{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:B7TAIHVJN7AJMG2W3LWQ3MMCRE","short_pith_number":"pith:B7TAIHVJ","canonical_record":{"source":{"id":"2607.12841","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2026-07-14T14:57:51Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"1c7ff95b8cd7196fa7ab0dc41277bcb3602c1912f92b9d870bfd8a8a8a0fb711","abstract_canon_sha256":"60aed866f514cae4fb41a1eb4c25f96a380e5cab332f11918a39591722c1d7cb"},"schema_version":"1.0"},"canonical_sha256":"0fe6041ea96fc0961b56daed0db1828923bd13eb28f60e1a6add8b7a8d20cad8","source":{"kind":"arxiv","id":"2607.12841","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.12841","created_at":"2026-07-15T01:22:21Z"},{"alias_kind":"arxiv_version","alias_value":"2607.12841v1","created_at":"2026-07-15T01:22:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.12841","created_at":"2026-07-15T01:22:21Z"},{"alias_kind":"pith_short_12","alias_value":"B7TAIHVJN7AJ","created_at":"2026-07-15T01:22:21Z"},{"alias_kind":"pith_short_16","alias_value":"B7TAIHVJN7AJMG2W","created_at":"2026-07-15T01:22:21Z"},{"alias_kind":"pith_short_8","alias_value":"B7TAIHVJ","created_at":"2026-07-15T01:22:21Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:B7TAIHVJN7AJMG2W3LWQ3MMCRE","target":"record","payload":{"canonical_record":{"source":{"id":"2607.12841","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2026-07-14T14:57:51Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"1c7ff95b8cd7196fa7ab0dc41277bcb3602c1912f92b9d870bfd8a8a8a0fb711","abstract_canon_sha256":"60aed866f514cae4fb41a1eb4c25f96a380e5cab332f11918a39591722c1d7cb"},"schema_version":"1.0"},"canonical_sha256":"0fe6041ea96fc0961b56daed0db1828923bd13eb28f60e1a6add8b7a8d20cad8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-15T01:22:21.316857Z","signature_b64":"2gD6ycpQ6gFMU76D2w5BlLhfdLS1/cuxqUeCxMHh8GtQvLJmf9T7ylxcQCuV5gQIaW+/brSL6g2nbLkBJD7tAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0fe6041ea96fc0961b56daed0db1828923bd13eb28f60e1a6add8b7a8d20cad8","last_reissued_at":"2026-07-15T01:22:21.316063Z","signature_status":"signed_v1","first_computed_at":"2026-07-15T01:22:21.316063Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.12841","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":"2026-07-15T01:22:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yKwK9iPVauAaBEzdxxnEA0lAR/JdGiqTs44bT6OPufTcYcL7amAGcP3+p8gcXQFmz/uVNeRnu9kdMOIsigNFCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T07:08:25.976712Z"},"content_sha256":"78eec10375ca5779cd4c6205df14f2d4087d913a72d954d6a4f18afa8e6a7b8c","schema_version":"1.0","event_id":"sha256:78eec10375ca5779cd4c6205df14f2d4087d913a72d954d6a4f18afa8e6a7b8c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:B7TAIHVJN7AJMG2W3LWQ3MMCRE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The exact region determined by Kendall's tau, Spearman's footrule and Blomqvist's beta","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Jacob Israel Orenday Lares, Marcus Rockel","submitted_at":"2026-07-14T14:57:51Z","abstract_excerpt":"We determine the exact region $\\Omega_{\\tau,\\phi,\\beta}:=\\{(\\tau(C),\\phi(C),\\beta(C)):C\\in\\mathcal{C}\\}$ of possible joint values of Kendall's tau, Spearman's footrule and Blomqvist's beta over the class $\\mathcal{C}$ of all bivariate copulas. The region consists precisely of all triples $(t,p,b)$ satisfying $-1\\le b\\le 1$, $\\frac{3}{16}(1+b)^2-\\frac12\\le p\\le 1-\\frac38(1-b)^2$ and $\\frac43 p-\\frac13\\le t\\le \\frac23 p+\\frac13$. In other words, the known exact $(\\phi,\\beta)$- and $(\\tau,\\phi)$-regions already characterize the joint region, so that, once the value of Spearman's footrule is fixed"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.12841","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.12841/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-15T01:22:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bffCdoUdTeuEtEi4Pm3kgzZGloZidcnEQOMmzCiAimdNyqcGzHyFI8tvD5E+UcAndO9LjoU1Oe7YZ/atxR6gDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T07:08:25.977050Z"},"content_sha256":"8f3db62429cce0a7d044e65db6b60d87d6622652be5dd93a9ea3a0fc65145bf5","schema_version":"1.0","event_id":"sha256:8f3db62429cce0a7d044e65db6b60d87d6622652be5dd93a9ea3a0fc65145bf5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/B7TAIHVJN7AJMG2W3LWQ3MMCRE/bundle.json","state_url":"https://pith.science/pith/B7TAIHVJN7AJMG2W3LWQ3MMCRE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/B7TAIHVJN7AJMG2W3LWQ3MMCRE/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":"2026-08-20T07:08:25Z","links":{"resolver":"https://pith.science/pith/B7TAIHVJN7AJMG2W3LWQ3MMCRE","bundle":"https://pith.science/pith/B7TAIHVJN7AJMG2W3LWQ3MMCRE/bundle.json","state":"https://pith.science/pith/B7TAIHVJN7AJMG2W3LWQ3MMCRE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/B7TAIHVJN7AJMG2W3LWQ3MMCRE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:B7TAIHVJN7AJMG2W3LWQ3MMCRE","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":"60aed866f514cae4fb41a1eb4c25f96a380e5cab332f11918a39591722c1d7cb","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2026-07-14T14:57:51Z","title_canon_sha256":"1c7ff95b8cd7196fa7ab0dc41277bcb3602c1912f92b9d870bfd8a8a8a0fb711"},"schema_version":"1.0","source":{"id":"2607.12841","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.12841","created_at":"2026-07-15T01:22:21Z"},{"alias_kind":"arxiv_version","alias_value":"2607.12841v1","created_at":"2026-07-15T01:22:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.12841","created_at":"2026-07-15T01:22:21Z"},{"alias_kind":"pith_short_12","alias_value":"B7TAIHVJN7AJ","created_at":"2026-07-15T01:22:21Z"},{"alias_kind":"pith_short_16","alias_value":"B7TAIHVJN7AJMG2W","created_at":"2026-07-15T01:22:21Z"},{"alias_kind":"pith_short_8","alias_value":"B7TAIHVJ","created_at":"2026-07-15T01:22:21Z"}],"graph_snapshots":[{"event_id":"sha256:8f3db62429cce0a7d044e65db6b60d87d6622652be5dd93a9ea3a0fc65145bf5","target":"graph","created_at":"2026-07-15T01:22:21Z","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/2607.12841/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We determine the exact region $\\Omega_{\\tau,\\phi,\\beta}:=\\{(\\tau(C),\\phi(C),\\beta(C)):C\\in\\mathcal{C}\\}$ of possible joint values of Kendall's tau, Spearman's footrule and Blomqvist's beta over the class $\\mathcal{C}$ of all bivariate copulas. The region consists precisely of all triples $(t,p,b)$ satisfying $-1\\le b\\le 1$, $\\frac{3}{16}(1+b)^2-\\frac12\\le p\\le 1-\\frac38(1-b)^2$ and $\\frac43 p-\\frac13\\le t\\le \\frac23 p+\\frac13$. In other words, the known exact $(\\phi,\\beta)$- and $(\\tau,\\phi)$-regions already characterize the joint region, so that, once the value of Spearman's footrule is fixed","authors_text":"Jacob Israel Orenday Lares, Marcus Rockel","cross_cats":["stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2026-07-14T14:57:51Z","title":"The exact region determined by Kendall's tau, Spearman's footrule and Blomqvist's beta"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.12841","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":"sha256:78eec10375ca5779cd4c6205df14f2d4087d913a72d954d6a4f18afa8e6a7b8c","target":"record","created_at":"2026-07-15T01:22:21Z","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":"60aed866f514cae4fb41a1eb4c25f96a380e5cab332f11918a39591722c1d7cb","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2026-07-14T14:57:51Z","title_canon_sha256":"1c7ff95b8cd7196fa7ab0dc41277bcb3602c1912f92b9d870bfd8a8a8a0fb711"},"schema_version":"1.0","source":{"id":"2607.12841","kind":"arxiv","version":1}},"canonical_sha256":"0fe6041ea96fc0961b56daed0db1828923bd13eb28f60e1a6add8b7a8d20cad8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0fe6041ea96fc0961b56daed0db1828923bd13eb28f60e1a6add8b7a8d20cad8","first_computed_at":"2026-07-15T01:22:21.316063Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-15T01:22:21.316063Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2gD6ycpQ6gFMU76D2w5BlLhfdLS1/cuxqUeCxMHh8GtQvLJmf9T7ylxcQCuV5gQIaW+/brSL6g2nbLkBJD7tAQ==","signature_status":"signed_v1","signed_at":"2026-07-15T01:22:21.316857Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.12841","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:78eec10375ca5779cd4c6205df14f2d4087d913a72d954d6a4f18afa8e6a7b8c","sha256:8f3db62429cce0a7d044e65db6b60d87d6622652be5dd93a9ea3a0fc65145bf5"],"state_sha256":"225f3a09e3ffea05d4a2b1437585023e7a1a2ff9dc1bff988271d5caf84ba88c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/t74jTDdftUvTPLDwfKJfbrkV1efyoiRv0UNbwxXW1BHGKLTIp215DR4pqmiER5sdW1V7HkD6fg7s9AAO+CXDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-20T07:08:25.980705Z","bundle_sha256":"f7f17c0512daed94ff66c4f2f3883990648db872b59da4a1e6c94f312ec6755b"}}