{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:LC5XH5SNC2D4FCHWCWVBWX3ZHB","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":"5a5e79ed87c3854c2559f407ac4b4d5db80879b3580c50f83922cdfae0854a75","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-03-13T17:11:09Z","title_canon_sha256":"110b35ed4b11a5b7c6e7847d6019c336fae29faa796af8f1041ce0602f4c7a0d"},"schema_version":"1.0","source":{"id":"1603.04049","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.04049","created_at":"2026-05-18T01:19:09Z"},{"alias_kind":"arxiv_version","alias_value":"1603.04049v1","created_at":"2026-05-18T01:19:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.04049","created_at":"2026-05-18T01:19:09Z"},{"alias_kind":"pith_short_12","alias_value":"LC5XH5SNC2D4","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_16","alias_value":"LC5XH5SNC2D4FCHW","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_8","alias_value":"LC5XH5SN","created_at":"2026-05-18T12:30:29Z"}],"graph_snapshots":[{"event_id":"sha256:3ae58f178dca051b536f2021b3c5c211516bcf8c94db59e03f005a372fe2ff53","target":"graph","created_at":"2026-05-18T01:19:09Z","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":"The metric dimension of a general metric space was defined in 1953, applied to the set of vertices of a graph metric in 1975, and developed further for metric spaces in 2013. It was then generalised in 2015 to the k-metric dimension of a graph for each positive integer k, where k=1 corresponds to the original definition. Here, we discuss the k-metric dimension of general metric spaces.","authors_text":"A. F. Beardon, J. A. Rodriguez-Velazquez","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-03-13T17:11:09Z","title":"On the k-metric Dimension of Metric Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.04049","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:afa06175332897e6b1879e2d59596efac0fdb6b20d49c42d8815b82579e6dc96","target":"record","created_at":"2026-05-18T01:19:09Z","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":"5a5e79ed87c3854c2559f407ac4b4d5db80879b3580c50f83922cdfae0854a75","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2016-03-13T17:11:09Z","title_canon_sha256":"110b35ed4b11a5b7c6e7847d6019c336fae29faa796af8f1041ce0602f4c7a0d"},"schema_version":"1.0","source":{"id":"1603.04049","kind":"arxiv","version":1}},"canonical_sha256":"58bb73f64d1687c288f615aa1b5f79384ded38dd7a8603a1eca1f380dd04727f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"58bb73f64d1687c288f615aa1b5f79384ded38dd7a8603a1eca1f380dd04727f","first_computed_at":"2026-05-18T01:19:09.566061Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:09.566061Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"x7wJF3jEif3EUSvoD6tbRebEHIOH+JnWXqJhzPmKa54oIh475AvSQfkjj0y/8B/b5amu8IdL7zl80i1S2d3tBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:09.566834Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.04049","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:afa06175332897e6b1879e2d59596efac0fdb6b20d49c42d8815b82579e6dc96","sha256:3ae58f178dca051b536f2021b3c5c211516bcf8c94db59e03f005a372fe2ff53"],"state_sha256":"2a878085e9cd0b3be8af9cf6e1d23770547e6d9fc89992d811a0ffb26f75da86"}