{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:OKYCZO47YYYCBBB3P56K5N3SQX","short_pith_number":"pith:OKYCZO47","canonical_record":{"source":{"id":"2608.05544","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CC","submitted_at":"2026-08-06T02:50:53Z","cross_cats_sorted":[],"title_canon_sha256":"44c5677591c3405a51460c661f9cf6d05fd4ce533a0a49f89063fba48382bd26","abstract_canon_sha256":"37fb2a4505d621adccd916dd0d8affbc862caeb27e10bd7c63cbd88c5c975482"},"schema_version":"1.0"},"canonical_sha256":"72b02cbb9fc63020843b7f7caeb77285dc03c7ad034124a58a23dda2bd700c19","source":{"kind":"arxiv","id":"2608.05544","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.05544","created_at":"2026-08-07T00:48:39Z"},{"alias_kind":"arxiv_version","alias_value":"2608.05544v1","created_at":"2026-08-07T00:48:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.05544","created_at":"2026-08-07T00:48:39Z"},{"alias_kind":"pith_short_12","alias_value":"OKYCZO47YYYC","created_at":"2026-08-07T00:48:39Z"},{"alias_kind":"pith_short_16","alias_value":"OKYCZO47YYYCBBB3","created_at":"2026-08-07T00:48:39Z"},{"alias_kind":"pith_short_8","alias_value":"OKYCZO47","created_at":"2026-08-07T00:48:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:OKYCZO47YYYCBBB3P56K5N3SQX","target":"record","payload":{"canonical_record":{"source":{"id":"2608.05544","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CC","submitted_at":"2026-08-06T02:50:53Z","cross_cats_sorted":[],"title_canon_sha256":"44c5677591c3405a51460c661f9cf6d05fd4ce533a0a49f89063fba48382bd26","abstract_canon_sha256":"37fb2a4505d621adccd916dd0d8affbc862caeb27e10bd7c63cbd88c5c975482"},"schema_version":"1.0"},"canonical_sha256":"72b02cbb9fc63020843b7f7caeb77285dc03c7ad034124a58a23dda2bd700c19","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-07T00:48:39.293707Z","signature_b64":"iVI1TYNASAOUqHUdgSctYuNix8ht1rUd2L1ta+2YoYMONMthm9ZzcDvWPW0JyccIuFukuUx7+88OAYLAPoNfCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"72b02cbb9fc63020843b7f7caeb77285dc03c7ad034124a58a23dda2bd700c19","last_reissued_at":"2026-08-07T00:48:39.292246Z","signature_status":"signed_v1","first_computed_at":"2026-08-07T00:48:39.292246Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2608.05544","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":"2026-08-07T00:48:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/adaDomCC6UGP8GHRYaEF3lrxcZXSlqw2V7txO7Ri54fpaivBXx5Qg7UQMoTDZxXAXWtt9txJ0hibg21cul4Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T19:29:10.664407Z"},"content_sha256":"0f1fcd446c5c9a243c296ca5f89568c071e1a7a76017843fd28e7797ad4271ce","schema_version":"1.0","event_id":"sha256:0f1fcd446c5c9a243c296ca5f89568c071e1a7a76017843fd28e7797ad4271ce"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:OKYCZO47YYYCBBB3P56K5N3SQX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ulam Median is NP-hard for Four Permutations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Mursalin Habib","submitted_at":"2026-08-06T02:50:53Z","abstract_excerpt":"We show that computing a median under the Ulam distance is NP-hard even when the input consists of exactly four permutations. Previously, NP-hardness was known only for an unbounded number of input permutations (Fischer et al., ESA '25). Our result is tight, since an Ulam median of three permutations can be computed in polynomial time (Chakraborty--Das--Krauthgamer, SODA '21)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.05544","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/2608.05544/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":"2026-08-07T00:48:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EkRmzdu5JQFI7Mv16PHz6qEsXy7mz+VBB9u56APF2Bv+5T1VgTFFEKnpwtEBE+4A5yRdHFbSgP2dfLynuZxqCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T19:29:10.665469Z"},"content_sha256":"59c11c8738473b4a58179831b01f54d542a5f68968656283eb65d1a07ee8f2bb","schema_version":"1.0","event_id":"sha256:59c11c8738473b4a58179831b01f54d542a5f68968656283eb65d1a07ee8f2bb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OKYCZO47YYYCBBB3P56K5N3SQX/bundle.json","state_url":"https://pith.science/pith/OKYCZO47YYYCBBB3P56K5N3SQX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OKYCZO47YYYCBBB3P56K5N3SQX/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":"2026-08-09T19:29:10Z","links":{"resolver":"https://pith.science/pith/OKYCZO47YYYCBBB3P56K5N3SQX","bundle":"https://pith.science/pith/OKYCZO47YYYCBBB3P56K5N3SQX/bundle.json","state":"https://pith.science/pith/OKYCZO47YYYCBBB3P56K5N3SQX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OKYCZO47YYYCBBB3P56K5N3SQX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:OKYCZO47YYYCBBB3P56K5N3SQX","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":"37fb2a4505d621adccd916dd0d8affbc862caeb27e10bd7c63cbd88c5c975482","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CC","submitted_at":"2026-08-06T02:50:53Z","title_canon_sha256":"44c5677591c3405a51460c661f9cf6d05fd4ce533a0a49f89063fba48382bd26"},"schema_version":"1.0","source":{"id":"2608.05544","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2608.05544","created_at":"2026-08-07T00:48:39Z"},{"alias_kind":"arxiv_version","alias_value":"2608.05544v1","created_at":"2026-08-07T00:48:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.05544","created_at":"2026-08-07T00:48:39Z"},{"alias_kind":"pith_short_12","alias_value":"OKYCZO47YYYC","created_at":"2026-08-07T00:48:39Z"},{"alias_kind":"pith_short_16","alias_value":"OKYCZO47YYYCBBB3","created_at":"2026-08-07T00:48:39Z"},{"alias_kind":"pith_short_8","alias_value":"OKYCZO47","created_at":"2026-08-07T00:48:39Z"}],"graph_snapshots":[{"event_id":"sha256:59c11c8738473b4a58179831b01f54d542a5f68968656283eb65d1a07ee8f2bb","target":"graph","created_at":"2026-08-07T00:48:39Z","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/2608.05544/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We show that computing a median under the Ulam distance is NP-hard even when the input consists of exactly four permutations. Previously, NP-hardness was known only for an unbounded number of input permutations (Fischer et al., ESA '25). Our result is tight, since an Ulam median of three permutations can be computed in polynomial time (Chakraborty--Das--Krauthgamer, SODA '21).","authors_text":"Mursalin Habib","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CC","submitted_at":"2026-08-06T02:50:53Z","title":"Ulam Median is NP-hard for Four Permutations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.05544","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":"sha256:0f1fcd446c5c9a243c296ca5f89568c071e1a7a76017843fd28e7797ad4271ce","target":"record","created_at":"2026-08-07T00:48:39Z","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":"37fb2a4505d621adccd916dd0d8affbc862caeb27e10bd7c63cbd88c5c975482","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CC","submitted_at":"2026-08-06T02:50:53Z","title_canon_sha256":"44c5677591c3405a51460c661f9cf6d05fd4ce533a0a49f89063fba48382bd26"},"schema_version":"1.0","source":{"id":"2608.05544","kind":"arxiv","version":1}},"canonical_sha256":"72b02cbb9fc63020843b7f7caeb77285dc03c7ad034124a58a23dda2bd700c19","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"72b02cbb9fc63020843b7f7caeb77285dc03c7ad034124a58a23dda2bd700c19","first_computed_at":"2026-08-07T00:48:39.292246Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-08-07T00:48:39.292246Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iVI1TYNASAOUqHUdgSctYuNix8ht1rUd2L1ta+2YoYMONMthm9ZzcDvWPW0JyccIuFukuUx7+88OAYLAPoNfCQ==","signature_status":"signed_v1","signed_at":"2026-08-07T00:48:39.293707Z","signed_message":"canonical_sha256_bytes"},"source_id":"2608.05544","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0f1fcd446c5c9a243c296ca5f89568c071e1a7a76017843fd28e7797ad4271ce","sha256:59c11c8738473b4a58179831b01f54d542a5f68968656283eb65d1a07ee8f2bb"],"state_sha256":"ae409d7a4c5ae5149f24eb7901058bf1f53dd91976889e707e64deea704cc661"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nngaPfptk9az3D1lgsW4fw1dimxGeTDZaNt51/eUlgwjcUxvuDpZHHCZKU7C0BVHLggV5iTQdlpjwDjGatTXAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T19:29:10.670074Z","bundle_sha256":"acf639fce84949bc15afd83cede84abbb751a6b1f4aa5637fb89060c7ab96f4a"}}