3e8f393840d1a1062400e2e253f514810f9e2a3e37a67c6c740"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JAAW3URS4YNXZ5STCF6RSBD27V/bundle.json","state_url":"https://pith.science/pith/JAAW3URS4YNXZ5STCF6RSBD27V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JAAW3URS4YNXZ5STCF6RSBD27V/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":"2026-06-23T16:21:35Z","links":{"resolver":"https://pith.science/pith/JAAW3URS4YNXZ5STCF6RSBD27V","bundle":"https://pith.science/pith/JAAW3URS4YNXZ5STCF6RSBD27V/bundle.json","state":"https://pith.science/pith/JAAW3URS4YNXZ5STCF6RSBD27V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JAAW3URS4YNXZ5STCF6RSBD27V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:JAAW3URS4YNXZ5STCF6RSBD27V","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":"7d0f7f109fa7d65cd035ebb3b2efc00ce9f5114e61622c4f3ecdcc189f7410c5","cross_cats_sorted":["math.CO","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-19T15:24:05Z","title_canon_sha256":"d8fca0a315cb0b806f2aefa13bfad3c55eaee284eac16403b5d897aa53df99c5"},"schema_version":"1.0","source":{"id":"1207.4705","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.4705","created_at":"2026-05-18T02:41:43Z"},{"alias_kind":"arxiv_version","alias_value":"1207.4705v3","created_at":"2026-05-18T02:41:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.4705","created_at":"2026-05-18T02:41:43Z"},{"alias_kind":"pith_short_12","alias_value":"JAAW3URS4YNX","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JAAW3URS4YNXZ5ST","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JAAW3URS","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:e06d3941708fa3e8f393840d1a1062400e2e253f514810f9e2a3e37a67c6c740","target":"graph","created_at":"2026-05-18T02:41:43Z","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":"Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a combinatorial construction of super-expanders, i.e., a sequence of 3-regular graphs that does not admit a coarse embedding into any uniformly convex normed space.","authors_text":"Assaf Naor, Manor Mendel","cross_cats":["math.CO","math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-19T15:24:05Z","title":"Nonlinear spectral calculus and super-expanders"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.4705","kind":"arxiv","version":3},"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":"sha256:26f35bb1aa6c14faa4af67fbb332d76989f36b16327314ba0fe4d5b592804d3c","target":"record","created_at":"2026-05-18T02:41:43Z","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":"7d0f7f109fa7d65cd035ebb3b2efc00ce9f5114e61622c4f3ecdcc189f7410c5","cross_cats_sorted":["math.CO","math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-07-19T15:24:05Z","title_canon_sha256":"d8fca0a315cb0b806f2aefa13bfad3c55eaee284eac16403b5d897aa53df99c5"},"schema_version":"1.0","source":{"id":"1207.4705","kind":"arxiv","version":3}},"canonical_sha256":"48016dd232e61b7cf653117d19047afd41e30b4093f0a1af6aa665680d4b715c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"48016dd232e61b7cf653117d19047afd41e30b4093f0a1af6aa665680d4b715c","first_computed_at":"2026-05-18T02:41:43.032728Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:41:43.032728Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DpLTgBlo1cUn9SzWU3XXfbqqXk7N3VGXZzR4q3hD8qV4xHobygDYnJGc2018Mmh8VuI13cS9VDIdUruglrSJBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:41:43.033348Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.4705","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:26f35bb1aa6c14faa4af67fbb332d76989f36b16327314ba0fe4d5b592804d3c","sha256:e06d3941708fa3e8f393840d1a1062400e2e253f514810f9e2a3e37a67c6c740"],"state_sha256":"eccad10355ccad0cde9582aa4d99f2ff90b831b5fc7230c963a9fc1c6279854d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jB4owgcHUfEbFRLLwjwM9yoidZJE9B1/VPyvbMBNU9Og0psltrjSdEineEL30vdsIA5b/4QS05jD6Vf6rxzIBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T16:21:35.160331Z","bundle_sha256":"5ede0822bed24851f786cd42730c4ee79f27c48daa0d2e334f01d687f17fb856"}}