{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:4K6PRJEUUWLUNON7AW4ZYI5YMY","short_pith_number":"pith:4K6PRJEU","canonical_record":{"source":{"id":"2606.13154","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-11T10:27:01Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"1e8813e1b3c6df9bad59620c03dbf3501f4e4b39fc4ed08232b2b84d1dba8306","abstract_canon_sha256":"6947f4ee8f28fc8c6d462018bc73af78c16918acab7559dc19c3a6b5fcad55f9"},"schema_version":"1.0"},"canonical_sha256":"e2bcf8a494a59746b9bf05b99c23b8661375c4d7eb2ef32e941942ba19571556","source":{"kind":"arxiv","id":"2606.13154","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.13154","created_at":"2026-06-12T01:09:43Z"},{"alias_kind":"arxiv_version","alias_value":"2606.13154v1","created_at":"2026-06-12T01:09:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.13154","created_at":"2026-06-12T01:09:43Z"},{"alias_kind":"pith_short_12","alias_value":"4K6PRJEUUWLU","created_at":"2026-06-12T01:09:43Z"},{"alias_kind":"pith_short_16","alias_value":"4K6PRJEUUWLUNON7","created_at":"2026-06-12T01:09:43Z"},{"alias_kind":"pith_short_8","alias_value":"4K6PRJEU","created_at":"2026-06-12T01:09:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:4K6PRJEUUWLUNON7AW4ZYI5YMY","target":"record","payload":{"canonical_record":{"source":{"id":"2606.13154","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-11T10:27:01Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"1e8813e1b3c6df9bad59620c03dbf3501f4e4b39fc4ed08232b2b84d1dba8306","abstract_canon_sha256":"6947f4ee8f28fc8c6d462018bc73af78c16918acab7559dc19c3a6b5fcad55f9"},"schema_version":"1.0"},"canonical_sha256":"e2bcf8a494a59746b9bf05b99c23b8661375c4d7eb2ef32e941942ba19571556","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-12T01:09:43.792721Z","signature_b64":"qKr4D6ky7BN5LenSg1jrjOtldVR83xm/GEJ2vxIJ+Ah9hIzIAo2+jTNX4y36K0BokPQuiWkKP2lEVRsIqOEfBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e2bcf8a494a59746b9bf05b99c23b8661375c4d7eb2ef32e941942ba19571556","last_reissued_at":"2026-06-12T01:09:43.791779Z","signature_status":"signed_v1","first_computed_at":"2026-06-12T01:09:43.791779Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.13154","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":"2026-06-12T01:09:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dXxCYCUfuFmIE7hAU6B1+gGtYY8+b+eWksDpcThA5VW9h5Kxe6DaIjQ31mRFjMLEhlHQDwuvIv3zp/rRabL4Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T23:33:42.787982Z"},"content_sha256":"6818186e4e6def9ae681b1b6e9b34f856ee25a84a3d5e160ccf9b91850be67df","schema_version":"1.0","event_id":"sha256:6818186e4e6def9ae681b1b6e9b34f856ee25a84a3d5e160ccf9b91850be67df"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:4K6PRJEUUWLUNON7AW4ZYI5YMY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Computing Joins in the Weak Order of Type B Coxeter Groups: an Algorithmic Approach","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Lorenzo Perrone, Riccardo Biagioli","submitted_at":"2026-06-11T10:27:01Z","abstract_excerpt":"We present an algorithm for computing the join of two elements in the weak order of the Coxeter group of type B. This extends Markowsky's algorithm for computing joins of standard permutations to signed permutations, and allows us to confirm a conjecture of Dyer concerning a geometric interpretation of these joins."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.13154","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/2606.13154/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":"2026-06-12T01:09:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MTEz8KU8CVInlslkUmuJOmH974cUQV/QoAzXG7739Ga0p3HezdZRODzHganm9ZgWe7qmmHT6xV0EmtjXGj8RBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T23:33:42.788363Z"},"content_sha256":"d596f1ab748f46b1ba23c07368fd5764f8791fa095924f927915547dcbefcad0","schema_version":"1.0","event_id":"sha256:d596f1ab748f46b1ba23c07368fd5764f8791fa095924f927915547dcbefcad0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4K6PRJEUUWLUNON7AW4ZYI5YMY/bundle.json","state_url":"https://pith.science/pith/4K6PRJEUUWLUNON7AW4ZYI5YMY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4K6PRJEUUWLUNON7AW4ZYI5YMY/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":"2026-06-28T23:33:42Z","links":{"resolver":"https://pith.science/pith/4K6PRJEUUWLUNON7AW4ZYI5YMY","bundle":"https://pith.science/pith/4K6PRJEUUWLUNON7AW4ZYI5YMY/bundle.json","state":"https://pith.science/pith/4K6PRJEUUWLUNON7AW4ZYI5YMY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4K6PRJEUUWLUNON7AW4ZYI5YMY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:4K6PRJEUUWLUNON7AW4ZYI5YMY","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":"6947f4ee8f28fc8c6d462018bc73af78c16918acab7559dc19c3a6b5fcad55f9","cross_cats_sorted":["math.GR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-11T10:27:01Z","title_canon_sha256":"1e8813e1b3c6df9bad59620c03dbf3501f4e4b39fc4ed08232b2b84d1dba8306"},"schema_version":"1.0","source":{"id":"2606.13154","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.13154","created_at":"2026-06-12T01:09:43Z"},{"alias_kind":"arxiv_version","alias_value":"2606.13154v1","created_at":"2026-06-12T01:09:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.13154","created_at":"2026-06-12T01:09:43Z"},{"alias_kind":"pith_short_12","alias_value":"4K6PRJEUUWLU","created_at":"2026-06-12T01:09:43Z"},{"alias_kind":"pith_short_16","alias_value":"4K6PRJEUUWLUNON7","created_at":"2026-06-12T01:09:43Z"},{"alias_kind":"pith_short_8","alias_value":"4K6PRJEU","created_at":"2026-06-12T01:09:43Z"}],"graph_snapshots":[{"event_id":"sha256:d596f1ab748f46b1ba23c07368fd5764f8791fa095924f927915547dcbefcad0","target":"graph","created_at":"2026-06-12T01:09:43Z","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/2606.13154/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We present an algorithm for computing the join of two elements in the weak order of the Coxeter group of type B. This extends Markowsky's algorithm for computing joins of standard permutations to signed permutations, and allows us to confirm a conjecture of Dyer concerning a geometric interpretation of these joins.","authors_text":"Lorenzo Perrone, Riccardo Biagioli","cross_cats":["math.GR"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-11T10:27:01Z","title":"Computing Joins in the Weak Order of Type B Coxeter Groups: an Algorithmic Approach"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.13154","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":"sha256:6818186e4e6def9ae681b1b6e9b34f856ee25a84a3d5e160ccf9b91850be67df","target":"record","created_at":"2026-06-12T01:09:43Z","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":"6947f4ee8f28fc8c6d462018bc73af78c16918acab7559dc19c3a6b5fcad55f9","cross_cats_sorted":["math.GR"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-11T10:27:01Z","title_canon_sha256":"1e8813e1b3c6df9bad59620c03dbf3501f4e4b39fc4ed08232b2b84d1dba8306"},"schema_version":"1.0","source":{"id":"2606.13154","kind":"arxiv","version":1}},"canonical_sha256":"e2bcf8a494a59746b9bf05b99c23b8661375c4d7eb2ef32e941942ba19571556","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e2bcf8a494a59746b9bf05b99c23b8661375c4d7eb2ef32e941942ba19571556","first_computed_at":"2026-06-12T01:09:43.791779Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-12T01:09:43.791779Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qKr4D6ky7BN5LenSg1jrjOtldVR83xm/GEJ2vxIJ+Ah9hIzIAo2+jTNX4y36K0BokPQuiWkKP2lEVRsIqOEfBw==","signature_status":"signed_v1","signed_at":"2026-06-12T01:09:43.792721Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.13154","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6818186e4e6def9ae681b1b6e9b34f856ee25a84a3d5e160ccf9b91850be67df","sha256:d596f1ab748f46b1ba23c07368fd5764f8791fa095924f927915547dcbefcad0"],"state_sha256":"7b54e4e4a5ddfe2dfcfbf166bcf3774224b248a8382039872c8f19214a945966"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tFRwwS1P7/JJkg+e1PO0xXkfMgL00KX8f7gWx8xR4kMojLvXuC6SMrChMucV/CCcaXYYo0v8IjYTSp6lVz0mDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T23:33:42.790276Z","bundle_sha256":"14143983907b1b2da79e31e9fbc18af4c97c5f313667abb4cc57b338b043271e"}}