{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:NLUEHMHFZRXJKKHGURY462UZZW","short_pith_number":"pith:NLUEHMHF","canonical_record":{"source":{"id":"1607.06135","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-07-20T21:48:17Z","cross_cats_sorted":["math.LO"],"title_canon_sha256":"6a13ee34120b6a2e5a2eb58c6f3b742d5a2a59edff242d212c85a0f6ff92b1b0","abstract_canon_sha256":"b302f71d1b1dd2fbe5f182dca6339969eea736d5f0877d6d083ba92bb92cf665"},"schema_version":"1.0"},"canonical_sha256":"6ae843b0e5cc6e9528e6a471cf6a99cda5becf8e5aa037efa3df6eb75b7ba49c","source":{"kind":"arxiv","id":"1607.06135","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.06135","created_at":"2026-05-18T00:48:01Z"},{"alias_kind":"arxiv_version","alias_value":"1607.06135v2","created_at":"2026-05-18T00:48:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.06135","created_at":"2026-05-18T00:48:01Z"},{"alias_kind":"pith_short_12","alias_value":"NLUEHMHFZRXJ","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"NLUEHMHFZRXJKKHG","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"NLUEHMHF","created_at":"2026-05-18T12:30:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:NLUEHMHFZRXJKKHGURY462UZZW","target":"record","payload":{"canonical_record":{"source":{"id":"1607.06135","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-07-20T21:48:17Z","cross_cats_sorted":["math.LO"],"title_canon_sha256":"6a13ee34120b6a2e5a2eb58c6f3b742d5a2a59edff242d212c85a0f6ff92b1b0","abstract_canon_sha256":"b302f71d1b1dd2fbe5f182dca6339969eea736d5f0877d6d083ba92bb92cf665"},"schema_version":"1.0"},"canonical_sha256":"6ae843b0e5cc6e9528e6a471cf6a99cda5becf8e5aa037efa3df6eb75b7ba49c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:48:01.131989Z","signature_b64":"BlESsUwj13HDa3eWIz9FjtPaFNG9YoYNtU0FNhTz+ykW57LnNCq05N3uEA+WTnmybUXCOZPlVQuI4Usq0GHwBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6ae843b0e5cc6e9528e6a471cf6a99cda5becf8e5aa037efa3df6eb75b7ba49c","last_reissued_at":"2026-05-18T00:48:01.131507Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:48:01.131507Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1607.06135","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:48:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4HBM9GAbFyogo2/2pidbVmFRUwxauPvFGiM7AHmoRrAVw/buCj/4OMWzDiBUfC0cwbVjRTRyCNOE9x3TOeA2Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T17:44:52.490246Z"},"content_sha256":"f7481843ab7113404ceb27763375d48506b90e25382053d3242b08ce3b86d482","schema_version":"1.0","event_id":"sha256:f7481843ab7113404ceb27763375d48506b90e25382053d3242b08ce3b86d482"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:NLUEHMHFZRXJKKHGURY462UZZW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Homogeneous families on trees and subsymmetric basic sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.FA","authors_text":"Christina Brech, Jordi Lopez-Abad, Stevo Todorcevic","submitted_at":"2016-07-20T21:48:17Z","abstract_excerpt":"We study density requirements on a given Banach space that guarantee the existence of subsymmetric basic sequences by extending Tsirelson's well-known space to larger index sets. We prove that for every cardinal $\\kappa$ smaller than the first Mahlo cardinal there is a reflexive Banach space of density $\\kappa$ without subsymmetric basic sequences. As for Tsirelson's space, our construction is based on the existence of a rich collection of homogeneous families on large index sets for which one can estimate the complexity on any given infinite set. This is used to describe detailedly the asympt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06135","kind":"arxiv","version":2},"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":"2026-05-18T00:48:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+FYKb+X74nsGG5bY86itcM5Y/FUj6Sf5Ic+41g+tug7+Zr28/5QwcWH9LOj1ss/gnOdx/Vb2sHLCxsUo0rHIDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T17:44:52.490620Z"},"content_sha256":"60464db79f884ee5e84c6c8e159e08c3a82bf35d8b594bbeea5b93d9b680bc0c","schema_version":"1.0","event_id":"sha256:60464db79f884ee5e84c6c8e159e08c3a82bf35d8b594bbeea5b93d9b680bc0c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NLUEHMHFZRXJKKHGURY462UZZW/bundle.json","state_url":"https://pith.science/pith/NLUEHMHFZRXJKKHGURY462UZZW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NLUEHMHFZRXJKKHGURY462UZZW/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":"2026-06-22T17:44:52Z","links":{"resolver":"https://pith.science/pith/NLUEHMHFZRXJKKHGURY462UZZW","bundle":"https://pith.science/pith/NLUEHMHFZRXJKKHGURY462UZZW/bundle.json","state":"https://pith.science/pith/NLUEHMHFZRXJKKHGURY462UZZW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NLUEHMHFZRXJKKHGURY462UZZW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:NLUEHMHFZRXJKKHGURY462UZZW","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":"b302f71d1b1dd2fbe5f182dca6339969eea736d5f0877d6d083ba92bb92cf665","cross_cats_sorted":["math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-07-20T21:48:17Z","title_canon_sha256":"6a13ee34120b6a2e5a2eb58c6f3b742d5a2a59edff242d212c85a0f6ff92b1b0"},"schema_version":"1.0","source":{"id":"1607.06135","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.06135","created_at":"2026-05-18T00:48:01Z"},{"alias_kind":"arxiv_version","alias_value":"1607.06135v2","created_at":"2026-05-18T00:48:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.06135","created_at":"2026-05-18T00:48:01Z"},{"alias_kind":"pith_short_12","alias_value":"NLUEHMHFZRXJ","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"NLUEHMHFZRXJKKHG","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"NLUEHMHF","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:60464db79f884ee5e84c6c8e159e08c3a82bf35d8b594bbeea5b93d9b680bc0c","target":"graph","created_at":"2026-05-18T00:48:01Z","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 study density requirements on a given Banach space that guarantee the existence of subsymmetric basic sequences by extending Tsirelson's well-known space to larger index sets. We prove that for every cardinal $\\kappa$ smaller than the first Mahlo cardinal there is a reflexive Banach space of density $\\kappa$ without subsymmetric basic sequences. As for Tsirelson's space, our construction is based on the existence of a rich collection of homogeneous families on large index sets for which one can estimate the complexity on any given infinite set. This is used to describe detailedly the asympt","authors_text":"Christina Brech, Jordi Lopez-Abad, Stevo Todorcevic","cross_cats":["math.LO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-07-20T21:48:17Z","title":"Homogeneous families on trees and subsymmetric basic sequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06135","kind":"arxiv","version":2},"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:f7481843ab7113404ceb27763375d48506b90e25382053d3242b08ce3b86d482","target":"record","created_at":"2026-05-18T00:48:01Z","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":"b302f71d1b1dd2fbe5f182dca6339969eea736d5f0877d6d083ba92bb92cf665","cross_cats_sorted":["math.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-07-20T21:48:17Z","title_canon_sha256":"6a13ee34120b6a2e5a2eb58c6f3b742d5a2a59edff242d212c85a0f6ff92b1b0"},"schema_version":"1.0","source":{"id":"1607.06135","kind":"arxiv","version":2}},"canonical_sha256":"6ae843b0e5cc6e9528e6a471cf6a99cda5becf8e5aa037efa3df6eb75b7ba49c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6ae843b0e5cc6e9528e6a471cf6a99cda5becf8e5aa037efa3df6eb75b7ba49c","first_computed_at":"2026-05-18T00:48:01.131507Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:48:01.131507Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BlESsUwj13HDa3eWIz9FjtPaFNG9YoYNtU0FNhTz+ykW57LnNCq05N3uEA+WTnmybUXCOZPlVQuI4Usq0GHwBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:48:01.131989Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.06135","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f7481843ab7113404ceb27763375d48506b90e25382053d3242b08ce3b86d482","sha256:60464db79f884ee5e84c6c8e159e08c3a82bf35d8b594bbeea5b93d9b680bc0c"],"state_sha256":"06dcd8621b80f08236d2b3722bf51b237ad925300150d7277e11f0d328850e4b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QjypNxAnrBMkQfTgOFsOHAHmns346/FpM4lKIPMGd96+S9Snt5ju1bSkZheDRxY+A9VOBpYDMfWLMB7r1f7yBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T17:44:52.492556Z","bundle_sha256":"59d84480fbf0c2e684ab217f36520fe32a6e0b32f68944884eaae4cf691bc504"}}