{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:OTK2Z2QVG4GYHQZAM5LVZB7ZWV","short_pith_number":"pith:OTK2Z2QV","canonical_record":{"source":{"id":"1502.01693","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-29T23:43:17Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"cb86274e54a6edc755e998e08df7c43112e9dcd57d9791b34018cf9243079dc4","abstract_canon_sha256":"a4c9b81bf9091a3e250542815595c230d5e3b5d97a8e9be4bb3c632615f9b84a"},"schema_version":"1.0"},"canonical_sha256":"74d5acea15370d83c32067575c87f9b546f985735b54ffa3f0e53fe93c0c7eb3","source":{"kind":"arxiv","id":"1502.01693","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.01693","created_at":"2026-05-18T02:27:51Z"},{"alias_kind":"arxiv_version","alias_value":"1502.01693v1","created_at":"2026-05-18T02:27:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.01693","created_at":"2026-05-18T02:27:51Z"},{"alias_kind":"pith_short_12","alias_value":"OTK2Z2QVG4GY","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_16","alias_value":"OTK2Z2QVG4GYHQZA","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_8","alias_value":"OTK2Z2QV","created_at":"2026-05-18T12:29:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:OTK2Z2QVG4GYHQZAM5LVZB7ZWV","target":"record","payload":{"canonical_record":{"source":{"id":"1502.01693","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-29T23:43:17Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"cb86274e54a6edc755e998e08df7c43112e9dcd57d9791b34018cf9243079dc4","abstract_canon_sha256":"a4c9b81bf9091a3e250542815595c230d5e3b5d97a8e9be4bb3c632615f9b84a"},"schema_version":"1.0"},"canonical_sha256":"74d5acea15370d83c32067575c87f9b546f985735b54ffa3f0e53fe93c0c7eb3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:27:51.513993Z","signature_b64":"kxdDAJVJaOZbLdTECX3u1pJ0jS9I4pIN73bit7f0kp2t1tgRlwzs8+JoASWPaVZq5fA65Tm5+47Sj0JRxZLCCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"74d5acea15370d83c32067575c87f9b546f985735b54ffa3f0e53fe93c0c7eb3","last_reissued_at":"2026-05-18T02:27:51.513514Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:27:51.513514Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1502.01693","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":"2026-05-18T02:27:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pZ+Be0bL9lL0Yj1G/Qdr20Zqf/3xo7hAflA3zXSHIw7KxKXtJpjDJwkNnrSD9vC7yWfLDDfhV3Ewh9A3mNUGDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T13:47:46.558695Z"},"content_sha256":"8f94bc1c0053c66eecd864b073d8f26e4beea3fa823733764405a4870d34794a","schema_version":"1.0","event_id":"sha256:8f94bc1c0053c66eecd864b073d8f26e4beea3fa823733764405a4870d34794a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:OTK2Z2QVG4GYHQZAM5LVZB7ZWV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Gaps Between Almost-Primes and a Construction of Almost-Ramanujan Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Adrian Dudek","submitted_at":"2015-01-29T23:43:17Z","abstract_excerpt":"For all $k \\geq 3$, we show how one can explicitly construct an infinite family of $k$-regular graphs all of which have second largest eigenvalue satisfying the bound $O(k^{1/2})$. This resolves an open problem of Reingold, Vadhan and Wigderson."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.01693","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":"2026-05-18T02:27:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3Zuf9zR3FgDV+UXTWlzcm5JxMcLVk5DPs1bEooFlKF0bhODlwpXQOVYTAumaAxLvePtPAc//9Wvv3wYReQweAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T13:47:46.559049Z"},"content_sha256":"fbd08e4eb13093ae79994b74bea1a6975c2ffe8a6dad47dbe2fc40060f5d7299","schema_version":"1.0","event_id":"sha256:fbd08e4eb13093ae79994b74bea1a6975c2ffe8a6dad47dbe2fc40060f5d7299"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OTK2Z2QVG4GYHQZAM5LVZB7ZWV/bundle.json","state_url":"https://pith.science/pith/OTK2Z2QVG4GYHQZAM5LVZB7ZWV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OTK2Z2QVG4GYHQZAM5LVZB7ZWV/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":"2026-06-27T13:47:46Z","links":{"resolver":"https://pith.science/pith/OTK2Z2QVG4GYHQZAM5LVZB7ZWV","bundle":"https://pith.science/pith/OTK2Z2QVG4GYHQZAM5LVZB7ZWV/bundle.json","state":"https://pith.science/pith/OTK2Z2QVG4GYHQZAM5LVZB7ZWV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OTK2Z2QVG4GYHQZAM5LVZB7ZWV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:OTK2Z2QVG4GYHQZAM5LVZB7ZWV","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":"a4c9b81bf9091a3e250542815595c230d5e3b5d97a8e9be4bb3c632615f9b84a","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-29T23:43:17Z","title_canon_sha256":"cb86274e54a6edc755e998e08df7c43112e9dcd57d9791b34018cf9243079dc4"},"schema_version":"1.0","source":{"id":"1502.01693","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.01693","created_at":"2026-05-18T02:27:51Z"},{"alias_kind":"arxiv_version","alias_value":"1502.01693v1","created_at":"2026-05-18T02:27:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.01693","created_at":"2026-05-18T02:27:51Z"},{"alias_kind":"pith_short_12","alias_value":"OTK2Z2QVG4GY","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_16","alias_value":"OTK2Z2QVG4GYHQZA","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_8","alias_value":"OTK2Z2QV","created_at":"2026-05-18T12:29:34Z"}],"graph_snapshots":[{"event_id":"sha256:fbd08e4eb13093ae79994b74bea1a6975c2ffe8a6dad47dbe2fc40060f5d7299","target":"graph","created_at":"2026-05-18T02:27:51Z","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":"For all $k \\geq 3$, we show how one can explicitly construct an infinite family of $k$-regular graphs all of which have second largest eigenvalue satisfying the bound $O(k^{1/2})$. This resolves an open problem of Reingold, Vadhan and Wigderson.","authors_text":"Adrian Dudek","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-29T23:43:17Z","title":"Gaps Between Almost-Primes and a Construction of Almost-Ramanujan Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.01693","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":"sha256:8f94bc1c0053c66eecd864b073d8f26e4beea3fa823733764405a4870d34794a","target":"record","created_at":"2026-05-18T02:27:51Z","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":"a4c9b81bf9091a3e250542815595c230d5e3b5d97a8e9be4bb3c632615f9b84a","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-29T23:43:17Z","title_canon_sha256":"cb86274e54a6edc755e998e08df7c43112e9dcd57d9791b34018cf9243079dc4"},"schema_version":"1.0","source":{"id":"1502.01693","kind":"arxiv","version":1}},"canonical_sha256":"74d5acea15370d83c32067575c87f9b546f985735b54ffa3f0e53fe93c0c7eb3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"74d5acea15370d83c32067575c87f9b546f985735b54ffa3f0e53fe93c0c7eb3","first_computed_at":"2026-05-18T02:27:51.513514Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:27:51.513514Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kxdDAJVJaOZbLdTECX3u1pJ0jS9I4pIN73bit7f0kp2t1tgRlwzs8+JoASWPaVZq5fA65Tm5+47Sj0JRxZLCCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:27:51.513993Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.01693","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8f94bc1c0053c66eecd864b073d8f26e4beea3fa823733764405a4870d34794a","sha256:fbd08e4eb13093ae79994b74bea1a6975c2ffe8a6dad47dbe2fc40060f5d7299"],"state_sha256":"cde7d2858b875afd7fb8f22970aab4be48c05b9c1f0e33550c8cc4b595246dd4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qmk6qzy3OhCRnrtwsypOo5tzBmhEIUTNcPiublqFUbMQwZyfgZw6H39cgkZziiLvDVuEc+V+55jQegJCg+2VAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T13:47:46.560877Z","bundle_sha256":"cba0f60904e1e59caae97b26f43ab0e58e483281d82a6679bc281b75ad9731fa"}}