{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:SFK7KNUPDM53U3YQEPUJZMY7GL","short_pith_number":"pith:SFK7KNUP","schema_version":"1.0","canonical_sha256":"9155f5368f1b3bba6f1023e89cb31f32c9d1a0a758c3a0da5cdde136aba0fc1a","source":{"kind":"arxiv","id":"1403.2225","version":3},"attestation_state":"computed","paper":{"title":"On the variable hierarchy of first-order spectra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"cs.LO","authors_text":"Eryk Kopczynski, Tony Tan","submitted_at":"2014-03-10T12:27:12Z","abstract_excerpt":"The spectrum of a first-order logic sentence is the set of natural numbers that are cardinalities of its finite models. In this paper we study the hierarchy of first-order spectra based on the number of variables. It has been conjectured that it collapses to three variable. We show the opposite: it forms an infinite hierarchy. However, despite the fact that more variables can express more spectra, we show that to establish whether the class of first-order spectra is closed under complement, it is sufficient to consider sentences using only three variables and binary relations."},"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":"1403.2225","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2014-03-10T12:27:12Z","cross_cats_sorted":["cs.CC"],"title_canon_sha256":"541affdd79c852ef8a99ecf1e1921368cd3f96513a707eddeeb9908accf99d7e","abstract_canon_sha256":"fdef6933f6b46eceaa430ed63abade23beda542b71bf78dcfd13579ec8a50a20"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:27:18.840707Z","signature_b64":"4yrFmu2GB8WkDFP93LdHbQbZ1qkprf+ZIcdSyLxWqIJ3Vxr1OtcF2Jsi7qmOi2ugmSDw8/4WBFBjP7s4HEsgAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9155f5368f1b3bba6f1023e89cb31f32c9d1a0a758c3a0da5cdde136aba0fc1a","last_reissued_at":"2026-05-18T02:27:18.840123Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:27:18.840123Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the variable hierarchy of first-order spectra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC"],"primary_cat":"cs.LO","authors_text":"Eryk Kopczynski, Tony Tan","submitted_at":"2014-03-10T12:27:12Z","abstract_excerpt":"The spectrum of a first-order logic sentence is the set of natural numbers that are cardinalities of its finite models. In this paper we study the hierarchy of first-order spectra based on the number of variables. It has been conjectured that it collapses to three variable. We show the opposite: it forms an infinite hierarchy. However, despite the fact that more variables can express more spectra, we show that to establish whether the class of first-order spectra is closed under complement, it is sufficient to consider sentences using only three variables and binary relations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.2225","kind":"arxiv","version":3},"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":"1403.2225","created_at":"2026-05-18T02:27:18.840208+00:00"},{"alias_kind":"arxiv_version","alias_value":"1403.2225v3","created_at":"2026-05-18T02:27:18.840208+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.2225","created_at":"2026-05-18T02:27:18.840208+00:00"},{"alias_kind":"pith_short_12","alias_value":"SFK7KNUPDM53","created_at":"2026-05-18T12:28:49.207871+00:00"},{"alias_kind":"pith_short_16","alias_value":"SFK7KNUPDM53U3YQ","created_at":"2026-05-18T12:28:49.207871+00:00"},{"alias_kind":"pith_short_8","alias_value":"SFK7KNUP","created_at":"2026-05-18T12:28:49.207871+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/SFK7KNUPDM53U3YQEPUJZMY7GL","json":"https://pith.science/pith/SFK7KNUPDM53U3YQEPUJZMY7GL.json","graph_json":"https://pith.science/api/pith-number/SFK7KNUPDM53U3YQEPUJZMY7GL/graph.json","events_json":"https://pith.science/api/pith-number/SFK7KNUPDM53U3YQEPUJZMY7GL/events.json","paper":"https://pith.science/paper/SFK7KNUP"},"agent_actions":{"view_html":"https://pith.science/pith/SFK7KNUPDM53U3YQEPUJZMY7GL","download_json":"https://pith.science/pith/SFK7KNUPDM53U3YQEPUJZMY7GL.json","view_paper":"https://pith.science/paper/SFK7KNUP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1403.2225&json=true","fetch_graph":"https://pith.science/api/pith-number/SFK7KNUPDM53U3YQEPUJZMY7GL/graph.json","fetch_events":"https://pith.science/api/pith-number/SFK7KNUPDM53U3YQEPUJZMY7GL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SFK7KNUPDM53U3YQEPUJZMY7GL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SFK7KNUPDM53U3YQEPUJZMY7GL/action/storage_attestation","attest_author":"https://pith.science/pith/SFK7KNUPDM53U3YQEPUJZMY7GL/action/author_attestation","sign_citation":"https://pith.science/pith/SFK7KNUPDM53U3YQEPUJZMY7GL/action/citation_signature","submit_replication":"https://pith.science/pith/SFK7KNUPDM53U3YQEPUJZMY7GL/action/replication_record"}},"created_at":"2026-05-18T02:27:18.840208+00:00","updated_at":"2026-05-18T02:27:18.840208+00:00"}