{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:UN4BK5KLPRYMOAK2F2IDNBLG5L","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":"2c28055bbace9631d40eaf07759937f18c0c09d74985440a6802b33b78076b2a","cross_cats_sorted":["math.MG","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-12-20T06:05:12Z","title_canon_sha256":"9679abe2df3dba5bfaa5e7e6cc97b5f302101767037ef60ac6987415085bdcc5"},"schema_version":"1.0","source":{"id":"1412.6612","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.6612","created_at":"2026-05-18T00:29:39Z"},{"alias_kind":"arxiv_version","alias_value":"1412.6612v3","created_at":"2026-05-18T00:29:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.6612","created_at":"2026-05-18T00:29:39Z"},{"alias_kind":"pith_short_12","alias_value":"UN4BK5KLPRYM","created_at":"2026-05-18T12:28:52Z"},{"alias_kind":"pith_short_16","alias_value":"UN4BK5KLPRYMOAK2","created_at":"2026-05-18T12:28:52Z"},{"alias_kind":"pith_short_8","alias_value":"UN4BK5KL","created_at":"2026-05-18T12:28:52Z"}],"graph_snapshots":[{"event_id":"sha256:c030f7f3015317467ff626d769c624ff6705e62cfddf29bca6c97c5a3af4e71d","target":"graph","created_at":"2026-05-18T00:29:39Z","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":"The Vapnik-Chervonenkis (VC) dimension of a collection of subsets of a set is an important combinatorial concept in settings such as discrete geometry and machine learning. In this paper we prove that the VC dimension of the family of $d$-dimensional cubes in $\\mathbb R^d$ is $\\lfloor(3d+1)/2\\rfloor$.","authors_text":"Christian J. J. Despres","cross_cats":["math.MG","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-12-20T06:05:12Z","title":"The Vapnik-Chervonenkis dimension of cubes in $\\mathbb{R}^d$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.6612","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:e5aa4b025d9a07592942bfb78d374c2efcfe52349070ad453593c65852a9aa0a","target":"record","created_at":"2026-05-18T00:29:39Z","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":"2c28055bbace9631d40eaf07759937f18c0c09d74985440a6802b33b78076b2a","cross_cats_sorted":["math.MG","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-12-20T06:05:12Z","title_canon_sha256":"9679abe2df3dba5bfaa5e7e6cc97b5f302101767037ef60ac6987415085bdcc5"},"schema_version":"1.0","source":{"id":"1412.6612","kind":"arxiv","version":3}},"canonical_sha256":"a37815754b7c70c7015a2e90368566ead7550f9a1a6d2ea1f4c83dc070d2f35a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a37815754b7c70c7015a2e90368566ead7550f9a1a6d2ea1f4c83dc070d2f35a","first_computed_at":"2026-05-18T00:29:39.894869Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:29:39.894869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ITM1tyCoa1Jq0pm0c4lB/ZROuEJqFRYsDUHyjDKzmAzdieXcHkhUyy/6dFlom2cm9vS/gyi/aXvFm85i8uKIBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:29:39.895815Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.6612","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e5aa4b025d9a07592942bfb78d374c2efcfe52349070ad453593c65852a9aa0a","sha256:c030f7f3015317467ff626d769c624ff6705e62cfddf29bca6c97c5a3af4e71d"],"state_sha256":"1c9e7459b1a9525ea7bb5bee33a508da4206676cab16d117f19d85353e4f1ef8"}