{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:CAGDC2AFPYSRFKJAZTUOMARRKA","short_pith_number":"pith:CAGDC2AF","canonical_record":{"source":{"id":"0912.0306","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-12-02T00:15:48Z","cross_cats_sorted":[],"title_canon_sha256":"8a8b0a2c2bcb2239476bd0eed47d2f3fba195c4a7efdb28ac3d0d4c3cd283deb","abstract_canon_sha256":"254ca45679335ecf1d57a447953ee76cacddf193fb2bf23448860e26a6143662"},"schema_version":"1.0"},"canonical_sha256":"100c3168057e2512a920cce8e602315027dad05e563b133210ad3c4bc50f949e","source":{"kind":"arxiv","id":"0912.0306","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0912.0306","created_at":"2026-05-18T04:39:19Z"},{"alias_kind":"arxiv_version","alias_value":"0912.0306v2","created_at":"2026-05-18T04:39:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.0306","created_at":"2026-05-18T04:39:19Z"},{"alias_kind":"pith_short_12","alias_value":"CAGDC2AFPYSR","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"CAGDC2AFPYSRFKJA","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"CAGDC2AF","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:CAGDC2AFPYSRFKJAZTUOMARRKA","target":"record","payload":{"canonical_record":{"source":{"id":"0912.0306","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-12-02T00:15:48Z","cross_cats_sorted":[],"title_canon_sha256":"8a8b0a2c2bcb2239476bd0eed47d2f3fba195c4a7efdb28ac3d0d4c3cd283deb","abstract_canon_sha256":"254ca45679335ecf1d57a447953ee76cacddf193fb2bf23448860e26a6143662"},"schema_version":"1.0"},"canonical_sha256":"100c3168057e2512a920cce8e602315027dad05e563b133210ad3c4bc50f949e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:19.599892Z","signature_b64":"7pCY26w+VRmmOD2vm+HM6bYNnAuVUozH6CEdzqocorLaW0omX6qDlvU1paxI8BefTz8dP/8zvZqPI0+NYU0/Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"100c3168057e2512a920cce8e602315027dad05e563b133210ad3c4bc50f949e","last_reissued_at":"2026-05-18T04:39:19.599424Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:19.599424Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0912.0306","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:39:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9Xg65O5S5Zj+2TuE6oM2qy8iPC94Tft/NDZrHgh5lqR/AKZtL9dvxZBcx4Zz6NQ/Ps59dieKD6y7wt/o+7DTDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T02:56:32.555689Z"},"content_sha256":"f946b51f9c7dcd083158b4d7ed05509fa0f3cef66e07a240a499dffd7f38697e","schema_version":"1.0","event_id":"sha256:f946b51f9c7dcd083158b4d7ed05509fa0f3cef66e07a240a499dffd7f38697e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:CAGDC2AFPYSRFKJAZTUOMARRKA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On a non-abelian Balog-Szemeredi-type lemma","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tom Sanders","submitted_at":"2009-12-02T00:15:48Z","abstract_excerpt":"We show that if G is a group and A is a finite subset of G with |A^2| < K|A|, then for all k there is a symmetric neighbourhood of the identity S with S^k a subset of A^2A^{-2} and |S| > exp(-K^{O(k)})|A|."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.0306","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:39:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FMa1nGjs8Smw/qrCYR0lJE4BpbkxjWbfNtkBiy8HAg3ViDu9Z0JhMQge26GZFNbhXPkh6BdA68YaF0ay/cq7BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T02:56:32.556227Z"},"content_sha256":"5073485fcfe2a7d88cf3533ffb08a2946dd73828bb517c482ff32685023e76f0","schema_version":"1.0","event_id":"sha256:5073485fcfe2a7d88cf3533ffb08a2946dd73828bb517c482ff32685023e76f0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CAGDC2AFPYSRFKJAZTUOMARRKA/bundle.json","state_url":"https://pith.science/pith/CAGDC2AFPYSRFKJAZTUOMARRKA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CAGDC2AFPYSRFKJAZTUOMARRKA/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-31T02:56:32Z","links":{"resolver":"https://pith.science/pith/CAGDC2AFPYSRFKJAZTUOMARRKA","bundle":"https://pith.science/pith/CAGDC2AFPYSRFKJAZTUOMARRKA/bundle.json","state":"https://pith.science/pith/CAGDC2AFPYSRFKJAZTUOMARRKA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CAGDC2AFPYSRFKJAZTUOMARRKA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:CAGDC2AFPYSRFKJAZTUOMARRKA","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":"254ca45679335ecf1d57a447953ee76cacddf193fb2bf23448860e26a6143662","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-12-02T00:15:48Z","title_canon_sha256":"8a8b0a2c2bcb2239476bd0eed47d2f3fba195c4a7efdb28ac3d0d4c3cd283deb"},"schema_version":"1.0","source":{"id":"0912.0306","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0912.0306","created_at":"2026-05-18T04:39:19Z"},{"alias_kind":"arxiv_version","alias_value":"0912.0306v2","created_at":"2026-05-18T04:39:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0912.0306","created_at":"2026-05-18T04:39:19Z"},{"alias_kind":"pith_short_12","alias_value":"CAGDC2AFPYSR","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"CAGDC2AFPYSRFKJA","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"CAGDC2AF","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:5073485fcfe2a7d88cf3533ffb08a2946dd73828bb517c482ff32685023e76f0","target":"graph","created_at":"2026-05-18T04:39:19Z","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 show that if G is a group and A is a finite subset of G with |A^2| < K|A|, then for all k there is a symmetric neighbourhood of the identity S with S^k a subset of A^2A^{-2} and |S| > exp(-K^{O(k)})|A|.","authors_text":"Tom Sanders","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-12-02T00:15:48Z","title":"On a non-abelian Balog-Szemeredi-type lemma"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0912.0306","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:f946b51f9c7dcd083158b4d7ed05509fa0f3cef66e07a240a499dffd7f38697e","target":"record","created_at":"2026-05-18T04:39:19Z","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":"254ca45679335ecf1d57a447953ee76cacddf193fb2bf23448860e26a6143662","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-12-02T00:15:48Z","title_canon_sha256":"8a8b0a2c2bcb2239476bd0eed47d2f3fba195c4a7efdb28ac3d0d4c3cd283deb"},"schema_version":"1.0","source":{"id":"0912.0306","kind":"arxiv","version":2}},"canonical_sha256":"100c3168057e2512a920cce8e602315027dad05e563b133210ad3c4bc50f949e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"100c3168057e2512a920cce8e602315027dad05e563b133210ad3c4bc50f949e","first_computed_at":"2026-05-18T04:39:19.599424Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:39:19.599424Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7pCY26w+VRmmOD2vm+HM6bYNnAuVUozH6CEdzqocorLaW0omX6qDlvU1paxI8BefTz8dP/8zvZqPI0+NYU0/Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:39:19.599892Z","signed_message":"canonical_sha256_bytes"},"source_id":"0912.0306","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f946b51f9c7dcd083158b4d7ed05509fa0f3cef66e07a240a499dffd7f38697e","sha256:5073485fcfe2a7d88cf3533ffb08a2946dd73828bb517c482ff32685023e76f0"],"state_sha256":"830d873adef03497e0c05306336af3c67db1cbc6def6ff184cbd4c0944f6a183"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2ROT7C0HdNiLFWw4PBZF8lAYBsVdI38uthA8ZFcb8UAADNBcbLOr8DDiLibLLCHq3jGZvjku0MZq1KBjvE7jAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T02:56:32.559051Z","bundle_sha256":"f2c111145d951ce5f8ccad5aa96e499d66fc23a4c53b6abddd539a8620b32210"}}