{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:VRO323OOJW7MT7TLXU33W4A7CE","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":"fd6b842a1c315fbcec49cc1a09bf89a8748abf79030deee841b58a02a9f1afc4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2014-10-15T16:57:38Z","title_canon_sha256":"6d988a44d471a51d4175c921f8b0c514a8392c13d3cf7dcf6aa4daa238938979"},"schema_version":"1.0","source":{"id":"1410.4135","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.4135","created_at":"2026-05-18T02:40:02Z"},{"alias_kind":"arxiv_version","alias_value":"1410.4135v1","created_at":"2026-05-18T02:40:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.4135","created_at":"2026-05-18T02:40:02Z"},{"alias_kind":"pith_short_12","alias_value":"VRO323OOJW7M","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_16","alias_value":"VRO323OOJW7MT7TL","created_at":"2026-05-18T12:28:54Z"},{"alias_kind":"pith_short_8","alias_value":"VRO323OO","created_at":"2026-05-18T12:28:54Z"}],"graph_snapshots":[{"event_id":"sha256:1e4e7318fff06b50cd553448d3b4a25e7b5fe965bee55e86b17d337f0cb43b3c","target":"graph","created_at":"2026-05-18T02:40:02Z","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 define the lower and upper mutual dimensions $mdim(x:y)$ and $Mdim(x:y)$ between any two points $x$ and $y$ in Euclidean space. Intuitively these are the lower and upper densities of the algorithmic information shared by $x$ and $y$. We show that these quantities satisfy the main desiderata for a satisfactory measure of mutual algorithmic information. Our main theorem, the data processing inequality for mutual dimension, says that, if $f:\\mathbb{R}^m \\rightarrow \\mathbb{R}^n$ is computable and Lipschitz, then the inequalities $mdim(f(x):y) \\leq mdim(x:y)$ and $Mdim(f(x):y) \\leq Mdim(x:y)$ h","authors_text":"Adam Case, Jack H. Lutz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2014-10-15T16:57:38Z","title":"Mutual Dimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.4135","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":"sha256:628a1bace73b393d9d6d0ad89ed08833aad4e7808d9b5527e68bc74073439bcc","target":"record","created_at":"2026-05-18T02:40:02Z","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":"fd6b842a1c315fbcec49cc1a09bf89a8748abf79030deee841b58a02a9f1afc4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2014-10-15T16:57:38Z","title_canon_sha256":"6d988a44d471a51d4175c921f8b0c514a8392c13d3cf7dcf6aa4daa238938979"},"schema_version":"1.0","source":{"id":"1410.4135","kind":"arxiv","version":1}},"canonical_sha256":"ac5dbd6dce4dbec9fe6bbd37bb701f1114a4308b51bbfdaddf4ed5b16fc9a982","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ac5dbd6dce4dbec9fe6bbd37bb701f1114a4308b51bbfdaddf4ed5b16fc9a982","first_computed_at":"2026-05-18T02:40:02.585640Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:40:02.585640Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fZ8/0z2uT1NEOht6U97m+2Kpr6R7hYhqH/fKGBJtryWVlm8BQrsuTE2GlOzTo6JTQJg2eTIbqDr7kpJjbdLyDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:40:02.586093Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.4135","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:628a1bace73b393d9d6d0ad89ed08833aad4e7808d9b5527e68bc74073439bcc","sha256:1e4e7318fff06b50cd553448d3b4a25e7b5fe965bee55e86b17d337f0cb43b3c"],"state_sha256":"8f1048122716bf3a39370c0ac667adf5907f83ead5109f1e1886e1657d24296a"}