{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:LULHPU3RRJZJ45FGY2OLHZFAIA","short_pith_number":"pith:LULHPU3R","canonical_record":{"source":{"id":"1502.07410","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-26T01:06:55Z","cross_cats_sorted":["cs.CC"],"title_canon_sha256":"c2499aaabfbc5eb15798e39b9935292e7028d3f549609b215dcefce6c45e45f6","abstract_canon_sha256":"eac12de462e522415fc1d47ff59de2a9aaf23086fd04dbab991c75da783652b3"},"schema_version":"1.0"},"canonical_sha256":"5d1677d3718a729e74a6c69cb3e4a0400c3dc845559e501c0f4ad2e54ece59cb","source":{"kind":"arxiv","id":"1502.07410","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.07410","created_at":"2026-05-18T01:34:35Z"},{"alias_kind":"arxiv_version","alias_value":"1502.07410v3","created_at":"2026-05-18T01:34:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.07410","created_at":"2026-05-18T01:34:35Z"},{"alias_kind":"pith_short_12","alias_value":"LULHPU3RRJZJ","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"LULHPU3RRJZJ45FG","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"LULHPU3R","created_at":"2026-05-18T12:29:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:LULHPU3RRJZJ45FGY2OLHZFAIA","target":"record","payload":{"canonical_record":{"source":{"id":"1502.07410","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-26T01:06:55Z","cross_cats_sorted":["cs.CC"],"title_canon_sha256":"c2499aaabfbc5eb15798e39b9935292e7028d3f549609b215dcefce6c45e45f6","abstract_canon_sha256":"eac12de462e522415fc1d47ff59de2a9aaf23086fd04dbab991c75da783652b3"},"schema_version":"1.0"},"canonical_sha256":"5d1677d3718a729e74a6c69cb3e4a0400c3dc845559e501c0f4ad2e54ece59cb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:35.036694Z","signature_b64":"Xe6Q+GhyRqqIHqXemt7DU6J1ogVVmu9bDfvwgpWm4lLmtTXg3mW6q0sFT1mGvm1xPZugeADg6+MKIH1bi93lCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5d1677d3718a729e74a6c69cb3e4a0400c3dc845559e501c0f4ad2e54ece59cb","last_reissued_at":"2026-05-18T01:34:35.036058Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:35.036058Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1502.07410","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":"2026-05-18T01:34:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oFW6I6iX0Y2x5+DIUE3zLAb9RbrAUqizBhM8hti5vqhTB5e4lnnnEN7qBl1ObiztChqBpYtf6+XG6xC0t8LBBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T11:33:49.406169Z"},"content_sha256":"d05428c7b391680ce62cd4c2bf1578ed3819baab3232e878665190a6fa2bda21","schema_version":"1.0","event_id":"sha256:d05428c7b391680ce62cd4c2bf1578ed3819baab3232e878665190a6fa2bda21"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:LULHPU3RRJZJ45FGY2OLHZFAIA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Towards Constructing Ramanujan Graphs Using Shift Lifts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"math.CO","authors_text":"Ameya Velingker, Karthekeyan Chandrasekaran","submitted_at":"2015-02-26T01:06:55Z","abstract_excerpt":"In a breakthrough work, Marcus-Spielman-Srivastava recently showed that every $d$-regular bipartite Ramanujan graph has a 2-lift that is also $d$-regular bipartite Ramanujan. As a consequence, a straightforward iterative brute-force search algorithm leads to the construction of a $d$-regular bipartite Ramanujan graph on $N$ vertices in time $2^{O(dN)}$. Shift $k$-lifts studied by Agarwal-Kolla-Madan lead to a natural approach for constructing Ramanujan graphs more efficiently. The number of possible shift $k$-lifts of a $d$-regular $n$-vertex graph is $k^{nd/2}$. Suppose the following holds fo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.07410","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":"2026-05-18T01:34:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U0pZC9ZEPBqdOJJKpeT4WuqT3idsm/WpKjJaMODxi21AluI5ypL6GOX7AEppn8lb0ZypdPNlSzRyCElRB6m3AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T11:33:49.406906Z"},"content_sha256":"a9d7545ab5fc339e77c4d649c764c6039c731360a100df654c4361a8d5000551","schema_version":"1.0","event_id":"sha256:a9d7545ab5fc339e77c4d649c764c6039c731360a100df654c4361a8d5000551"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LULHPU3RRJZJ45FGY2OLHZFAIA/bundle.json","state_url":"https://pith.science/pith/LULHPU3RRJZJ45FGY2OLHZFAIA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LULHPU3RRJZJ45FGY2OLHZFAIA/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":"2026-05-23T11:33:49Z","links":{"resolver":"https://pith.science/pith/LULHPU3RRJZJ45FGY2OLHZFAIA","bundle":"https://pith.science/pith/LULHPU3RRJZJ45FGY2OLHZFAIA/bundle.json","state":"https://pith.science/pith/LULHPU3RRJZJ45FGY2OLHZFAIA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LULHPU3RRJZJ45FGY2OLHZFAIA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:LULHPU3RRJZJ45FGY2OLHZFAIA","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":"eac12de462e522415fc1d47ff59de2a9aaf23086fd04dbab991c75da783652b3","cross_cats_sorted":["cs.CC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-26T01:06:55Z","title_canon_sha256":"c2499aaabfbc5eb15798e39b9935292e7028d3f549609b215dcefce6c45e45f6"},"schema_version":"1.0","source":{"id":"1502.07410","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.07410","created_at":"2026-05-18T01:34:35Z"},{"alias_kind":"arxiv_version","alias_value":"1502.07410v3","created_at":"2026-05-18T01:34:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.07410","created_at":"2026-05-18T01:34:35Z"},{"alias_kind":"pith_short_12","alias_value":"LULHPU3RRJZJ","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"LULHPU3RRJZJ45FG","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"LULHPU3R","created_at":"2026-05-18T12:29:29Z"}],"graph_snapshots":[{"event_id":"sha256:a9d7545ab5fc339e77c4d649c764c6039c731360a100df654c4361a8d5000551","target":"graph","created_at":"2026-05-18T01:34:35Z","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 a breakthrough work, Marcus-Spielman-Srivastava recently showed that every $d$-regular bipartite Ramanujan graph has a 2-lift that is also $d$-regular bipartite Ramanujan. As a consequence, a straightforward iterative brute-force search algorithm leads to the construction of a $d$-regular bipartite Ramanujan graph on $N$ vertices in time $2^{O(dN)}$. Shift $k$-lifts studied by Agarwal-Kolla-Madan lead to a natural approach for constructing Ramanujan graphs more efficiently. The number of possible shift $k$-lifts of a $d$-regular $n$-vertex graph is $k^{nd/2}$. Suppose the following holds fo","authors_text":"Ameya Velingker, Karthekeyan Chandrasekaran","cross_cats":["cs.CC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-26T01:06:55Z","title":"Towards Constructing Ramanujan Graphs Using Shift Lifts"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.07410","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:d05428c7b391680ce62cd4c2bf1578ed3819baab3232e878665190a6fa2bda21","target":"record","created_at":"2026-05-18T01:34:35Z","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":"eac12de462e522415fc1d47ff59de2a9aaf23086fd04dbab991c75da783652b3","cross_cats_sorted":["cs.CC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-26T01:06:55Z","title_canon_sha256":"c2499aaabfbc5eb15798e39b9935292e7028d3f549609b215dcefce6c45e45f6"},"schema_version":"1.0","source":{"id":"1502.07410","kind":"arxiv","version":3}},"canonical_sha256":"5d1677d3718a729e74a6c69cb3e4a0400c3dc845559e501c0f4ad2e54ece59cb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5d1677d3718a729e74a6c69cb3e4a0400c3dc845559e501c0f4ad2e54ece59cb","first_computed_at":"2026-05-18T01:34:35.036058Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:34:35.036058Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Xe6Q+GhyRqqIHqXemt7DU6J1ogVVmu9bDfvwgpWm4lLmtTXg3mW6q0sFT1mGvm1xPZugeADg6+MKIH1bi93lCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:34:35.036694Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.07410","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d05428c7b391680ce62cd4c2bf1578ed3819baab3232e878665190a6fa2bda21","sha256:a9d7545ab5fc339e77c4d649c764c6039c731360a100df654c4361a8d5000551"],"state_sha256":"4d60cb1d6ef0980aed91df1031cec026da4676c17d8c36376488bdf3e45a3e8c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AZyZh2YRstoRPNVL6rkWo8fPqXP61M6ndiGGskRUXC1aGTbDb1MEg9Q6iB/7FMM3IgC8aaKmlukmsM4AYiUBBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T11:33:49.411326Z","bundle_sha256":"f1500886ad3ba35a468e57d4f70cef1906dd1c702a1a88d9c78e4e4f85d460c6"}}