{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:AB55MXGJ3CUBLR6B5E3PMV2JJD","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":"198dcc2e70dc5840f165658f01ccc12994c6ad9ac4b1efea9f216ebdea4fe6ff","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-04-29T17:01:21Z","title_canon_sha256":"25d7fcf8287b7be79c417c0748a9563722842763ad7a535a6ddf74241581f179"},"schema_version":"1.0","source":{"id":"1004.5341","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.5341","created_at":"2026-05-18T02:24:14Z"},{"alias_kind":"arxiv_version","alias_value":"1004.5341v1","created_at":"2026-05-18T02:24:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.5341","created_at":"2026-05-18T02:24:14Z"},{"alias_kind":"pith_short_12","alias_value":"AB55MXGJ3CUB","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"AB55MXGJ3CUBLR6B","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"AB55MXGJ","created_at":"2026-05-18T12:26:05Z"}],"graph_snapshots":[{"event_id":"sha256:d46c6fd07987141727d84a8c2e5b316a044333e96e6bebc574499e256b9c6f66","target":"graph","created_at":"2026-05-18T02:24:14Z","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":"Understanding the complemented subspaces of $L_p$ has been an interesting topic of research in Banach space theory since 1960. 1999, Alspach proposed a systematic approach to classifying the subspaces of $L_p$ by introducing a norm given by partitions and weights. This paper shows that with Alspach Norm we are able to classify some complemented subspaces of $L_p, 2<p<\\infty$.","authors_text":"Isaac DeFrain, Mitch Phillipson, Simei Tong","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-04-29T17:01:21Z","title":"Classifying Complemented Subspaces of $L_p$ with Alspach Norm"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.5341","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":"sha256:17f0f6b12a1b58e5e2cdb8786b0364b8432ad2f633538bfe89ca1afa036a5715","target":"record","created_at":"2026-05-18T02:24:14Z","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":"198dcc2e70dc5840f165658f01ccc12994c6ad9ac4b1efea9f216ebdea4fe6ff","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2010-04-29T17:01:21Z","title_canon_sha256":"25d7fcf8287b7be79c417c0748a9563722842763ad7a535a6ddf74241581f179"},"schema_version":"1.0","source":{"id":"1004.5341","kind":"arxiv","version":1}},"canonical_sha256":"007bd65cc9d8a815c7c1e936f6574948d270768900d883a49a38270b2f73089b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"007bd65cc9d8a815c7c1e936f6574948d270768900d883a49a38270b2f73089b","first_computed_at":"2026-05-18T02:24:14.936727Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:24:14.936727Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iJK40KpiSv7HkDxNgl+1wd37GnfN/elqYtnN4u8mGBYUJqjKlAF99QbWfJyVoSI1RSFKnqHsTY5galg0l0QzDw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:24:14.937273Z","signed_message":"canonical_sha256_bytes"},"source_id":"1004.5341","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:17f0f6b12a1b58e5e2cdb8786b0364b8432ad2f633538bfe89ca1afa036a5715","sha256:d46c6fd07987141727d84a8c2e5b316a044333e96e6bebc574499e256b9c6f66"],"state_sha256":"699a8a18a3ce533a70c3f09f2802596c348bfe304c8413bbc1fe54a7062fb70e"}