{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:H4C6CCSXSE77AVMIQCDWPXEPB5","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":"0c4cbcc8846b76ed22a08d354e9688b79c5969adad481fb6d68f6ac5d9c9c3f5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-11-02T11:04:35Z","title_canon_sha256":"5194acb965a16e2d63d00f6af3e88ae502d6038aa36cf1c3058e2594570f75c4"},"schema_version":"1.0","source":{"id":"1511.00442","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.00442","created_at":"2026-05-18T00:56:06Z"},{"alias_kind":"arxiv_version","alias_value":"1511.00442v4","created_at":"2026-05-18T00:56:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.00442","created_at":"2026-05-18T00:56:06Z"},{"alias_kind":"pith_short_12","alias_value":"H4C6CCSXSE77","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_16","alias_value":"H4C6CCSXSE77AVMI","created_at":"2026-05-18T12:29:22Z"},{"alias_kind":"pith_short_8","alias_value":"H4C6CCSX","created_at":"2026-05-18T12:29:22Z"}],"graph_snapshots":[{"event_id":"sha256:a1815a3f2b83e5ee086a0063033d6c242d50f87f8d58635b1995117e4e3baab0","target":"graph","created_at":"2026-05-18T00:56:06Z","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":"We formulate the conditional Kolmogorov complexity of x given y at precision r, where x and y are points in Euclidean spaces and r is a natural number. We demonstrate the utility of this notion in two ways.\n  1. We prove a point-to-set principle that enables one to use the (relativized, constructive) dimension of a single point in a set E in a Euclidean space to establish a lower bound on the (classical) Hausdorff dimension of E. We then use this principle, together with conditional Kolmogorov complexity in Euclidean spaces, to give a new proof of the known, two-dimensional case of the Kakeya ","authors_text":"Jack H. Lutz, Neil Lutz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-11-02T11:04:35Z","title":"Algorithmic information, plane Kakeya sets, and conditional dimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.00442","kind":"arxiv","version":4},"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":"sha256:642513000809f03e05f8e99fdc7ca1936cdd3648772cc62f3006e963fbbdc34a","target":"record","created_at":"2026-05-18T00:56:06Z","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":"0c4cbcc8846b76ed22a08d354e9688b79c5969adad481fb6d68f6ac5d9c9c3f5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2015-11-02T11:04:35Z","title_canon_sha256":"5194acb965a16e2d63d00f6af3e88ae502d6038aa36cf1c3058e2594570f75c4"},"schema_version":"1.0","source":{"id":"1511.00442","kind":"arxiv","version":4}},"canonical_sha256":"3f05e10a57913ff05588808767dc8f0f5bde682e6ebdac52492d0d57fef90498","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3f05e10a57913ff05588808767dc8f0f5bde682e6ebdac52492d0d57fef90498","first_computed_at":"2026-05-18T00:56:06.537259Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:56:06.537259Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZZwCgBocLRRupBiJ8z50G/WyUOQ5xSBW9u8L5Lh4sCLiG9GSx75P9v5mWejx96HSWaGcBT04gbFeLqSJ/N7hAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:56:06.537851Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.00442","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:642513000809f03e05f8e99fdc7ca1936cdd3648772cc62f3006e963fbbdc34a","sha256:a1815a3f2b83e5ee086a0063033d6c242d50f87f8d58635b1995117e4e3baab0"],"state_sha256":"2d6e2c931730f20ee56c200934df71955dc9bdddc5c84f2a2564caed19ae36db"}