{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:V3CS7ZCJ4GZQTZZII3KQR33VX6","short_pith_number":"pith:V3CS7ZCJ","schema_version":"1.0","canonical_sha256":"aec52fe449e1b309e72846d508ef75bfbdd4eead5da2e143db445571781a78c0","source":{"kind":"arxiv","id":"1612.00785","version":2},"attestation_state":"computed","paper":{"title":"How to avoid a compact set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.LO","authors_text":"Antongiulio Fornasiero, Erik Walsberg, Philipp Hieronymi","submitted_at":"2016-12-02T18:35:52Z","abstract_excerpt":"A first-order expansion of the $\\mathbb{R}$-vector space structure on $\\mathbb{R}$ does not define every compact subset of every $\\mathbb{R}^n$ if and only if topological and Hausdorff dimension coincide on all closed definable sets. Equivalently, if $A \\subseteq \\mathbb{R}^k$ is closed and the Hausdorff dimension of $A$ exceeds the topological dimension of $A$, then every compact subset of every $\\mathbb{R}^n$ can be constructed from $A$ using finitely many boolean operations, cartesian products, and linear operations. The same statement fails when Hausdorff dimension is replaced by packing d"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1612.00785","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2016-12-02T18:35:52Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"a763a0b6495b3c79a48de87aec27fdfe737c0be86015d69d1b0b0a105562ebcf","abstract_canon_sha256":"580eed560d8824eac60f3969678335cd5139845d23cb6e51793158dbfb20aaf4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:13.555578Z","signature_b64":"uU9XTBa/ouxO6Jd8cRaNgJ6Fb1FcnrY5XXp4B+LKvShrndpwfLb4uXYqln4lpQDKEf7R6ZEGwOrdr9ia/MveBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aec52fe449e1b309e72846d508ef75bfbdd4eead5da2e143db445571781a78c0","last_reissued_at":"2026-05-18T00:40:13.555020Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:13.555020Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"How to avoid a compact set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.LO","authors_text":"Antongiulio Fornasiero, Erik Walsberg, Philipp Hieronymi","submitted_at":"2016-12-02T18:35:52Z","abstract_excerpt":"A first-order expansion of the $\\mathbb{R}$-vector space structure on $\\mathbb{R}$ does not define every compact subset of every $\\mathbb{R}^n$ if and only if topological and Hausdorff dimension coincide on all closed definable sets. Equivalently, if $A \\subseteq \\mathbb{R}^k$ is closed and the Hausdorff dimension of $A$ exceeds the topological dimension of $A$, then every compact subset of every $\\mathbb{R}^n$ can be constructed from $A$ using finitely many boolean operations, cartesian products, and linear operations. The same statement fails when Hausdorff dimension is replaced by packing d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.00785","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1612.00785","created_at":"2026-05-18T00:40:13.555088+00:00"},{"alias_kind":"arxiv_version","alias_value":"1612.00785v2","created_at":"2026-05-18T00:40:13.555088+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.00785","created_at":"2026-05-18T00:40:13.555088+00:00"},{"alias_kind":"pith_short_12","alias_value":"V3CS7ZCJ4GZQ","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_16","alias_value":"V3CS7ZCJ4GZQTZZI","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_8","alias_value":"V3CS7ZCJ","created_at":"2026-05-18T12:30:46.583412+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/V3CS7ZCJ4GZQTZZII3KQR33VX6","json":"https://pith.science/pith/V3CS7ZCJ4GZQTZZII3KQR33VX6.json","graph_json":"https://pith.science/api/pith-number/V3CS7ZCJ4GZQTZZII3KQR33VX6/graph.json","events_json":"https://pith.science/api/pith-number/V3CS7ZCJ4GZQTZZII3KQR33VX6/events.json","paper":"https://pith.science/paper/V3CS7ZCJ"},"agent_actions":{"view_html":"https://pith.science/pith/V3CS7ZCJ4GZQTZZII3KQR33VX6","download_json":"https://pith.science/pith/V3CS7ZCJ4GZQTZZII3KQR33VX6.json","view_paper":"https://pith.science/paper/V3CS7ZCJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1612.00785&json=true","fetch_graph":"https://pith.science/api/pith-number/V3CS7ZCJ4GZQTZZII3KQR33VX6/graph.json","fetch_events":"https://pith.science/api/pith-number/V3CS7ZCJ4GZQTZZII3KQR33VX6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V3CS7ZCJ4GZQTZZII3KQR33VX6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V3CS7ZCJ4GZQTZZII3KQR33VX6/action/storage_attestation","attest_author":"https://pith.science/pith/V3CS7ZCJ4GZQTZZII3KQR33VX6/action/author_attestation","sign_citation":"https://pith.science/pith/V3CS7ZCJ4GZQTZZII3KQR33VX6/action/citation_signature","submit_replication":"https://pith.science/pith/V3CS7ZCJ4GZQTZZII3KQR33VX6/action/replication_record"}},"created_at":"2026-05-18T00:40:13.555088+00:00","updated_at":"2026-05-18T00:40:13.555088+00:00"}