{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:AHTEPHYRMTQGYPP3D5VANTAQPK","short_pith_number":"pith:AHTEPHYR","canonical_record":{"source":{"id":"1603.06236","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2016-03-20T16:34:03Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"5272dd0979aac5278deacb240225ae05d845cd05deadf2aed064871ca9f24abd","abstract_canon_sha256":"eaf972f8643795337fb38daeb0bb7c7b1cf29bf45eb028c4e72e8da7acdb7dd5"},"schema_version":"1.0"},"canonical_sha256":"01e6479f1164e06c3dfb1f6a06cc107a9a8200047081643c39c7d2fc798b882a","source":{"kind":"arxiv","id":"1603.06236","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.06236","created_at":"2026-05-18T01:03:50Z"},{"alias_kind":"arxiv_version","alias_value":"1603.06236v2","created_at":"2026-05-18T01:03:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.06236","created_at":"2026-05-18T01:03:50Z"},{"alias_kind":"pith_short_12","alias_value":"AHTEPHYRMTQG","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"AHTEPHYRMTQGYPP3","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"AHTEPHYR","created_at":"2026-05-18T12:30:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:AHTEPHYRMTQGYPP3D5VANTAQPK","target":"record","payload":{"canonical_record":{"source":{"id":"1603.06236","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2016-03-20T16:34:03Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"5272dd0979aac5278deacb240225ae05d845cd05deadf2aed064871ca9f24abd","abstract_canon_sha256":"eaf972f8643795337fb38daeb0bb7c7b1cf29bf45eb028c4e72e8da7acdb7dd5"},"schema_version":"1.0"},"canonical_sha256":"01e6479f1164e06c3dfb1f6a06cc107a9a8200047081643c39c7d2fc798b882a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:50.293363Z","signature_b64":"qVo3f2MM80ub+mvMJG6EbFUrZsFtJAgFFED+9/VvQ1df+TA+QT98QhVjeKeOwZQYnqiZkprs0d5WjoXfoykpDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"01e6479f1164e06c3dfb1f6a06cc107a9a8200047081643c39c7d2fc798b882a","last_reissued_at":"2026-05-18T01:03:50.292747Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:50.292747Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1603.06236","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":"2026-05-18T01:03:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vYhx5wOdzY54DQIF7tW22pOBYGJUexxf9vofcqCwq3a+cEGlXzyZdI9GMchbUgsRIkwAT0c+LT4JCnsNl4k0BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T07:05:25.100052Z"},"content_sha256":"5b23230e507f24a18c19d01bd43282b1b170219b461610fea432da10e4aa2fc1","schema_version":"1.0","event_id":"sha256:5b23230e507f24a18c19d01bd43282b1b170219b461610fea432da10e4aa2fc1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:AHTEPHYRMTQGYPP3D5VANTAQPK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the star-height of factor counting languages and their relationship to Rees zero-matrix semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"cs.FL","authors_text":"Nik Ruskuc, Tom Bourne","submitted_at":"2016-03-20T16:34:03Z","abstract_excerpt":"Given a word $w$ over a finite alphabet, we consider, in three special cases, the generalised star-height of the languages in which $w$ occurs as a contiguous subword (factor) an exact number of times and of the languages in which $w$ occurs as a contiguous subword modulo a fixed number, and prove that in each case it is at most one. We use these combinatorial results to show that any language recognised by a Rees (zero-)matrix semigroup over an abelian group is of generalised star-height at most one."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06236","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":"2026-05-18T01:03:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"G8l2TOKE4/Z/cOxvBYJ59gYiVFRS39dFuBQrgFOAVY1mDlNjfKLpKRilWIZ76V93Abrc+jnvmI/qhcr/i+gAAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T07:05:25.100410Z"},"content_sha256":"b7af49918cadc972dc3098cd0d73b87475eb919ce22986d02452d160bc4bcba0","schema_version":"1.0","event_id":"sha256:b7af49918cadc972dc3098cd0d73b87475eb919ce22986d02452d160bc4bcba0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AHTEPHYRMTQGYPP3D5VANTAQPK/bundle.json","state_url":"https://pith.science/pith/AHTEPHYRMTQGYPP3D5VANTAQPK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AHTEPHYRMTQGYPP3D5VANTAQPK/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":"2026-06-03T07:05:25Z","links":{"resolver":"https://pith.science/pith/AHTEPHYRMTQGYPP3D5VANTAQPK","bundle":"https://pith.science/pith/AHTEPHYRMTQGYPP3D5VANTAQPK/bundle.json","state":"https://pith.science/pith/AHTEPHYRMTQGYPP3D5VANTAQPK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AHTEPHYRMTQGYPP3D5VANTAQPK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:AHTEPHYRMTQGYPP3D5VANTAQPK","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":"eaf972f8643795337fb38daeb0bb7c7b1cf29bf45eb028c4e72e8da7acdb7dd5","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2016-03-20T16:34:03Z","title_canon_sha256":"5272dd0979aac5278deacb240225ae05d845cd05deadf2aed064871ca9f24abd"},"schema_version":"1.0","source":{"id":"1603.06236","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.06236","created_at":"2026-05-18T01:03:50Z"},{"alias_kind":"arxiv_version","alias_value":"1603.06236v2","created_at":"2026-05-18T01:03:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.06236","created_at":"2026-05-18T01:03:50Z"},{"alias_kind":"pith_short_12","alias_value":"AHTEPHYRMTQG","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_16","alias_value":"AHTEPHYRMTQGYPP3","created_at":"2026-05-18T12:30:07Z"},{"alias_kind":"pith_short_8","alias_value":"AHTEPHYR","created_at":"2026-05-18T12:30:07Z"}],"graph_snapshots":[{"event_id":"sha256:b7af49918cadc972dc3098cd0d73b87475eb919ce22986d02452d160bc4bcba0","target":"graph","created_at":"2026-05-18T01:03:50Z","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":"Given a word $w$ over a finite alphabet, we consider, in three special cases, the generalised star-height of the languages in which $w$ occurs as a contiguous subword (factor) an exact number of times and of the languages in which $w$ occurs as a contiguous subword modulo a fixed number, and prove that in each case it is at most one. We use these combinatorial results to show that any language recognised by a Rees (zero-)matrix semigroup over an abelian group is of generalised star-height at most one.","authors_text":"Nik Ruskuc, Tom Bourne","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2016-03-20T16:34:03Z","title":"On the star-height of factor counting languages and their relationship to Rees zero-matrix semigroups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06236","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":"sha256:5b23230e507f24a18c19d01bd43282b1b170219b461610fea432da10e4aa2fc1","target":"record","created_at":"2026-05-18T01:03:50Z","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":"eaf972f8643795337fb38daeb0bb7c7b1cf29bf45eb028c4e72e8da7acdb7dd5","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.FL","submitted_at":"2016-03-20T16:34:03Z","title_canon_sha256":"5272dd0979aac5278deacb240225ae05d845cd05deadf2aed064871ca9f24abd"},"schema_version":"1.0","source":{"id":"1603.06236","kind":"arxiv","version":2}},"canonical_sha256":"01e6479f1164e06c3dfb1f6a06cc107a9a8200047081643c39c7d2fc798b882a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"01e6479f1164e06c3dfb1f6a06cc107a9a8200047081643c39c7d2fc798b882a","first_computed_at":"2026-05-18T01:03:50.292747Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:03:50.292747Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qVo3f2MM80ub+mvMJG6EbFUrZsFtJAgFFED+9/VvQ1df+TA+QT98QhVjeKeOwZQYnqiZkprs0d5WjoXfoykpDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:03:50.293363Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.06236","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5b23230e507f24a18c19d01bd43282b1b170219b461610fea432da10e4aa2fc1","sha256:b7af49918cadc972dc3098cd0d73b87475eb919ce22986d02452d160bc4bcba0"],"state_sha256":"71be078f69c932401221ffe7e99c3807095264c0532fddba8cb4da710ff77a52"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"058dWbkP0teikIICS95WzdgfPUoUUX2X9swTV2gjptSUhSxJoy1FblSevG6dX6t/4nEQe7dNkiHm1E56DsF2BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T07:05:25.102357Z","bundle_sha256":"e23ca751a1db61216ce1582a12d55414b098e7b413d6ffec922ca257951cda83"}}