{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:PBURDCU22J5KNSP57NYNL5NVWK","short_pith_number":"pith:PBURDCU2","canonical_record":{"source":{"id":"1307.5027","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-18T17:54:24Z","cross_cats_sorted":[],"title_canon_sha256":"c39bfcdec81cafa68553e8259eee58821c423057b2e80da5dd8ba4ec45a5e071","abstract_canon_sha256":"44e485766ed2406f9219ccf43c8c475ce0c7ba6e55dfc48d576ac6d401f6b397"},"schema_version":"1.0"},"canonical_sha256":"7869118a9ad27aa6c9fdfb70d5f5b5b2a1e38424a58d19eef8ca4e2e05d56e4c","source":{"kind":"arxiv","id":"1307.5027","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.5027","created_at":"2026-05-18T03:18:06Z"},{"alias_kind":"arxiv_version","alias_value":"1307.5027v1","created_at":"2026-05-18T03:18:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.5027","created_at":"2026-05-18T03:18:06Z"},{"alias_kind":"pith_short_12","alias_value":"PBURDCU22J5K","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"PBURDCU22J5KNSP5","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"PBURDCU2","created_at":"2026-05-18T12:27:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:PBURDCU22J5KNSP57NYNL5NVWK","target":"record","payload":{"canonical_record":{"source":{"id":"1307.5027","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-18T17:54:24Z","cross_cats_sorted":[],"title_canon_sha256":"c39bfcdec81cafa68553e8259eee58821c423057b2e80da5dd8ba4ec45a5e071","abstract_canon_sha256":"44e485766ed2406f9219ccf43c8c475ce0c7ba6e55dfc48d576ac6d401f6b397"},"schema_version":"1.0"},"canonical_sha256":"7869118a9ad27aa6c9fdfb70d5f5b5b2a1e38424a58d19eef8ca4e2e05d56e4c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:06.894881Z","signature_b64":"3umhZ3by7GFbJVW1gK5MlZIXQJKQNTNqeDd5pNorR/u8+q2E8aAvS1wHsI+hyT3W5l+FsHXxMNzfha3FazD6DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7869118a9ad27aa6c9fdfb70d5f5b5b2a1e38424a58d19eef8ca4e2e05d56e4c","last_reissued_at":"2026-05-18T03:18:06.894180Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:06.894180Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1307.5027","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":"2026-05-18T03:18:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bh3uKIeXTO8gHclTW7G2xAT5LwFgHV5NauiAypJnmYEuMXvWijgbDMXr4JeeOkLlzJWiODo8MN5bc17lqOflBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T20:18:12.999933Z"},"content_sha256":"cc01d8b0fe2071c4643fabee70f7fa35c0c0f97ce6d9b68f48c42177d255be03","schema_version":"1.0","event_id":"sha256:cc01d8b0fe2071c4643fabee70f7fa35c0c0f97ce6d9b68f48c42177d255be03"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:PBURDCU22J5KNSP57NYNL5NVWK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The indecomposable tournaments $T$ with $\\mid W_{5}(T) \\mid = \\mid T \\mid -2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Houmem Belkhechine, Imed Boudabbous, Kaouthar Hzami","submitted_at":"2013-07-18T17:54:24Z","abstract_excerpt":"We consider a tournament $T=(V, A)$. For $X\\subseteq V$, the subtournament of $T$ induced by $X$ is $T[X] = (X, A \\cap (X \\times X))$. An interval of $T$ is a subset $X$ of $V$ such that for $a, b\\in X$ and $ x\\in V\\setminus X$, $(a,x)\\in A$ if and only if $(b,x)\\in A$. The trivial intervals of $T$ are $\\emptyset$, $\\{x\\}(x\\in V)$ and $V$. A tournament is indecomposable if all its intervals are trivial. For $n\\geq 2$, $W_{2n+1}$ denotes the unique indecomposable tournament defined on $\\{0,\\dots,2n\\}$ such that $W_{2n+1}[\\{0,\\dots,2n-1\\}]$ is the usual total order. Given an indecomposable tourn"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.5027","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":"2026-05-18T03:18:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pQwFkcafVFjH9ruWkRlt9yWEILXlKza2q0Rbdeir9kZMV30dVSs6RBUHVDgRg1p/P6pzxtcrTaLFygwIsdvcBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T20:18:13.000531Z"},"content_sha256":"cbc446242ea7701cfaccde5339e0d95a86499772c34ca465dab9a329d7e1e82c","schema_version":"1.0","event_id":"sha256:cbc446242ea7701cfaccde5339e0d95a86499772c34ca465dab9a329d7e1e82c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PBURDCU22J5KNSP57NYNL5NVWK/bundle.json","state_url":"https://pith.science/pith/PBURDCU22J5KNSP57NYNL5NVWK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PBURDCU22J5KNSP57NYNL5NVWK/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":"2026-05-27T20:18:13Z","links":{"resolver":"https://pith.science/pith/PBURDCU22J5KNSP57NYNL5NVWK","bundle":"https://pith.science/pith/PBURDCU22J5KNSP57NYNL5NVWK/bundle.json","state":"https://pith.science/pith/PBURDCU22J5KNSP57NYNL5NVWK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PBURDCU22J5KNSP57NYNL5NVWK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:PBURDCU22J5KNSP57NYNL5NVWK","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":"44e485766ed2406f9219ccf43c8c475ce0c7ba6e55dfc48d576ac6d401f6b397","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-18T17:54:24Z","title_canon_sha256":"c39bfcdec81cafa68553e8259eee58821c423057b2e80da5dd8ba4ec45a5e071"},"schema_version":"1.0","source":{"id":"1307.5027","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.5027","created_at":"2026-05-18T03:18:06Z"},{"alias_kind":"arxiv_version","alias_value":"1307.5027v1","created_at":"2026-05-18T03:18:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.5027","created_at":"2026-05-18T03:18:06Z"},{"alias_kind":"pith_short_12","alias_value":"PBURDCU22J5K","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"PBURDCU22J5KNSP5","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"PBURDCU2","created_at":"2026-05-18T12:27:54Z"}],"graph_snapshots":[{"event_id":"sha256:cbc446242ea7701cfaccde5339e0d95a86499772c34ca465dab9a329d7e1e82c","target":"graph","created_at":"2026-05-18T03:18:06Z","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 consider a tournament $T=(V, A)$. For $X\\subseteq V$, the subtournament of $T$ induced by $X$ is $T[X] = (X, A \\cap (X \\times X))$. An interval of $T$ is a subset $X$ of $V$ such that for $a, b\\in X$ and $ x\\in V\\setminus X$, $(a,x)\\in A$ if and only if $(b,x)\\in A$. The trivial intervals of $T$ are $\\emptyset$, $\\{x\\}(x\\in V)$ and $V$. A tournament is indecomposable if all its intervals are trivial. For $n\\geq 2$, $W_{2n+1}$ denotes the unique indecomposable tournament defined on $\\{0,\\dots,2n\\}$ such that $W_{2n+1}[\\{0,\\dots,2n-1\\}]$ is the usual total order. Given an indecomposable tourn","authors_text":"Houmem Belkhechine, Imed Boudabbous, Kaouthar Hzami","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-18T17:54:24Z","title":"The indecomposable tournaments $T$ with $\\mid W_{5}(T) \\mid = \\mid T \\mid -2$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.5027","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:cc01d8b0fe2071c4643fabee70f7fa35c0c0f97ce6d9b68f48c42177d255be03","target":"record","created_at":"2026-05-18T03:18:06Z","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":"44e485766ed2406f9219ccf43c8c475ce0c7ba6e55dfc48d576ac6d401f6b397","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-07-18T17:54:24Z","title_canon_sha256":"c39bfcdec81cafa68553e8259eee58821c423057b2e80da5dd8ba4ec45a5e071"},"schema_version":"1.0","source":{"id":"1307.5027","kind":"arxiv","version":1}},"canonical_sha256":"7869118a9ad27aa6c9fdfb70d5f5b5b2a1e38424a58d19eef8ca4e2e05d56e4c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7869118a9ad27aa6c9fdfb70d5f5b5b2a1e38424a58d19eef8ca4e2e05d56e4c","first_computed_at":"2026-05-18T03:18:06.894180Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:18:06.894180Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3umhZ3by7GFbJVW1gK5MlZIXQJKQNTNqeDd5pNorR/u8+q2E8aAvS1wHsI+hyT3W5l+FsHXxMNzfha3FazD6DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:18:06.894881Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.5027","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cc01d8b0fe2071c4643fabee70f7fa35c0c0f97ce6d9b68f48c42177d255be03","sha256:cbc446242ea7701cfaccde5339e0d95a86499772c34ca465dab9a329d7e1e82c"],"state_sha256":"50d87b18064e393b19960b1c38de08eae576c00a7d45d3c9199c3104a6566ed3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sStyP4BEm5WaJnIw5DeBLNOrKl2M0Asn1cFDKYdMsUniSVemelaislgZorwJv8W5ib6dfeRc/srg0KwSA10MDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T20:18:13.003858Z","bundle_sha256":"6711b6c7785d6ddb573a004b77c96558e6e501f93562286132130ec2516adaf2"}}