{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:2TFCCNA7LQNDVCSETZLPAXWBKX","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":"cde613ed20ae1dd0d59ebc10069f70cd987b1cd483636d4f59fe96af7dcdac40","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2019-07-23T06:23:24Z","title_canon_sha256":"0bc942cf5c02f85fa9fa55824b3fed10e1c702ea149c4e9c30100f22d7a51b0b"},"schema_version":"1.0","source":{"id":"1907.09711","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.09711","created_at":"2026-05-17T23:39:50Z"},{"alias_kind":"arxiv_version","alias_value":"1907.09711v1","created_at":"2026-05-17T23:39:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.09711","created_at":"2026-05-17T23:39:50Z"},{"alias_kind":"pith_short_12","alias_value":"2TFCCNA7LQND","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"2TFCCNA7LQNDVCSE","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"2TFCCNA7","created_at":"2026-05-18T12:33:07Z"}],"graph_snapshots":[{"event_id":"sha256:7f0d904513ba0d5f42eaa949b5faf15812ec7aa76c1f582e6b372c6454cb40aa","target":"graph","created_at":"2026-05-17T23:39:50Z","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":"The voting rules of the European Council (EU) under the Treaty of Lisbon became effective on 1 November 2014. Kurz \\& Napel (2015) showed that the dimension of this voting system is between $7$ and $13,368$. The lower bound $7$ actually sets a new world record for the dimension of the real-world voting bodies. In this article, by finding a new way to represent the union of two weighted games as an intersection of certain weighted games (Theorem 1), we greatly reduce the upper bound $13,368$ to just $25$. We also consider what will happen to our upper bound and Kurz \\& Napel's lower bound if th","authors_text":"Tuen Wai Ng, Wai Shun Cheung, Yuming Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2019-07-23T06:23:24Z","title":"An upper bound on the dimension of the voting system of the European Union Council under the Lisbon rules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.09711","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:cc495c5eb528e21294a96ac436312d62fef0b1522e691fd6054214241e07b4cf","target":"record","created_at":"2026-05-17T23:39:50Z","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":"cde613ed20ae1dd0d59ebc10069f70cd987b1cd483636d4f59fe96af7dcdac40","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.GT","submitted_at":"2019-07-23T06:23:24Z","title_canon_sha256":"0bc942cf5c02f85fa9fa55824b3fed10e1c702ea149c4e9c30100f22d7a51b0b"},"schema_version":"1.0","source":{"id":"1907.09711","kind":"arxiv","version":1}},"canonical_sha256":"d4ca21341f5c1a3a8a449e56f05ec155c22b2517ec6c77065d37c9e42698b386","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d4ca21341f5c1a3a8a449e56f05ec155c22b2517ec6c77065d37c9e42698b386","first_computed_at":"2026-05-17T23:39:50.308386Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:39:50.308386Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"S9pQdDjLI0UJeCmvhIx2Z44F2HXqZ6VszARnzFRmHP76r1CT0+QEQNln9D51ZyHo8y8z+67SSGyfRu1AR3hYDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:39:50.309061Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.09711","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cc495c5eb528e21294a96ac436312d62fef0b1522e691fd6054214241e07b4cf","sha256:7f0d904513ba0d5f42eaa949b5faf15812ec7aa76c1f582e6b372c6454cb40aa"],"state_sha256":"c47dea2a99868f506954e3498eb0cc77d2038104fb66d1380212259866c5ef3b"}