{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:NBWO6XA7SP4QFFL2EG2RFSEIMS","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":"d0589f8b8ee35f79084200c361e202648e7ee6c14bc7ab95086e8a552fb977a9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-20T14:37:25Z","title_canon_sha256":"1aa1b69848d7c996cbd8abd9fe5c34747f10333970c3914156c3b7d36e15e797"},"schema_version":"1.0","source":{"id":"1212.5056","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.5056","created_at":"2026-05-18T03:38:04Z"},{"alias_kind":"arxiv_version","alias_value":"1212.5056v1","created_at":"2026-05-18T03:38:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.5056","created_at":"2026-05-18T03:38:04Z"},{"alias_kind":"pith_short_12","alias_value":"NBWO6XA7SP4Q","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"NBWO6XA7SP4QFFL2","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"NBWO6XA7","created_at":"2026-05-18T12:27:16Z"}],"graph_snapshots":[{"event_id":"sha256:ce88b24c5f401a4fb000cec0b0ed6d9908d31f05549db01e08e752d1c44b26e1","target":"graph","created_at":"2026-05-18T03:38:04Z","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":"There is a parallelism between growth in arithmetic combinatorics and growth in a geometric context. While, over $\\mathbb{R}$ or $\\mathbb{C}$, geometric statements on growth often have geometric proofs, what little is known over finite fields rests on arithmetic proofs. We discuss strategies for geometric proofs of growth over finite fields, and show that growth can be defined and proven in an abstract projective plane -- even one with weak axioms.","authors_text":"H. A. Helfgott, Misha Rudnev, Nick Gill","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-20T14:37:25Z","title":"On growth in an abstract plane"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.5056","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":"sha256:46c0f972c0757cf83a0eace229c181961248b41f815ae622137b4de4516c0384","target":"record","created_at":"2026-05-18T03:38:04Z","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":"d0589f8b8ee35f79084200c361e202648e7ee6c14bc7ab95086e8a552fb977a9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-12-20T14:37:25Z","title_canon_sha256":"1aa1b69848d7c996cbd8abd9fe5c34747f10333970c3914156c3b7d36e15e797"},"schema_version":"1.0","source":{"id":"1212.5056","kind":"arxiv","version":1}},"canonical_sha256":"686cef5c1f93f902957a21b512c8886484b6b37bea8e7fa9e67972e7955e5ce3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"686cef5c1f93f902957a21b512c8886484b6b37bea8e7fa9e67972e7955e5ce3","first_computed_at":"2026-05-18T03:38:04.018645Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:38:04.018645Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yxFZ+eVE0aTkclezZS9BygDjl20VYpV/1ZHHeXlLJX8rkLpSkjkLIjuqU+1vZOeAyGC+ewKCEW5OeqSOSkyDDg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:38:04.019353Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.5056","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:46c0f972c0757cf83a0eace229c181961248b41f815ae622137b4de4516c0384","sha256:ce88b24c5f401a4fb000cec0b0ed6d9908d31f05549db01e08e752d1c44b26e1"],"state_sha256":"8dfb045d98e235e52437539241b8efcca8dcde406e1e3cb996d85165f785226f"}