{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:MSUFU45EUSOH3CBYBILGRNBLK4","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":"d952f23da0dab47216ae6ddce243363e1fc49c4d4e2e55e9be190f5fd250268d","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"cs.CL","submitted_at":"2021-04-19T05:45:44Z","title_canon_sha256":"6a4a84bd1c033071d8c541aec06f6b8387f65d41e755c13034082cb4926bf3f6"},"schema_version":"1.0","source":{"id":"2104.09063","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2104.09063","created_at":"2026-07-05T02:32:58Z"},{"alias_kind":"arxiv_version","alias_value":"2104.09063v1","created_at":"2026-07-05T02:32:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2104.09063","created_at":"2026-07-05T02:32:58Z"},{"alias_kind":"pith_short_12","alias_value":"MSUFU45EUSOH","created_at":"2026-07-05T02:32:58Z"},{"alias_kind":"pith_short_16","alias_value":"MSUFU45EUSOH3CBY","created_at":"2026-07-05T02:32:58Z"},{"alias_kind":"pith_short_8","alias_value":"MSUFU45E","created_at":"2026-07-05T02:32:58Z"}],"graph_snapshots":[{"event_id":"sha256:f3f1c4375da718644873bd334944fb559d74e074f16ec2d240ad81260a1c60ea","target":"graph","created_at":"2026-07-05T02:32:58Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2104.09063/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Scattered factor (circular) universality was firstly introduced by Barker et al. in 2020. A word $w$ is called $k$-universal for some natural number $k$, if every word of length $k$ of $w$'s alphabet occurs as a scattered factor in $w$; it is called circular $k$-universal if a conjugate of $w$ is $k$-universal. Here, a word $u=u_1\\cdots u_n$ is called a scattered factor of $w$ if $u$ is obtained from $w$ by deleting parts of $w$, i.e. there exists (possibly empty) words $v_1,\\dots,v_{n+1}$ with $w=v_1u_1v_2\\cdots v_nu_nv_{n+1}$. In this work, we prove two problems, left open in the aforementio","authors_text":"Dirk Nowotka, Pamela Fleischmann, Sebastian Bernhard Germann","cross_cats":["math.CO"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"cs.CL","submitted_at":"2021-04-19T05:45:44Z","title":"Scattered Factor Universality -- The Power of the Remainder"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2104.09063","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":"sha256:1266974533e446e6b481e8593e703847f526aa853f84385133c0e0ebbe315b1b","target":"record","created_at":"2026-07-05T02:32:58Z","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":"d952f23da0dab47216ae6ddce243363e1fc49c4d4e2e55e9be190f5fd250268d","cross_cats_sorted":["math.CO"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"cs.CL","submitted_at":"2021-04-19T05:45:44Z","title_canon_sha256":"6a4a84bd1c033071d8c541aec06f6b8387f65d41e755c13034082cb4926bf3f6"},"schema_version":"1.0","source":{"id":"2104.09063","kind":"arxiv","version":1}},"canonical_sha256":"64a85a73a4a49c7d88380a1668b42b570a6660399f65d2e258ae3b1d3259070f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"64a85a73a4a49c7d88380a1668b42b570a6660399f65d2e258ae3b1d3259070f","first_computed_at":"2026-07-05T02:32:58.613988Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:32:58.613988Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"74faqD27lWpApgnho/EhrYVYZMHufyzO6wls7Qu9smo9dRGCXQE/e8Rau2H7jwsZRX8640RWmnUbU7EzLI+MDw==","signature_status":"signed_v1","signed_at":"2026-07-05T02:32:58.614639Z","signed_message":"canonical_sha256_bytes"},"source_id":"2104.09063","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1266974533e446e6b481e8593e703847f526aa853f84385133c0e0ebbe315b1b","sha256:f3f1c4375da718644873bd334944fb559d74e074f16ec2d240ad81260a1c60ea"],"state_sha256":"402a629dbfacf9676c0cab3c02fb484ce331bbcd086d3325b183367f64e6be40"}