{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:OND5YFGA3IZFAE2N6JMWMMWIVR","short_pith_number":"pith:OND5YFGA","canonical_record":{"source":{"id":"1807.04012","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-07-11T09:05:52Z","cross_cats_sorted":[],"title_canon_sha256":"ceab03515230f1d4cf0c7c9afa9f5906f644a19023cc58db636b047509ed9f98","abstract_canon_sha256":"c3bfbb222ea684caee701436abbe5f73233049d223b2989d973fde5ee405a251"},"schema_version":"1.0"},"canonical_sha256":"7347dc14c0da3250134df2596632c8ac56ee1b5de863d1b9d416b0f8c7cfe3ce","source":{"kind":"arxiv","id":"1807.04012","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.04012","created_at":"2026-05-18T00:10:58Z"},{"alias_kind":"arxiv_version","alias_value":"1807.04012v1","created_at":"2026-05-18T00:10:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.04012","created_at":"2026-05-18T00:10:58Z"},{"alias_kind":"pith_short_12","alias_value":"OND5YFGA3IZF","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_16","alias_value":"OND5YFGA3IZFAE2N","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_8","alias_value":"OND5YFGA","created_at":"2026-05-18T12:32:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:OND5YFGA3IZFAE2N6JMWMMWIVR","target":"record","payload":{"canonical_record":{"source":{"id":"1807.04012","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-07-11T09:05:52Z","cross_cats_sorted":[],"title_canon_sha256":"ceab03515230f1d4cf0c7c9afa9f5906f644a19023cc58db636b047509ed9f98","abstract_canon_sha256":"c3bfbb222ea684caee701436abbe5f73233049d223b2989d973fde5ee405a251"},"schema_version":"1.0"},"canonical_sha256":"7347dc14c0da3250134df2596632c8ac56ee1b5de863d1b9d416b0f8c7cfe3ce","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:58.292176Z","signature_b64":"PYq9FdExWKfP1hZIxqqnwb+wcImW/qaOqaknTR/nrE5ZjpyhRspXijiIS+UR9zQq02/y0gAWPMnXa1c3w7QyDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7347dc14c0da3250134df2596632c8ac56ee1b5de863d1b9d416b0f8c7cfe3ce","last_reissued_at":"2026-05-18T00:10:58.291383Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:58.291383Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1807.04012","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":"2026-05-18T00:10:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pvwsgPNoglcpcVFeOeJOKRozfRz6Lq1dih2sIMi9LIFvfzkDPzfmQMucC2Hx0wXcCn39wTbqJ7r502mAUhl8Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T21:16:31.396334Z"},"content_sha256":"319134cd6ed242bd7f5b71d97228959e050a2199d7dc8d9da66ef95250f55e14","schema_version":"1.0","event_id":"sha256:319134cd6ed242bd7f5b71d97228959e050a2199d7dc8d9da66ef95250f55e14"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:OND5YFGA3IZFAE2N6JMWMMWIVR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the existence of vectors dual to a set of linear functionals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Shibo Liu","submitted_at":"2018-07-11T09:05:52Z","abstract_excerpt":"We give a simple proof of a crucial lemma that is established in [1, Lemma 2.1] by induction, and plays important roles in that paper and [2]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.04012","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":"2026-05-18T00:10:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DCjWeOz3wtCHW60fyjPYddfWJEJ1/0FJUJx1XAAFpcXKdurH3YS4L/O+hlfmdtRPBcteQLfecsvVeG+AyabtCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T21:16:31.396948Z"},"content_sha256":"dee5533cdc95388e129da8beb614e10a405dd4f5fc2ac7e33b9083f634047aed","schema_version":"1.0","event_id":"sha256:dee5533cdc95388e129da8beb614e10a405dd4f5fc2ac7e33b9083f634047aed"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OND5YFGA3IZFAE2N6JMWMMWIVR/bundle.json","state_url":"https://pith.science/pith/OND5YFGA3IZFAE2N6JMWMMWIVR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OND5YFGA3IZFAE2N6JMWMMWIVR/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":"2026-06-06T21:16:31Z","links":{"resolver":"https://pith.science/pith/OND5YFGA3IZFAE2N6JMWMMWIVR","bundle":"https://pith.science/pith/OND5YFGA3IZFAE2N6JMWMMWIVR/bundle.json","state":"https://pith.science/pith/OND5YFGA3IZFAE2N6JMWMMWIVR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OND5YFGA3IZFAE2N6JMWMMWIVR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:OND5YFGA3IZFAE2N6JMWMMWIVR","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":"c3bfbb222ea684caee701436abbe5f73233049d223b2989d973fde5ee405a251","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-07-11T09:05:52Z","title_canon_sha256":"ceab03515230f1d4cf0c7c9afa9f5906f644a19023cc58db636b047509ed9f98"},"schema_version":"1.0","source":{"id":"1807.04012","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.04012","created_at":"2026-05-18T00:10:58Z"},{"alias_kind":"arxiv_version","alias_value":"1807.04012v1","created_at":"2026-05-18T00:10:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.04012","created_at":"2026-05-18T00:10:58Z"},{"alias_kind":"pith_short_12","alias_value":"OND5YFGA3IZF","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_16","alias_value":"OND5YFGA3IZFAE2N","created_at":"2026-05-18T12:32:43Z"},{"alias_kind":"pith_short_8","alias_value":"OND5YFGA","created_at":"2026-05-18T12:32:43Z"}],"graph_snapshots":[{"event_id":"sha256:dee5533cdc95388e129da8beb614e10a405dd4f5fc2ac7e33b9083f634047aed","target":"graph","created_at":"2026-05-18T00:10:58Z","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 give a simple proof of a crucial lemma that is established in [1, Lemma 2.1] by induction, and plays important roles in that paper and [2].","authors_text":"Shibo Liu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-07-11T09:05:52Z","title":"On the existence of vectors dual to a set of linear functionals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.04012","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":"sha256:319134cd6ed242bd7f5b71d97228959e050a2199d7dc8d9da66ef95250f55e14","target":"record","created_at":"2026-05-18T00:10:58Z","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":"c3bfbb222ea684caee701436abbe5f73233049d223b2989d973fde5ee405a251","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-07-11T09:05:52Z","title_canon_sha256":"ceab03515230f1d4cf0c7c9afa9f5906f644a19023cc58db636b047509ed9f98"},"schema_version":"1.0","source":{"id":"1807.04012","kind":"arxiv","version":1}},"canonical_sha256":"7347dc14c0da3250134df2596632c8ac56ee1b5de863d1b9d416b0f8c7cfe3ce","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7347dc14c0da3250134df2596632c8ac56ee1b5de863d1b9d416b0f8c7cfe3ce","first_computed_at":"2026-05-18T00:10:58.291383Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:10:58.291383Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PYq9FdExWKfP1hZIxqqnwb+wcImW/qaOqaknTR/nrE5ZjpyhRspXijiIS+UR9zQq02/y0gAWPMnXa1c3w7QyDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:10:58.292176Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.04012","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:319134cd6ed242bd7f5b71d97228959e050a2199d7dc8d9da66ef95250f55e14","sha256:dee5533cdc95388e129da8beb614e10a405dd4f5fc2ac7e33b9083f634047aed"],"state_sha256":"1a0dd7f8b8554ca934d3ca15284cd7df2ce73815dcbc6222ea62fb8db2879eb8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5SgD/8vMGp7gUYvzuGi6wbpNjjJS+5RrXvr2uJNgEiqSJgda0Kv33LgjtoeTGurbts1GmlteUJGbDEnTNibPBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T21:16:31.401194Z","bundle_sha256":"4d09de6dc050a011223eab92dd5da57e5b1b256d382e1d56da6e7566050daa6c"}}