{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:QR4WGKGMRGZ4XNQOFKYJBNJB6F","short_pith_number":"pith:QR4WGKGM","canonical_record":{"source":{"id":"1104.3543","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-04-18T17:04:43Z","cross_cats_sorted":["math.LO"],"title_canon_sha256":"4bab47b160aa2a69e83b9097ecec7676ff3c5c4d327af299d6be0720040e3fe1","abstract_canon_sha256":"4bba5fc7a75be526f295decd16da7dee48ff94a8cc580b8e10803cdac61182c5"},"schema_version":"1.0"},"canonical_sha256":"84796328cc89b3cbb60e2ab090b521f150610e4b6f0425bc2e39d36833c52289","source":{"kind":"arxiv","id":"1104.3543","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.3543","created_at":"2026-05-18T04:24:04Z"},{"alias_kind":"arxiv_version","alias_value":"1104.3543v1","created_at":"2026-05-18T04:24:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.3543","created_at":"2026-05-18T04:24:04Z"},{"alias_kind":"pith_short_12","alias_value":"QR4WGKGMRGZ4","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"QR4WGKGMRGZ4XNQO","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"QR4WGKGM","created_at":"2026-05-18T12:26:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:QR4WGKGMRGZ4XNQOFKYJBNJB6F","target":"record","payload":{"canonical_record":{"source":{"id":"1104.3543","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-04-18T17:04:43Z","cross_cats_sorted":["math.LO"],"title_canon_sha256":"4bab47b160aa2a69e83b9097ecec7676ff3c5c4d327af299d6be0720040e3fe1","abstract_canon_sha256":"4bba5fc7a75be526f295decd16da7dee48ff94a8cc580b8e10803cdac61182c5"},"schema_version":"1.0"},"canonical_sha256":"84796328cc89b3cbb60e2ab090b521f150610e4b6f0425bc2e39d36833c52289","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:24:04.521465Z","signature_b64":"o5533cVEtDFRxp+381PXuPqBmwelio47hh1Nto3EJMva/JlAe8D34c2Ghg/RgRt6Vv715ov7Er2FjhoG1JGyBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"84796328cc89b3cbb60e2ab090b521f150610e4b6f0425bc2e39d36833c52289","last_reissued_at":"2026-05-18T04:24:04.520845Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:24:04.520845Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1104.3543","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":"2026-05-18T04:24:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n9P363WtrB1c3JDGCP/rJxYVZMrMP75/0HJXAshslOQCRSewAdJM4PTnXGPS8llVyfocBTuoFgNPBB3BhrrqCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T14:39:58.352779Z"},"content_sha256":"4b78231179f8afac6eb40c6386032f1b83e1e1f5e5745f6d757abccad69ce175","schema_version":"1.0","event_id":"sha256:4b78231179f8afac6eb40c6386032f1b83e1e1f5e5745f6d757abccad69ce175"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:QR4WGKGMRGZ4XNQOFKYJBNJB6F","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"$\\alpha$-minimal Banach spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.FA","authors_text":"Christian Rosendal","submitted_at":"2011-04-18T17:04:43Z","abstract_excerpt":"A Banach space with a Schauder basis is said to be $\\alpha$-minimal for some countable ordinal $\\alpha$ if, for any two block subspaces, the Bourgain embeddability index of one into the other is at least $\\alpha$. We prove a dichotomy that characterises when a Banach space has an $\\alpha$-minimal subspace, which contributes to the ongoing project, initiated by W. T. Gowers, of classifying separable Banach spaces by identifying characteristic subspaces."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.3543","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":"2026-05-18T04:24:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gEV58ASFf5LpQAwXfqWEx8qAmZpnVS07eMlWrvmNEjgmDQ7lS/lZR/VnEpxV2D/QwaPsCOaNcnCKhQDK8Ft7Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T14:39:58.353123Z"},"content_sha256":"9b65c23d31d0ee3f55b491dc4ad840cda9d65f2eed7182651640e716d4e5689a","schema_version":"1.0","event_id":"sha256:9b65c23d31d0ee3f55b491dc4ad840cda9d65f2eed7182651640e716d4e5689a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QR4WGKGMRGZ4XNQOFKYJBNJB6F/bundle.json","state_url":"https://pith.science/pith/QR4WGKGMRGZ4XNQOFKYJBNJB6F/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QR4WGKGMRGZ4XNQOFKYJBNJB6F/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":"2026-06-05T14:39:58Z","links":{"resolver":"https://pith.science/pith/QR4WGKGMRGZ4XNQOFKYJBNJB6F","bundle":"https://pith.science/pith/QR4WGKGMRGZ4XNQOFKYJBNJB6F/bundle.json","state":"https://pith.science/pith/QR4WGKGMRGZ4XNQOFKYJBNJB6F/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QR4WGKGMRGZ4XNQOFKYJBNJB6F/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:QR4WGKGMRGZ4XNQOFKYJBNJB6F","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":"4bba5fc7a75be526f295decd16da7dee48ff94a8cc580b8e10803cdac61182c5","cross_cats_sorted":["math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-04-18T17:04:43Z","title_canon_sha256":"4bab47b160aa2a69e83b9097ecec7676ff3c5c4d327af299d6be0720040e3fe1"},"schema_version":"1.0","source":{"id":"1104.3543","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.3543","created_at":"2026-05-18T04:24:04Z"},{"alias_kind":"arxiv_version","alias_value":"1104.3543v1","created_at":"2026-05-18T04:24:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.3543","created_at":"2026-05-18T04:24:04Z"},{"alias_kind":"pith_short_12","alias_value":"QR4WGKGMRGZ4","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"QR4WGKGMRGZ4XNQO","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"QR4WGKGM","created_at":"2026-05-18T12:26:39Z"}],"graph_snapshots":[{"event_id":"sha256:9b65c23d31d0ee3f55b491dc4ad840cda9d65f2eed7182651640e716d4e5689a","target":"graph","created_at":"2026-05-18T04:24:04Z","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":"A Banach space with a Schauder basis is said to be $\\alpha$-minimal for some countable ordinal $\\alpha$ if, for any two block subspaces, the Bourgain embeddability index of one into the other is at least $\\alpha$. We prove a dichotomy that characterises when a Banach space has an $\\alpha$-minimal subspace, which contributes to the ongoing project, initiated by W. T. Gowers, of classifying separable Banach spaces by identifying characteristic subspaces.","authors_text":"Christian Rosendal","cross_cats":["math.LO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-04-18T17:04:43Z","title":"$\\alpha$-minimal Banach spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.3543","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":"sha256:4b78231179f8afac6eb40c6386032f1b83e1e1f5e5745f6d757abccad69ce175","target":"record","created_at":"2026-05-18T04:24:04Z","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":"4bba5fc7a75be526f295decd16da7dee48ff94a8cc580b8e10803cdac61182c5","cross_cats_sorted":["math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-04-18T17:04:43Z","title_canon_sha256":"4bab47b160aa2a69e83b9097ecec7676ff3c5c4d327af299d6be0720040e3fe1"},"schema_version":"1.0","source":{"id":"1104.3543","kind":"arxiv","version":1}},"canonical_sha256":"84796328cc89b3cbb60e2ab090b521f150610e4b6f0425bc2e39d36833c52289","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"84796328cc89b3cbb60e2ab090b521f150610e4b6f0425bc2e39d36833c52289","first_computed_at":"2026-05-18T04:24:04.520845Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:24:04.520845Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"o5533cVEtDFRxp+381PXuPqBmwelio47hh1Nto3EJMva/JlAe8D34c2Ghg/RgRt6Vv715ov7Er2FjhoG1JGyBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:24:04.521465Z","signed_message":"canonical_sha256_bytes"},"source_id":"1104.3543","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4b78231179f8afac6eb40c6386032f1b83e1e1f5e5745f6d757abccad69ce175","sha256:9b65c23d31d0ee3f55b491dc4ad840cda9d65f2eed7182651640e716d4e5689a"],"state_sha256":"31e96efafff99b3643892dfad2339867b28518040280648c18ef1aeac81f7139"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CrPXjyri00x22ydd+U3mlcdRfZncwVbCSBknLbcOl+Zk0KaJyQP+ooYmvCqIVN16cM9ztAPAJHr3iIAX7cu2Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T14:39:58.355212Z","bundle_sha256":"b4d6dd8faae4e127fc877687824778dc4d273772cc3989d61ccbd4471999c91b"}}