{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:4TCWXJAT4MPTUM3APUZYO5ZNR3","short_pith_number":"pith:4TCWXJAT","canonical_record":{"source":{"id":"1302.5506","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-02-22T07:55:34Z","cross_cats_sorted":[],"title_canon_sha256":"7492640847d1483eed74ba0705b652ba7477fc070030dcae2741a6119b869a05","abstract_canon_sha256":"09bca76e88de3884252a6fa9ae2220ff20662a23e03b3d5383e1deaba5a457be"},"schema_version":"1.0"},"canonical_sha256":"e4c56ba413e31f3a33607d3387772d8ecbfce30c7605d8175196beddba56cef0","source":{"kind":"arxiv","id":"1302.5506","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.5506","created_at":"2026-05-18T03:32:46Z"},{"alias_kind":"arxiv_version","alias_value":"1302.5506v1","created_at":"2026-05-18T03:32:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.5506","created_at":"2026-05-18T03:32:46Z"},{"alias_kind":"pith_short_12","alias_value":"4TCWXJAT4MPT","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"4TCWXJAT4MPTUM3A","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"4TCWXJAT","created_at":"2026-05-18T12:27:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:4TCWXJAT4MPTUM3APUZYO5ZNR3","target":"record","payload":{"canonical_record":{"source":{"id":"1302.5506","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-02-22T07:55:34Z","cross_cats_sorted":[],"title_canon_sha256":"7492640847d1483eed74ba0705b652ba7477fc070030dcae2741a6119b869a05","abstract_canon_sha256":"09bca76e88de3884252a6fa9ae2220ff20662a23e03b3d5383e1deaba5a457be"},"schema_version":"1.0"},"canonical_sha256":"e4c56ba413e31f3a33607d3387772d8ecbfce30c7605d8175196beddba56cef0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:32:46.557491Z","signature_b64":"hMyxaTEXhMlerJBdTU6zXg3qypa6w6bfMXCWkvdkXqEi65XemOFmdFsYTLmwbwDvvLpwv80stq39Dl+Z+iscAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e4c56ba413e31f3a33607d3387772d8ecbfce30c7605d8175196beddba56cef0","last_reissued_at":"2026-05-18T03:32:46.556702Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:32:46.556702Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1302.5506","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":"2026-05-18T03:32:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TNMwkfDsfoCcQCNlb1wHYRlKtETiEhQLAy0KbQ0f25xCZWhI2w+78L91IgxfZASngm7dusbvVDi1Kv6fC3zuCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T09:40:24.200875Z"},"content_sha256":"00ec3e20b0c9579186332b34b8c0bdb8fdf9180fda261fd79d69e7ba6bbecd7c","schema_version":"1.0","event_id":"sha256:00ec3e20b0c9579186332b34b8c0bdb8fdf9180fda261fd79d69e7ba6bbecd7c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:4TCWXJAT4MPTUM3APUZYO5ZNR3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the endomorphisms of some sheaves of functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Patrice P. Ntumba","submitted_at":"2013-02-22T07:55:34Z","abstract_excerpt":"Given a $C^\\infty$ real manifold $X$ and $\\mathcal{C}^m_X$ its sheaf of $m$-times differentiable real-valued functions, we prove that the sheaf $\\mathcal{D}^{m, r}_X$ of differential operators of order $\\leq m$ with coefficient functions of class $C^r$ can be obtained in terms of the sheaf $\\mathcal{H}om_{\\mathbb{R}_X}(\\mathcal{C}^m_X, \\mathcal{C}^r_X)$ of morphisms of $\\mathcal{C}^m_X$ into $\\mathcal{C}^r_X$. The superscripts $m$ and $r$ are integers."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.5506","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":"2026-05-18T03:32:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hCsUujew+mKZzt9UB5rRsSlYXsdrQo5DUfwfbfygNFJ0DYu52IgtQa+b/vhiSfsz+nD8W719TyG9+WaImUpRDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T09:40:24.201204Z"},"content_sha256":"282010e61eb13aacc872206114a76832f48e6d35e0a9381f3c985e9b2b8e3e3b","schema_version":"1.0","event_id":"sha256:282010e61eb13aacc872206114a76832f48e6d35e0a9381f3c985e9b2b8e3e3b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4TCWXJAT4MPTUM3APUZYO5ZNR3/bundle.json","state_url":"https://pith.science/pith/4TCWXJAT4MPTUM3APUZYO5ZNR3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4TCWXJAT4MPTUM3APUZYO5ZNR3/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":"2026-06-09T09:40:24Z","links":{"resolver":"https://pith.science/pith/4TCWXJAT4MPTUM3APUZYO5ZNR3","bundle":"https://pith.science/pith/4TCWXJAT4MPTUM3APUZYO5ZNR3/bundle.json","state":"https://pith.science/pith/4TCWXJAT4MPTUM3APUZYO5ZNR3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4TCWXJAT4MPTUM3APUZYO5ZNR3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:4TCWXJAT4MPTUM3APUZYO5ZNR3","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":"09bca76e88de3884252a6fa9ae2220ff20662a23e03b3d5383e1deaba5a457be","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-02-22T07:55:34Z","title_canon_sha256":"7492640847d1483eed74ba0705b652ba7477fc070030dcae2741a6119b869a05"},"schema_version":"1.0","source":{"id":"1302.5506","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.5506","created_at":"2026-05-18T03:32:46Z"},{"alias_kind":"arxiv_version","alias_value":"1302.5506v1","created_at":"2026-05-18T03:32:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.5506","created_at":"2026-05-18T03:32:46Z"},{"alias_kind":"pith_short_12","alias_value":"4TCWXJAT4MPT","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_16","alias_value":"4TCWXJAT4MPTUM3A","created_at":"2026-05-18T12:27:34Z"},{"alias_kind":"pith_short_8","alias_value":"4TCWXJAT","created_at":"2026-05-18T12:27:34Z"}],"graph_snapshots":[{"event_id":"sha256:282010e61eb13aacc872206114a76832f48e6d35e0a9381f3c985e9b2b8e3e3b","target":"graph","created_at":"2026-05-18T03:32:46Z","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":"Given a $C^\\infty$ real manifold $X$ and $\\mathcal{C}^m_X$ its sheaf of $m$-times differentiable real-valued functions, we prove that the sheaf $\\mathcal{D}^{m, r}_X$ of differential operators of order $\\leq m$ with coefficient functions of class $C^r$ can be obtained in terms of the sheaf $\\mathcal{H}om_{\\mathbb{R}_X}(\\mathcal{C}^m_X, \\mathcal{C}^r_X)$ of morphisms of $\\mathcal{C}^m_X$ into $\\mathcal{C}^r_X$. The superscripts $m$ and $r$ are integers.","authors_text":"Patrice P. Ntumba","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-02-22T07:55:34Z","title":"On the endomorphisms of some sheaves of functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.5506","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":"sha256:00ec3e20b0c9579186332b34b8c0bdb8fdf9180fda261fd79d69e7ba6bbecd7c","target":"record","created_at":"2026-05-18T03:32:46Z","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":"09bca76e88de3884252a6fa9ae2220ff20662a23e03b3d5383e1deaba5a457be","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-02-22T07:55:34Z","title_canon_sha256":"7492640847d1483eed74ba0705b652ba7477fc070030dcae2741a6119b869a05"},"schema_version":"1.0","source":{"id":"1302.5506","kind":"arxiv","version":1}},"canonical_sha256":"e4c56ba413e31f3a33607d3387772d8ecbfce30c7605d8175196beddba56cef0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e4c56ba413e31f3a33607d3387772d8ecbfce30c7605d8175196beddba56cef0","first_computed_at":"2026-05-18T03:32:46.556702Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:32:46.556702Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hMyxaTEXhMlerJBdTU6zXg3qypa6w6bfMXCWkvdkXqEi65XemOFmdFsYTLmwbwDvvLpwv80stq39Dl+Z+iscAA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:32:46.557491Z","signed_message":"canonical_sha256_bytes"},"source_id":"1302.5506","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:00ec3e20b0c9579186332b34b8c0bdb8fdf9180fda261fd79d69e7ba6bbecd7c","sha256:282010e61eb13aacc872206114a76832f48e6d35e0a9381f3c985e9b2b8e3e3b"],"state_sha256":"ace4dfcc7622655c8b999512940431c0fce730ab2467c1720c27f20186c3148b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rKwStOtVx++H86tmkA1WBP6I/m4ZeolFJkm2pz10cNb5BCpqePapPcBb0wkgOErVy9l98adh1N02ncLc2Z6uBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T09:40:24.202987Z","bundle_sha256":"558b82e797d2f3f5ffc058d81dde6e529c0b8531de11fdfbed4bcca1e934ac91"}}