{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:D2SCZ253Z4TH4MHQHVOFZ7ZOBP","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":"843bf3a0cc5aadd2f8d5d209caff6fae61cdb446728b75b2a0f8c50196d63df7","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-02-04T15:02:28Z","title_canon_sha256":"114392c30c419b191a8c01ab40163ce8a1043019e9daf273ebb3fc2401ff63b1"},"schema_version":"1.0","source":{"id":"1002.1009","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1002.1009","created_at":"2026-05-18T04:33:15Z"},{"alias_kind":"arxiv_version","alias_value":"1002.1009v2","created_at":"2026-05-18T04:33:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1002.1009","created_at":"2026-05-18T04:33:15Z"},{"alias_kind":"pith_short_12","alias_value":"D2SCZ253Z4TH","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"D2SCZ253Z4TH4MHQ","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"D2SCZ253","created_at":"2026-05-18T12:26:06Z"}],"graph_snapshots":[{"event_id":"sha256:ccd964eedabe8a4eafa16f426c89b77665059e706efe5b5f8d21a2174e793f1c","target":"graph","created_at":"2026-05-18T04:33:15Z","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 study the numeration system with negative basis, introduced by Ito and Sadahiro. We focus on arithmetic operations in the set ${\\rm Fin}(-\\beta)$ and $\\Z_{-\\beta}$ of numbers having finite resp. integer $(-\\beta)$-expansions. We show that ${\\rm Fin}(-\\beta)$ is trivial if $\\beta$ is smaller than the golden ratio $\\frac12(1+\\sqrt5)$. For $\\beta\\geq\\frac12(1+\\sqrt5)$ we prove that ${\\rm Fin}(-\\beta)$ is a ring, only if $\\beta$ is a Pisot or Salem number with no negative conjugates. We prove the conjecture of Ito and Sadahiro that ${\\rm Fin}(-\\beta)$ is a ring if $\\beta$ is a quadratic Pisot n","authors_text":"E. Pelantov\\'a, T. V\\'avra, Z. Mas\\'akov\\'a","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-02-04T15:02:28Z","title":"Arithmetics in number systems with negative base"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.1009","kind":"arxiv","version":2},"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:a14762fcc00c4ba728d2b33fb4588ac061ceb6bb3e34538616e3c86b2f2560f7","target":"record","created_at":"2026-05-18T04:33:15Z","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":"843bf3a0cc5aadd2f8d5d209caff6fae61cdb446728b75b2a0f8c50196d63df7","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-02-04T15:02:28Z","title_canon_sha256":"114392c30c419b191a8c01ab40163ce8a1043019e9daf273ebb3fc2401ff63b1"},"schema_version":"1.0","source":{"id":"1002.1009","kind":"arxiv","version":2}},"canonical_sha256":"1ea42cebbbcf267e30f03d5c5cff2e0bd763a8c14ccf3f5bf049528900a5b0aa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1ea42cebbbcf267e30f03d5c5cff2e0bd763a8c14ccf3f5bf049528900a5b0aa","first_computed_at":"2026-05-18T04:33:15.836914Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:33:15.836914Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lWlcpIY89dudLgWCXx+kgckr1pwf7XsiPVryfiBgsBTHA0DTYSQQtj1Bxwmg7BqaWVaQv2J/3dPVt2KX0uzfBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:33:15.837484Z","signed_message":"canonical_sha256_bytes"},"source_id":"1002.1009","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a14762fcc00c4ba728d2b33fb4588ac061ceb6bb3e34538616e3c86b2f2560f7","sha256:ccd964eedabe8a4eafa16f426c89b77665059e706efe5b5f8d21a2174e793f1c"],"state_sha256":"8f02e3cbd57e2f8373307bef29b1e655ffe64b4782ae3690efba3a29df3d2b24"}