{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:X7JQYHKRGAPPBVEDFK5WPUYWUV","short_pith_number":"pith:X7JQYHKR","canonical_record":{"source":{"id":"1110.1097","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-05T20:28:53Z","cross_cats_sorted":[],"title_canon_sha256":"f0b41e4d3f135d000d5664a8b574bcc1bcbcd7ca0601d2c67bb92d7aacb8c783","abstract_canon_sha256":"ca7c7eedca25dbd59bf4ea045ad9a0c5bd3e177af792b57909b0a68d7b0fa463"},"schema_version":"1.0"},"canonical_sha256":"bfd30c1d51301ef0d4832abb67d316a56afa18a85325e572e9cf5719c6b3770f","source":{"kind":"arxiv","id":"1110.1097","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.1097","created_at":"2026-05-18T04:11:39Z"},{"alias_kind":"arxiv_version","alias_value":"1110.1097v1","created_at":"2026-05-18T04:11:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.1097","created_at":"2026-05-18T04:11:39Z"},{"alias_kind":"pith_short_12","alias_value":"X7JQYHKRGAPP","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"X7JQYHKRGAPPBVED","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"X7JQYHKR","created_at":"2026-05-18T12:26:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:X7JQYHKRGAPPBVEDFK5WPUYWUV","target":"record","payload":{"canonical_record":{"source":{"id":"1110.1097","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-05T20:28:53Z","cross_cats_sorted":[],"title_canon_sha256":"f0b41e4d3f135d000d5664a8b574bcc1bcbcd7ca0601d2c67bb92d7aacb8c783","abstract_canon_sha256":"ca7c7eedca25dbd59bf4ea045ad9a0c5bd3e177af792b57909b0a68d7b0fa463"},"schema_version":"1.0"},"canonical_sha256":"bfd30c1d51301ef0d4832abb67d316a56afa18a85325e572e9cf5719c6b3770f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:39.302042Z","signature_b64":"gcGVGpz8q1P9CEWFFLVMtV4HNA0sJwpHZZn4XI3jS86ZuIEfqW+tUYX6nN9OyQQoyMTbrifptvflv7gwSyUJDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bfd30c1d51301ef0d4832abb67d316a56afa18a85325e572e9cf5719c6b3770f","last_reissued_at":"2026-05-18T04:11:39.301372Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:39.301372Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.1097","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":"2026-05-18T04:11:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uDF5E/fjGKIfW/IXl3TTNg/7UZ03vxFUk62G0ZvNQ4oJhxCVCOiInzbth/zRGxtyzA5ZavuwLBibfrdIZek0Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T02:25:10.904939Z"},"content_sha256":"0aa6e8c414f1f4397f31e50e8c5a02996f1bba837daaf51b6120adcdec06d6a2","schema_version":"1.0","event_id":"sha256:0aa6e8c414f1f4397f31e50e8c5a02996f1bba837daaf51b6120adcdec06d6a2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:X7JQYHKRGAPPBVEDFK5WPUYWUV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"New Bounds on the Minimum Density of a Vertex Identifying Code for the Infinite Hexagonal Grid","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ari Cukierman, Gexin Yu","submitted_at":"2011-10-05T20:28:53Z","abstract_excerpt":"For a graph, $G$, and a vertex $v \\in V(G)$, let $N[v]$ be the set of vertices adjacent to and including $v$. A set $D \\subseteq V(G)$ is a vertex identifying code if for any two distinct vertices $v_1, v_2 \\in V(G)$, the vertex sets $N[v_1] \\cap D$ and $N[v_2] \\cap D$ are distinct and non-empty. We consider the minimum density of a vertex identifying code for the infinite hexagonal grid. In 2000, Cohen et al. constructed two codes with a density of $3/7 \\approx 0.428571$, and this remains the best known upper bound. Until now, the best known lower bound was $12/29 \\approx 0.413793$ and was pr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.1097","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":"2026-05-18T04:11:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BPj/yr8ZYUuDfh5em8rTvoqWU4YquyGmENHS6EtLRBribDrFjt5Nkwi+gm5gdp6riPvwzNJefnF2oWh81saoDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T02:25:10.905275Z"},"content_sha256":"721031e89351ba264b2e9819778ae3481aa8e61f562589b1932fea9687efc048","schema_version":"1.0","event_id":"sha256:721031e89351ba264b2e9819778ae3481aa8e61f562589b1932fea9687efc048"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/X7JQYHKRGAPPBVEDFK5WPUYWUV/bundle.json","state_url":"https://pith.science/pith/X7JQYHKRGAPPBVEDFK5WPUYWUV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/X7JQYHKRGAPPBVEDFK5WPUYWUV/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":"2026-06-03T02:25:10Z","links":{"resolver":"https://pith.science/pith/X7JQYHKRGAPPBVEDFK5WPUYWUV","bundle":"https://pith.science/pith/X7JQYHKRGAPPBVEDFK5WPUYWUV/bundle.json","state":"https://pith.science/pith/X7JQYHKRGAPPBVEDFK5WPUYWUV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/X7JQYHKRGAPPBVEDFK5WPUYWUV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:X7JQYHKRGAPPBVEDFK5WPUYWUV","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":"ca7c7eedca25dbd59bf4ea045ad9a0c5bd3e177af792b57909b0a68d7b0fa463","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-05T20:28:53Z","title_canon_sha256":"f0b41e4d3f135d000d5664a8b574bcc1bcbcd7ca0601d2c67bb92d7aacb8c783"},"schema_version":"1.0","source":{"id":"1110.1097","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.1097","created_at":"2026-05-18T04:11:39Z"},{"alias_kind":"arxiv_version","alias_value":"1110.1097v1","created_at":"2026-05-18T04:11:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.1097","created_at":"2026-05-18T04:11:39Z"},{"alias_kind":"pith_short_12","alias_value":"X7JQYHKRGAPP","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"X7JQYHKRGAPPBVED","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"X7JQYHKR","created_at":"2026-05-18T12:26:44Z"}],"graph_snapshots":[{"event_id":"sha256:721031e89351ba264b2e9819778ae3481aa8e61f562589b1932fea9687efc048","target":"graph","created_at":"2026-05-18T04:11:39Z","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 a graph, $G$, and a vertex $v \\in V(G)$, let $N[v]$ be the set of vertices adjacent to and including $v$. A set $D \\subseteq V(G)$ is a vertex identifying code if for any two distinct vertices $v_1, v_2 \\in V(G)$, the vertex sets $N[v_1] \\cap D$ and $N[v_2] \\cap D$ are distinct and non-empty. We consider the minimum density of a vertex identifying code for the infinite hexagonal grid. In 2000, Cohen et al. constructed two codes with a density of $3/7 \\approx 0.428571$, and this remains the best known upper bound. Until now, the best known lower bound was $12/29 \\approx 0.413793$ and was pr","authors_text":"Ari Cukierman, Gexin Yu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-05T20:28:53Z","title":"New Bounds on the Minimum Density of a Vertex Identifying Code for the Infinite Hexagonal Grid"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.1097","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":"sha256:0aa6e8c414f1f4397f31e50e8c5a02996f1bba837daaf51b6120adcdec06d6a2","target":"record","created_at":"2026-05-18T04:11:39Z","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":"ca7c7eedca25dbd59bf4ea045ad9a0c5bd3e177af792b57909b0a68d7b0fa463","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-05T20:28:53Z","title_canon_sha256":"f0b41e4d3f135d000d5664a8b574bcc1bcbcd7ca0601d2c67bb92d7aacb8c783"},"schema_version":"1.0","source":{"id":"1110.1097","kind":"arxiv","version":1}},"canonical_sha256":"bfd30c1d51301ef0d4832abb67d316a56afa18a85325e572e9cf5719c6b3770f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bfd30c1d51301ef0d4832abb67d316a56afa18a85325e572e9cf5719c6b3770f","first_computed_at":"2026-05-18T04:11:39.301372Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:11:39.301372Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gcGVGpz8q1P9CEWFFLVMtV4HNA0sJwpHZZn4XI3jS86ZuIEfqW+tUYX6nN9OyQQoyMTbrifptvflv7gwSyUJDA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:11:39.302042Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.1097","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0aa6e8c414f1f4397f31e50e8c5a02996f1bba837daaf51b6120adcdec06d6a2","sha256:721031e89351ba264b2e9819778ae3481aa8e61f562589b1932fea9687efc048"],"state_sha256":"7d58a4526faca8c544187524d5133c237547eb03d07a6858fb115047af16c98e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gdfmjJuNw+Mu99/G4iOHIzrM/eVlhR3ILlnCpZQvM1RQbLbTJ0ohla502Ji5lX1I4i26xRHS7VQ1TOSmXo5rBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T02:25:10.907403Z","bundle_sha256":"26763d9802c48b81dee0a932d1dd1468c2ba189e27b2ff11478cfcdac9fbbda4"}}