{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:3LYK3I7LYRP4ODDTXPUFYOU5ZK","short_pith_number":"pith:3LYK3I7L","canonical_record":{"source":{"id":"1111.1608","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-11-07T15:08:12Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"bd91a84f654f186c1933720c1d8ace21069ad2042f6b36fa677f7f6e6c5ffc5e","abstract_canon_sha256":"145449e17efb571472e1335a990ea161a837ce28cea8db47f88e3dbba1c418ee"},"schema_version":"1.0"},"canonical_sha256":"daf0ada3ebc45fc70c73bbe85c3a9dcab199699eb2d312c7f20a277222d25736","source":{"kind":"arxiv","id":"1111.1608","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.1608","created_at":"2026-05-18T02:57:36Z"},{"alias_kind":"arxiv_version","alias_value":"1111.1608v3","created_at":"2026-05-18T02:57:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.1608","created_at":"2026-05-18T02:57:36Z"},{"alias_kind":"pith_short_12","alias_value":"3LYK3I7LYRP4","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"3LYK3I7LYRP4ODDT","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"3LYK3I7L","created_at":"2026-05-18T12:26:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:3LYK3I7LYRP4ODDTXPUFYOU5ZK","target":"record","payload":{"canonical_record":{"source":{"id":"1111.1608","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-11-07T15:08:12Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"bd91a84f654f186c1933720c1d8ace21069ad2042f6b36fa677f7f6e6c5ffc5e","abstract_canon_sha256":"145449e17efb571472e1335a990ea161a837ce28cea8db47f88e3dbba1c418ee"},"schema_version":"1.0"},"canonical_sha256":"daf0ada3ebc45fc70c73bbe85c3a9dcab199699eb2d312c7f20a277222d25736","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:36.196889Z","signature_b64":"k5CkXTI7wBS975rX0eLp+ew661JKlWEEIQyvmvUxdxRSPzKUueL+gJFo9yIZmA+isLx8/4HlKAs25Bxtn48LBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"daf0ada3ebc45fc70c73bbe85c3a9dcab199699eb2d312c7f20a277222d25736","last_reissued_at":"2026-05-18T02:57:36.196267Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:36.196267Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.1608","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:57:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0izdrSERvZNOJhI3TIy75HCbBpVQhuXzlIfR7je6w0FAXwjVKSB1E2iYSLRlfXWBzLiXXEeWDUfWAiPMSCZvAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T06:10:44.152967Z"},"content_sha256":"e6e73b3023a4b8b992199721ff29b9d0405084836b0232d4930d937d80bb69a6","schema_version":"1.0","event_id":"sha256:e6e73b3023a4b8b992199721ff29b9d0405084836b0232d4930d937d80bb69a6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:3LYK3I7LYRP4ODDTXPUFYOU5ZK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Computational topology of equipartitions by hyperplanes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Rade T. Zivaljevic","submitted_at":"2011-11-07T15:08:12Z","abstract_excerpt":"We compute a primary cohomological obstruction to the existence of an equipartition for j mass distributions in R^d by two hyperplanes in the case 2d-3j = 1. The central new result is that such an equipartition always exists if d=6 2^k +2 and j=4 2^k+1 which for k=0 reduces to the main result of the paper P. Mani-Levitska et al., Topology and combinatorics of partitions of masses by hyperplanes, Adv. Math. 207 (2006), 266-296. This is an example of a genuine combinatorial geometric result which involves Z_4-torsion in an essential way and cannot be obtained by the application of either Stiefel"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.1608","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:57:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZRiMKrFFh9wYlvaeH1CddIAmmDxcYKxoLVlpqXp2zq2mm3pGiKoz8dsB1v8ILGBdmYSQxRA5KS5Ox/a+wSPKAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T06:10:44.153373Z"},"content_sha256":"8fe8f7f655e8f8f8251f113baad7846a12bd96a01dce121f2a18d75b29ecc10a","schema_version":"1.0","event_id":"sha256:8fe8f7f655e8f8f8251f113baad7846a12bd96a01dce121f2a18d75b29ecc10a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3LYK3I7LYRP4ODDTXPUFYOU5ZK/bundle.json","state_url":"https://pith.science/pith/3LYK3I7LYRP4ODDTXPUFYOU5ZK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3LYK3I7LYRP4ODDTXPUFYOU5ZK/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":"2026-06-08T06:10:44Z","links":{"resolver":"https://pith.science/pith/3LYK3I7LYRP4ODDTXPUFYOU5ZK","bundle":"https://pith.science/pith/3LYK3I7LYRP4ODDTXPUFYOU5ZK/bundle.json","state":"https://pith.science/pith/3LYK3I7LYRP4ODDTXPUFYOU5ZK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3LYK3I7LYRP4ODDTXPUFYOU5ZK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:3LYK3I7LYRP4ODDTXPUFYOU5ZK","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":"145449e17efb571472e1335a990ea161a837ce28cea8db47f88e3dbba1c418ee","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-11-07T15:08:12Z","title_canon_sha256":"bd91a84f654f186c1933720c1d8ace21069ad2042f6b36fa677f7f6e6c5ffc5e"},"schema_version":"1.0","source":{"id":"1111.1608","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.1608","created_at":"2026-05-18T02:57:36Z"},{"alias_kind":"arxiv_version","alias_value":"1111.1608v3","created_at":"2026-05-18T02:57:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.1608","created_at":"2026-05-18T02:57:36Z"},{"alias_kind":"pith_short_12","alias_value":"3LYK3I7LYRP4","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"3LYK3I7LYRP4ODDT","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"3LYK3I7L","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:8fe8f7f655e8f8f8251f113baad7846a12bd96a01dce121f2a18d75b29ecc10a","target":"graph","created_at":"2026-05-18T02:57:36Z","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 compute a primary cohomological obstruction to the existence of an equipartition for j mass distributions in R^d by two hyperplanes in the case 2d-3j = 1. The central new result is that such an equipartition always exists if d=6 2^k +2 and j=4 2^k+1 which for k=0 reduces to the main result of the paper P. Mani-Levitska et al., Topology and combinatorics of partitions of masses by hyperplanes, Adv. Math. 207 (2006), 266-296. This is an example of a genuine combinatorial geometric result which involves Z_4-torsion in an essential way and cannot be obtained by the application of either Stiefel","authors_text":"Rade T. Zivaljevic","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-11-07T15:08:12Z","title":"Computational topology of equipartitions by hyperplanes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.1608","kind":"arxiv","version":3},"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:e6e73b3023a4b8b992199721ff29b9d0405084836b0232d4930d937d80bb69a6","target":"record","created_at":"2026-05-18T02:57:36Z","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":"145449e17efb571472e1335a990ea161a837ce28cea8db47f88e3dbba1c418ee","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2011-11-07T15:08:12Z","title_canon_sha256":"bd91a84f654f186c1933720c1d8ace21069ad2042f6b36fa677f7f6e6c5ffc5e"},"schema_version":"1.0","source":{"id":"1111.1608","kind":"arxiv","version":3}},"canonical_sha256":"daf0ada3ebc45fc70c73bbe85c3a9dcab199699eb2d312c7f20a277222d25736","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"daf0ada3ebc45fc70c73bbe85c3a9dcab199699eb2d312c7f20a277222d25736","first_computed_at":"2026-05-18T02:57:36.196267Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:57:36.196267Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"k5CkXTI7wBS975rX0eLp+ew661JKlWEEIQyvmvUxdxRSPzKUueL+gJFo9yIZmA+isLx8/4HlKAs25Bxtn48LBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:57:36.196889Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.1608","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e6e73b3023a4b8b992199721ff29b9d0405084836b0232d4930d937d80bb69a6","sha256:8fe8f7f655e8f8f8251f113baad7846a12bd96a01dce121f2a18d75b29ecc10a"],"state_sha256":"b37bdcccd1d51ec2027ed822828e514de2a4c04107fb7f7d766fecfe6f914b84"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8Ycf8sM014zZUmYJBmfh6bGzXMK4gnTKPnaoOM90mUTzfdxUvvaSKYpEr/fQZNwmfo1cgPBhEy+6DKG/ZC5eBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T06:10:44.156947Z","bundle_sha256":"a38285df523c0778748d3b104f09c6f8534f5ca43d55627513325ae4ee2632e3"}}