{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:7B3AKMR5MSU7SECV2GHFM5OIJU","short_pith_number":"pith:7B3AKMR5","canonical_record":{"source":{"id":"1405.1590","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2014-05-07T12:40:41Z","cross_cats_sorted":["cs.CC","math.LO"],"title_canon_sha256":"a407c50732eae9d632df51d369eb8cd7a334a716eae4426f98e78ebc9623cfc7","abstract_canon_sha256":"b6cfee639d7edd490222c6ebac61e9776bc0dc9ed349b982a220d503d579768e"},"schema_version":"1.0"},"canonical_sha256":"f87605323d64a9f91055d18e5675c84d247b4a3fbefbe5b64a1df874e7b7aa92","source":{"kind":"arxiv","id":"1405.1590","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.1590","created_at":"2026-05-18T02:52:14Z"},{"alias_kind":"arxiv_version","alias_value":"1405.1590v1","created_at":"2026-05-18T02:52:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.1590","created_at":"2026-05-18T02:52:14Z"},{"alias_kind":"pith_short_12","alias_value":"7B3AKMR5MSU7","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"7B3AKMR5MSU7SECV","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"7B3AKMR5","created_at":"2026-05-18T12:28:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:7B3AKMR5MSU7SECV2GHFM5OIJU","target":"record","payload":{"canonical_record":{"source":{"id":"1405.1590","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2014-05-07T12:40:41Z","cross_cats_sorted":["cs.CC","math.LO"],"title_canon_sha256":"a407c50732eae9d632df51d369eb8cd7a334a716eae4426f98e78ebc9623cfc7","abstract_canon_sha256":"b6cfee639d7edd490222c6ebac61e9776bc0dc9ed349b982a220d503d579768e"},"schema_version":"1.0"},"canonical_sha256":"f87605323d64a9f91055d18e5675c84d247b4a3fbefbe5b64a1df874e7b7aa92","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:52:14.838086Z","signature_b64":"2e2+aVaUgX9goDXE5STBta2EheGBii5Y4TWVXo+PV1waaSrLAJS1fzVWe0MLDzHWBkQaDovGIJ+oqa9XIi6jBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f87605323d64a9f91055d18e5675c84d247b4a3fbefbe5b64a1df874e7b7aa92","last_reissued_at":"2026-05-18T02:52:14.837408Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:52:14.837408Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1405.1590","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":"2026-05-18T02:52:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H7KSZImzx5m0b7SNRvMRSihPC+sNyLVQ30YubA81TONaGDQf0uKRQCsOKXHxrWlVnFMUgQkzjuOT77lHjmC8Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T22:59:44.694471Z"},"content_sha256":"79f4732fb74bd83a7a00597da522db43aa965cadc68f900bd355f342a24605a2","schema_version":"1.0","event_id":"sha256:79f4732fb74bd83a7a00597da522db43aa965cadc68f900bd355f342a24605a2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:7B3AKMR5MSU7SECV2GHFM5OIJU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Computability and Complexity over the Product Topology of Real Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","math.LO"],"primary_cat":"cs.LO","authors_text":"Walid Gomaa","submitted_at":"2014-05-07T12:40:41Z","abstract_excerpt":"Kawamura and Cook have developed a framework for studying the computability and complexity theoretic problems over \"large\" topological spaces. This framework has been applied to study the complexity of the differential operator and the complexity of functionals over the space of continuous functions on the unit interval $C[0,1]$. In this paper we apply the ideas of Kawamura and Cook to the product space of the real numbers endowed with the product topology. We show that no computable norm can be defined over such topology. We investigate computability and complexity of total functions over the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.1590","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":"2026-05-18T02:52:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DpwGzGd08nd3frLKyc1tT4dZJ7Xii/tlFW/jtx4eTYjf8G0/VM/vslbdWiDYbnAP+/mbUWnRt4Cdv5plihbkAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T22:59:44.694844Z"},"content_sha256":"8be3dd9b0bc38bee56e7e389f39f2c409d6cb26d48708b3e918b97462877bab8","schema_version":"1.0","event_id":"sha256:8be3dd9b0bc38bee56e7e389f39f2c409d6cb26d48708b3e918b97462877bab8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7B3AKMR5MSU7SECV2GHFM5OIJU/bundle.json","state_url":"https://pith.science/pith/7B3AKMR5MSU7SECV2GHFM5OIJU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7B3AKMR5MSU7SECV2GHFM5OIJU/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":"2026-06-02T22:59:44Z","links":{"resolver":"https://pith.science/pith/7B3AKMR5MSU7SECV2GHFM5OIJU","bundle":"https://pith.science/pith/7B3AKMR5MSU7SECV2GHFM5OIJU/bundle.json","state":"https://pith.science/pith/7B3AKMR5MSU7SECV2GHFM5OIJU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7B3AKMR5MSU7SECV2GHFM5OIJU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:7B3AKMR5MSU7SECV2GHFM5OIJU","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":"b6cfee639d7edd490222c6ebac61e9776bc0dc9ed349b982a220d503d579768e","cross_cats_sorted":["cs.CC","math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2014-05-07T12:40:41Z","title_canon_sha256":"a407c50732eae9d632df51d369eb8cd7a334a716eae4426f98e78ebc9623cfc7"},"schema_version":"1.0","source":{"id":"1405.1590","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.1590","created_at":"2026-05-18T02:52:14Z"},{"alias_kind":"arxiv_version","alias_value":"1405.1590v1","created_at":"2026-05-18T02:52:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.1590","created_at":"2026-05-18T02:52:14Z"},{"alias_kind":"pith_short_12","alias_value":"7B3AKMR5MSU7","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_16","alias_value":"7B3AKMR5MSU7SECV","created_at":"2026-05-18T12:28:16Z"},{"alias_kind":"pith_short_8","alias_value":"7B3AKMR5","created_at":"2026-05-18T12:28:16Z"}],"graph_snapshots":[{"event_id":"sha256:8be3dd9b0bc38bee56e7e389f39f2c409d6cb26d48708b3e918b97462877bab8","target":"graph","created_at":"2026-05-18T02:52:14Z","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":"Kawamura and Cook have developed a framework for studying the computability and complexity theoretic problems over \"large\" topological spaces. This framework has been applied to study the complexity of the differential operator and the complexity of functionals over the space of continuous functions on the unit interval $C[0,1]$. In this paper we apply the ideas of Kawamura and Cook to the product space of the real numbers endowed with the product topology. We show that no computable norm can be defined over such topology. We investigate computability and complexity of total functions over the","authors_text":"Walid Gomaa","cross_cats":["cs.CC","math.LO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2014-05-07T12:40:41Z","title":"Computability and Complexity over the Product Topology of Real Numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.1590","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":"sha256:79f4732fb74bd83a7a00597da522db43aa965cadc68f900bd355f342a24605a2","target":"record","created_at":"2026-05-18T02:52:14Z","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":"b6cfee639d7edd490222c6ebac61e9776bc0dc9ed349b982a220d503d579768e","cross_cats_sorted":["cs.CC","math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2014-05-07T12:40:41Z","title_canon_sha256":"a407c50732eae9d632df51d369eb8cd7a334a716eae4426f98e78ebc9623cfc7"},"schema_version":"1.0","source":{"id":"1405.1590","kind":"arxiv","version":1}},"canonical_sha256":"f87605323d64a9f91055d18e5675c84d247b4a3fbefbe5b64a1df874e7b7aa92","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f87605323d64a9f91055d18e5675c84d247b4a3fbefbe5b64a1df874e7b7aa92","first_computed_at":"2026-05-18T02:52:14.837408Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:52:14.837408Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2e2+aVaUgX9goDXE5STBta2EheGBii5Y4TWVXo+PV1waaSrLAJS1fzVWe0MLDzHWBkQaDovGIJ+oqa9XIi6jBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:52:14.838086Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.1590","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:79f4732fb74bd83a7a00597da522db43aa965cadc68f900bd355f342a24605a2","sha256:8be3dd9b0bc38bee56e7e389f39f2c409d6cb26d48708b3e918b97462877bab8"],"state_sha256":"1a762ed7d1eff201e81fbd8749baff938b60333e1553a0c2763f71c800ff3df1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RtK5m1fA9cfIgBP9MugWakfEkTe2vRCjGIxEKpuThwo9L8P/YIp1eiiwTMja97yvFocwNsyqi97uU7VBEQznDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T22:59:44.696799Z","bundle_sha256":"987d79057956c78f5783baf56f31a29f8eb070494e7cffe6e929dff119eecd08"}}