{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:MPXCGHOAROWEH5IJFCJG6AJNAP","short_pith_number":"pith:MPXCGHOA","canonical_record":{"source":{"id":"1104.3362","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2011-04-18T00:47:41Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"b2fd3f4af8bbdc52f299b96d202cff4bec5fb3d529b4308a9a4523766a3cd0e0","abstract_canon_sha256":"3993aa319524d45fc15db89f2f4e0e9277f92bc785bca300c778d063fa73942f"},"schema_version":"1.0"},"canonical_sha256":"63ee231dc08bac43f50928926f012d03f76b6ef7197158d2a60327919404e13a","source":{"kind":"arxiv","id":"1104.3362","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.3362","created_at":"2026-05-18T04:24:05Z"},{"alias_kind":"arxiv_version","alias_value":"1104.3362v1","created_at":"2026-05-18T04:24:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.3362","created_at":"2026-05-18T04:24:05Z"},{"alias_kind":"pith_short_12","alias_value":"MPXCGHOAROWE","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"MPXCGHOAROWEH5IJ","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"MPXCGHOA","created_at":"2026-05-18T12:26:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:MPXCGHOAROWEH5IJFCJG6AJNAP","target":"record","payload":{"canonical_record":{"source":{"id":"1104.3362","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2011-04-18T00:47:41Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"b2fd3f4af8bbdc52f299b96d202cff4bec5fb3d529b4308a9a4523766a3cd0e0","abstract_canon_sha256":"3993aa319524d45fc15db89f2f4e0e9277f92bc785bca300c778d063fa73942f"},"schema_version":"1.0"},"canonical_sha256":"63ee231dc08bac43f50928926f012d03f76b6ef7197158d2a60327919404e13a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:24:05.237747Z","signature_b64":"MFUyExONFHV52gyVuyW6453Q91dhorZb9p8GkwwqdOXsruw2xMxVfeIeuOr1HDldRI+55Xmojg34jAlMLRNPCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"63ee231dc08bac43f50928926f012d03f76b6ef7197158d2a60327919404e13a","last_reissued_at":"2026-05-18T04:24:05.237097Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:24:05.237097Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1104.3362","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:24:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dsu4IQmpBLoUavr/Znvc8dQ9bAgQzcZi40+FoiOi+Edgkr9YJhBGvVQCtuyGJogS8VtW66SJnYZaRXDAQEoCAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T01:00:30.239271Z"},"content_sha256":"4039f4a152bdb54cd413243be0ac7096c4125e8c410d523857ab35f17e7102b2","schema_version":"1.0","event_id":"sha256:4039f4a152bdb54cd413243be0ac7096c4125e8c410d523857ab35f17e7102b2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:MPXCGHOAROWEH5IJFCJG6AJNAP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Packing numbers of rational ruled 4-manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.SG","authors_text":"Martin Pinsonnault, Olguta Buse","submitted_at":"2011-04-18T00:47:41Z","abstract_excerpt":"We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic 4-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov widths, the stability numbers, and the corresponding obstructing exceptional classes. As a corollary, we give explicit values for when an ellipsoid of type $E(a, b)$, with $\\frac{b}{a} \\in \\N$, embeds in a polydisc $P(s,t)$. Under this integrality assumption, we also give an alternative proof of a recent result of M. Hutchings showing that the ECH capacities give sharp inequalities for embedding el"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.3362","kind":"arxiv","version":1},"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":"2026-05-18T04:24:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uDVFaP7nRCty3BgdlhVHkxwB4tWqmFGjEyNg+Itl1ZjRoJN3mf2Ft3i0VlMQZ/XZNezqdvBmsSUJ+YD9p0zaDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T01:00:30.239859Z"},"content_sha256":"0c2e2b84adb239cde420b4c8f3e20e13879c64de0c0e32eb3d921a91122d3883","schema_version":"1.0","event_id":"sha256:0c2e2b84adb239cde420b4c8f3e20e13879c64de0c0e32eb3d921a91122d3883"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MPXCGHOAROWEH5IJFCJG6AJNAP/bundle.json","state_url":"https://pith.science/pith/MPXCGHOAROWEH5IJFCJG6AJNAP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MPXCGHOAROWEH5IJFCJG6AJNAP/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":"2026-06-01T01:00:30Z","links":{"resolver":"https://pith.science/pith/MPXCGHOAROWEH5IJFCJG6AJNAP","bundle":"https://pith.science/pith/MPXCGHOAROWEH5IJFCJG6AJNAP/bundle.json","state":"https://pith.science/pith/MPXCGHOAROWEH5IJFCJG6AJNAP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MPXCGHOAROWEH5IJFCJG6AJNAP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:MPXCGHOAROWEH5IJFCJG6AJNAP","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":"3993aa319524d45fc15db89f2f4e0e9277f92bc785bca300c778d063fa73942f","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2011-04-18T00:47:41Z","title_canon_sha256":"b2fd3f4af8bbdc52f299b96d202cff4bec5fb3d529b4308a9a4523766a3cd0e0"},"schema_version":"1.0","source":{"id":"1104.3362","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.3362","created_at":"2026-05-18T04:24:05Z"},{"alias_kind":"arxiv_version","alias_value":"1104.3362v1","created_at":"2026-05-18T04:24:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.3362","created_at":"2026-05-18T04:24:05Z"},{"alias_kind":"pith_short_12","alias_value":"MPXCGHOAROWE","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"MPXCGHOAROWEH5IJ","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"MPXCGHOA","created_at":"2026-05-18T12:26:34Z"}],"graph_snapshots":[{"event_id":"sha256:0c2e2b84adb239cde420b4c8f3e20e13879c64de0c0e32eb3d921a91122d3883","target":"graph","created_at":"2026-05-18T04:24:05Z","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 completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic 4-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov widths, the stability numbers, and the corresponding obstructing exceptional classes. As a corollary, we give explicit values for when an ellipsoid of type $E(a, b)$, with $\\frac{b}{a} \\in \\N$, embeds in a polydisc $P(s,t)$. Under this integrality assumption, we also give an alternative proof of a recent result of M. Hutchings showing that the ECH capacities give sharp inequalities for embedding el","authors_text":"Martin Pinsonnault, Olguta Buse","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2011-04-18T00:47:41Z","title":"Packing numbers of rational ruled 4-manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.3362","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":"sha256:4039f4a152bdb54cd413243be0ac7096c4125e8c410d523857ab35f17e7102b2","target":"record","created_at":"2026-05-18T04:24:05Z","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":"3993aa319524d45fc15db89f2f4e0e9277f92bc785bca300c778d063fa73942f","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2011-04-18T00:47:41Z","title_canon_sha256":"b2fd3f4af8bbdc52f299b96d202cff4bec5fb3d529b4308a9a4523766a3cd0e0"},"schema_version":"1.0","source":{"id":"1104.3362","kind":"arxiv","version":1}},"canonical_sha256":"63ee231dc08bac43f50928926f012d03f76b6ef7197158d2a60327919404e13a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"63ee231dc08bac43f50928926f012d03f76b6ef7197158d2a60327919404e13a","first_computed_at":"2026-05-18T04:24:05.237097Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:24:05.237097Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MFUyExONFHV52gyVuyW6453Q91dhorZb9p8GkwwqdOXsruw2xMxVfeIeuOr1HDldRI+55Xmojg34jAlMLRNPCA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:24:05.237747Z","signed_message":"canonical_sha256_bytes"},"source_id":"1104.3362","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4039f4a152bdb54cd413243be0ac7096c4125e8c410d523857ab35f17e7102b2","sha256:0c2e2b84adb239cde420b4c8f3e20e13879c64de0c0e32eb3d921a91122d3883"],"state_sha256":"c593e5d6df9144e461960d09c18a89cfca225f55368dd0115591b614d43459b3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zkgP5m+K2jMjcBi9ZYY7emu280ang8RFj8fjQAr6bwGa9ED9jmbpw7Lsg2D4yCOuT6GpbbpOF06d6YvEudk7CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T01:00:30.243675Z","bundle_sha256":"89a4db14e7bda73c6536b497966b5057429da59833b4a6e5c5d67376748776be"}}