{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:E6LHFDBTDC3BNWNLZ37SAAMQFF","short_pith_number":"pith:E6LHFDBT","canonical_record":{"source":{"id":"1106.3462","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-06-17T12:18:47Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"dc987af5a03212acba6249c7e2771c1a0626a880c4a253b52e544efe4be6564d","abstract_canon_sha256":"fff3eca7111101c349cd8c8cd2444309fe1f111d4c947a85f2ce23d7c200d35c"},"schema_version":"1.0"},"canonical_sha256":"2796728c3318b616d9abceff200190297a63be76ff01accc100222bb9963eb1a","source":{"kind":"arxiv","id":"1106.3462","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1106.3462","created_at":"2026-05-18T01:37:29Z"},{"alias_kind":"arxiv_version","alias_value":"1106.3462v2","created_at":"2026-05-18T01:37:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.3462","created_at":"2026-05-18T01:37:29Z"},{"alias_kind":"pith_short_12","alias_value":"E6LHFDBTDC3B","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"E6LHFDBTDC3BNWNL","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"E6LHFDBT","created_at":"2026-05-18T12:26:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:E6LHFDBTDC3BNWNLZ37SAAMQFF","target":"record","payload":{"canonical_record":{"source":{"id":"1106.3462","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-06-17T12:18:47Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"dc987af5a03212acba6249c7e2771c1a0626a880c4a253b52e544efe4be6564d","abstract_canon_sha256":"fff3eca7111101c349cd8c8cd2444309fe1f111d4c947a85f2ce23d7c200d35c"},"schema_version":"1.0"},"canonical_sha256":"2796728c3318b616d9abceff200190297a63be76ff01accc100222bb9963eb1a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:29.101434Z","signature_b64":"wUTKaBFC1h0Cq2vsyiTsFuTV5OXUZNKA7SuvEIkQ/IrQbPw7kSTMWKMMdFUX7AdgjNaRxPO2kN1VhFkUjam8Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2796728c3318b616d9abceff200190297a63be76ff01accc100222bb9963eb1a","last_reissued_at":"2026-05-18T01:37:29.100744Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:29.100744Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1106.3462","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":"2026-05-18T01:37:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ylsFO5n0wyxY9xYjWZ6v6nCXpmuf+eY6bBmS6qcMivHZui3neus0h33rNRaBUhmuxdRKiYoi0FMJjA965tlFDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T15:51:07.941203Z"},"content_sha256":"0648462c1a7bd90ee64e232832b184abbe1134fb6baf2298ab506a7be07def0e","schema_version":"1.0","event_id":"sha256:0648462c1a7bd90ee64e232832b184abbe1134fb6baf2298ab506a7be07def0e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:E6LHFDBTDC3BNWNLZ37SAAMQFF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Continuous closure, axes closure, and natural closure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.AC","authors_text":"Melvin Hochster, Neil Epstein","submitted_at":"2011-06-17T12:18:47Z","abstract_excerpt":"Let $R$ be a reduced affine $\\mathbb C$-algebra, with corresponding affine algebraic set $X$. Let $\\mathcal C(X)$ be the ring of continuous (Euclidean topology) $\\mathbb C$-valued functions on $X$. Brenner defined the \\emph{continuous closure} $I^{\\rm cont}$ of an ideal $I$ as $I\\mathcal C(X) \\cap R$. He also introduced an algebraic notion of \\emph{axes closure} $I^{\\rm ax}$ that always contains $I^{\\rm cont}$, and asked whether they coincide. We extend the notion of axes closure to general Noetherian rings, defining $f \\in I^{\\rm ax}$ if its image is in $IS$ for every homomorphism $R \\to S$, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.3462","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":"2026-05-18T01:37:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"moQCginU23OukmemfAG12X8InHumAF2V+jTKIA5HeYqPAhmtO49y2TU0hqPusrvQTp1V3HqdBEo4cwkeIsAEBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T15:51:07.941573Z"},"content_sha256":"130ff34bf137f7e1d81013626625f6fea23464a292de51a06a63b7392b110f03","schema_version":"1.0","event_id":"sha256:130ff34bf137f7e1d81013626625f6fea23464a292de51a06a63b7392b110f03"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/E6LHFDBTDC3BNWNLZ37SAAMQFF/bundle.json","state_url":"https://pith.science/pith/E6LHFDBTDC3BNWNLZ37SAAMQFF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/E6LHFDBTDC3BNWNLZ37SAAMQFF/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":"2026-06-01T15:51:07Z","links":{"resolver":"https://pith.science/pith/E6LHFDBTDC3BNWNLZ37SAAMQFF","bundle":"https://pith.science/pith/E6LHFDBTDC3BNWNLZ37SAAMQFF/bundle.json","state":"https://pith.science/pith/E6LHFDBTDC3BNWNLZ37SAAMQFF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/E6LHFDBTDC3BNWNLZ37SAAMQFF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:E6LHFDBTDC3BNWNLZ37SAAMQFF","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":"fff3eca7111101c349cd8c8cd2444309fe1f111d4c947a85f2ce23d7c200d35c","cross_cats_sorted":["math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-06-17T12:18:47Z","title_canon_sha256":"dc987af5a03212acba6249c7e2771c1a0626a880c4a253b52e544efe4be6564d"},"schema_version":"1.0","source":{"id":"1106.3462","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1106.3462","created_at":"2026-05-18T01:37:29Z"},{"alias_kind":"arxiv_version","alias_value":"1106.3462v2","created_at":"2026-05-18T01:37:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.3462","created_at":"2026-05-18T01:37:29Z"},{"alias_kind":"pith_short_12","alias_value":"E6LHFDBTDC3B","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"E6LHFDBTDC3BNWNL","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"E6LHFDBT","created_at":"2026-05-18T12:26:26Z"}],"graph_snapshots":[{"event_id":"sha256:130ff34bf137f7e1d81013626625f6fea23464a292de51a06a63b7392b110f03","target":"graph","created_at":"2026-05-18T01:37:29Z","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":"Let $R$ be a reduced affine $\\mathbb C$-algebra, with corresponding affine algebraic set $X$. Let $\\mathcal C(X)$ be the ring of continuous (Euclidean topology) $\\mathbb C$-valued functions on $X$. Brenner defined the \\emph{continuous closure} $I^{\\rm cont}$ of an ideal $I$ as $I\\mathcal C(X) \\cap R$. He also introduced an algebraic notion of \\emph{axes closure} $I^{\\rm ax}$ that always contains $I^{\\rm cont}$, and asked whether they coincide. We extend the notion of axes closure to general Noetherian rings, defining $f \\in I^{\\rm ax}$ if its image is in $IS$ for every homomorphism $R \\to S$, ","authors_text":"Melvin Hochster, Neil Epstein","cross_cats":["math.CV"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-06-17T12:18:47Z","title":"Continuous closure, axes closure, and natural closure"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.3462","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":"sha256:0648462c1a7bd90ee64e232832b184abbe1134fb6baf2298ab506a7be07def0e","target":"record","created_at":"2026-05-18T01:37:29Z","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":"fff3eca7111101c349cd8c8cd2444309fe1f111d4c947a85f2ce23d7c200d35c","cross_cats_sorted":["math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-06-17T12:18:47Z","title_canon_sha256":"dc987af5a03212acba6249c7e2771c1a0626a880c4a253b52e544efe4be6564d"},"schema_version":"1.0","source":{"id":"1106.3462","kind":"arxiv","version":2}},"canonical_sha256":"2796728c3318b616d9abceff200190297a63be76ff01accc100222bb9963eb1a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2796728c3318b616d9abceff200190297a63be76ff01accc100222bb9963eb1a","first_computed_at":"2026-05-18T01:37:29.100744Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:37:29.100744Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wUTKaBFC1h0Cq2vsyiTsFuTV5OXUZNKA7SuvEIkQ/IrQbPw7kSTMWKMMdFUX7AdgjNaRxPO2kN1VhFkUjam8Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:37:29.101434Z","signed_message":"canonical_sha256_bytes"},"source_id":"1106.3462","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0648462c1a7bd90ee64e232832b184abbe1134fb6baf2298ab506a7be07def0e","sha256:130ff34bf137f7e1d81013626625f6fea23464a292de51a06a63b7392b110f03"],"state_sha256":"fd41461edb14ec2311fe11b8bb82f08dad0b0c5e62b251e4710719538224c977"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+Rwk99d8QdKs9PGMg+SywcVmB6ldKC1BUfrwTPkCBncCyXrz06hQIb4Ty1DqxJ9MnQqsycVaT9RuGDhcPoBwDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T15:51:07.943782Z","bundle_sha256":"9b56d48af7c4860708d421476c9ccfc0ac02b861f411e45f7c1f2c28df65d82a"}}