{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:7NPKGOV6PT5DLPBDLGQB36RHJ2","short_pith_number":"pith:7NPKGOV6","canonical_record":{"source":{"id":"1810.07979","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2018-10-18T10:15:20Z","cross_cats_sorted":["math.CT","math.MG"],"title_canon_sha256":"2d1aeaf82ee3855fddb09aed908e5621bda80de81de6f87ce2265b2edb88cf29","abstract_canon_sha256":"ade29411b304a310085a8be1be51e0336dad41e0bc00817e08f398c871c7acb5"},"schema_version":"1.0"},"canonical_sha256":"fb5ea33abe7cfa35bc2359a01dfa274e8f07ae1822f245ceba0aa7d6bc378015","source":{"kind":"arxiv","id":"1810.07979","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.07979","created_at":"2026-05-18T00:01:26Z"},{"alias_kind":"arxiv_version","alias_value":"1810.07979v4","created_at":"2026-05-18T00:01:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.07979","created_at":"2026-05-18T00:01:26Z"},{"alias_kind":"pith_short_12","alias_value":"7NPKGOV6PT5D","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"7NPKGOV6PT5DLPBD","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"7NPKGOV6","created_at":"2026-05-18T12:32:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:7NPKGOV6PT5DLPBDLGQB36RHJ2","target":"record","payload":{"canonical_record":{"source":{"id":"1810.07979","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2018-10-18T10:15:20Z","cross_cats_sorted":["math.CT","math.MG"],"title_canon_sha256":"2d1aeaf82ee3855fddb09aed908e5621bda80de81de6f87ce2265b2edb88cf29","abstract_canon_sha256":"ade29411b304a310085a8be1be51e0336dad41e0bc00817e08f398c871c7acb5"},"schema_version":"1.0"},"canonical_sha256":"fb5ea33abe7cfa35bc2359a01dfa274e8f07ae1822f245ceba0aa7d6bc378015","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:26.080432Z","signature_b64":"wEMPDlTCWVWXz7ZsUHxsTferPFm1RxYYao0YmxJfKiRMmn5mdAv9xWAcx73+vrM5jncM/DnS9a/PaG+oWr2iCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fb5ea33abe7cfa35bc2359a01dfa274e8f07ae1822f245ceba0aa7d6bc378015","last_reissued_at":"2026-05-18T00:01:26.080032Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:26.080032Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1810.07979","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:01:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ilZ8BGJmq6c73f+v08qwIEIWnfC72ot7NJeNr3BARFpI2g2p6sB10eb3HMgzHgnvtvfTSGeZu6uAgaHlf7wZBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T11:40:48.898251Z"},"content_sha256":"3aa63c74c4b0d27b8bc8acbe1f2037347b6dafcd510508f55be40ca1c827a002","schema_version":"1.0","event_id":"sha256:3aa63c74c4b0d27b8bc8acbe1f2037347b6dafcd510508f55be40ca1c827a002"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:7NPKGOV6PT5DLPBDLGQB36RHJ2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The normality and bounded growth of balleans","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT","math.MG"],"primary_cat":"math.GN","authors_text":"Igor Protasov, Taras Banakh","submitted_at":"2018-10-18T10:15:20Z","abstract_excerpt":"By a ballean we understand a set $X$ endowed with a family of entourages which is a base of some coarse structure on $X$. Given two unbounded ballean $X,Y$ with normal product $X\\times Y$, we prove that the balleans $X,Y$ have bounded growth and the bornology of $X\\times Y$ has a linearly ordered base. A ballean $(X,\\mathcal E_X)$ is defined to have bounded growth if there exists a function $G$ assigning to each point $x\\in X$ a bounded subset $G[x]\\subset X$ so that for any bounded set $B\\subset X$ the union $\\bigcup_{x\\in B}G[x]$ is bounded and for any entourage $E\\in\\mathcal E_X$ there exis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.07979","kind":"arxiv","version":4},"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":"2026-05-18T00:01:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"osTwZM+xgXQdUHTZTtdGX8craqh8ZM1IYyG1xRvgOAdQNfBAuBXrGT0xjEu3/GALgF0CjpUcgv1EX5vP+FtnAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T11:40:48.898599Z"},"content_sha256":"bd165cf7d0b38be9d2fea276d6a0f70f8becab3dd3bb2e9f0a019fe29a744ab3","schema_version":"1.0","event_id":"sha256:bd165cf7d0b38be9d2fea276d6a0f70f8becab3dd3bb2e9f0a019fe29a744ab3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7NPKGOV6PT5DLPBDLGQB36RHJ2/bundle.json","state_url":"https://pith.science/pith/7NPKGOV6PT5DLPBDLGQB36RHJ2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7NPKGOV6PT5DLPBDLGQB36RHJ2/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":"2026-06-29T11:40:48Z","links":{"resolver":"https://pith.science/pith/7NPKGOV6PT5DLPBDLGQB36RHJ2","bundle":"https://pith.science/pith/7NPKGOV6PT5DLPBDLGQB36RHJ2/bundle.json","state":"https://pith.science/pith/7NPKGOV6PT5DLPBDLGQB36RHJ2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7NPKGOV6PT5DLPBDLGQB36RHJ2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:7NPKGOV6PT5DLPBDLGQB36RHJ2","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":"ade29411b304a310085a8be1be51e0336dad41e0bc00817e08f398c871c7acb5","cross_cats_sorted":["math.CT","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2018-10-18T10:15:20Z","title_canon_sha256":"2d1aeaf82ee3855fddb09aed908e5621bda80de81de6f87ce2265b2edb88cf29"},"schema_version":"1.0","source":{"id":"1810.07979","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.07979","created_at":"2026-05-18T00:01:26Z"},{"alias_kind":"arxiv_version","alias_value":"1810.07979v4","created_at":"2026-05-18T00:01:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.07979","created_at":"2026-05-18T00:01:26Z"},{"alias_kind":"pith_short_12","alias_value":"7NPKGOV6PT5D","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"7NPKGOV6PT5DLPBD","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"7NPKGOV6","created_at":"2026-05-18T12:32:11Z"}],"graph_snapshots":[{"event_id":"sha256:bd165cf7d0b38be9d2fea276d6a0f70f8becab3dd3bb2e9f0a019fe29a744ab3","target":"graph","created_at":"2026-05-18T00:01:26Z","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":"By a ballean we understand a set $X$ endowed with a family of entourages which is a base of some coarse structure on $X$. Given two unbounded ballean $X,Y$ with normal product $X\\times Y$, we prove that the balleans $X,Y$ have bounded growth and the bornology of $X\\times Y$ has a linearly ordered base. A ballean $(X,\\mathcal E_X)$ is defined to have bounded growth if there exists a function $G$ assigning to each point $x\\in X$ a bounded subset $G[x]\\subset X$ so that for any bounded set $B\\subset X$ the union $\\bigcup_{x\\in B}G[x]$ is bounded and for any entourage $E\\in\\mathcal E_X$ there exis","authors_text":"Igor Protasov, Taras Banakh","cross_cats":["math.CT","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2018-10-18T10:15:20Z","title":"The normality and bounded growth of balleans"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.07979","kind":"arxiv","version":4},"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":"sha256:3aa63c74c4b0d27b8bc8acbe1f2037347b6dafcd510508f55be40ca1c827a002","target":"record","created_at":"2026-05-18T00:01:26Z","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":"ade29411b304a310085a8be1be51e0336dad41e0bc00817e08f398c871c7acb5","cross_cats_sorted":["math.CT","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2018-10-18T10:15:20Z","title_canon_sha256":"2d1aeaf82ee3855fddb09aed908e5621bda80de81de6f87ce2265b2edb88cf29"},"schema_version":"1.0","source":{"id":"1810.07979","kind":"arxiv","version":4}},"canonical_sha256":"fb5ea33abe7cfa35bc2359a01dfa274e8f07ae1822f245ceba0aa7d6bc378015","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fb5ea33abe7cfa35bc2359a01dfa274e8f07ae1822f245ceba0aa7d6bc378015","first_computed_at":"2026-05-18T00:01:26.080032Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:01:26.080032Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wEMPDlTCWVWXz7ZsUHxsTferPFm1RxYYao0YmxJfKiRMmn5mdAv9xWAcx73+vrM5jncM/DnS9a/PaG+oWr2iCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:01:26.080432Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.07979","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3aa63c74c4b0d27b8bc8acbe1f2037347b6dafcd510508f55be40ca1c827a002","sha256:bd165cf7d0b38be9d2fea276d6a0f70f8becab3dd3bb2e9f0a019fe29a744ab3"],"state_sha256":"ce953b185ab205e9d192360f4935e8e773759cd406c9e4647b0e589571891e9e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2Zi7Tp5kEZgDaqf52v16B3ya024T93S1WFKFDwJyor7qHnYQtUp7ZEXR/ejRnbIVde1EVtcLAtfUkouzkM7/Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T11:40:48.900460Z","bundle_sha256":"7b61bbccb26203116b213c041fbf9c5aa5294ff481a5d1e8347399bae8373402"}}