{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:JFWAVOV5FZECYTY4AT7JEQ3JJ5","short_pith_number":"pith:JFWAVOV5","canonical_record":{"source":{"id":"2605.14902","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T14:40:16Z","cross_cats_sorted":["cs.DM","cs.DS"],"title_canon_sha256":"0dca85ea299f3aa0a5529a204a7bf682c5dcea2a3fe315ce7c6821c7674ca00d","abstract_canon_sha256":"3dcc84423f0fb47e079146fde560dfea9f7c0436889d6f1b8dd20f0f09f6154e"},"schema_version":"1.0"},"canonical_sha256":"496c0ababd2e482c4f1c04fe9243694f72acf186b003d99bfeb68096225e6fb3","source":{"kind":"arxiv","id":"2605.14902","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.14902","created_at":"2026-05-17T23:38:55Z"},{"alias_kind":"arxiv_version","alias_value":"2605.14902v1","created_at":"2026-05-17T23:38:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.14902","created_at":"2026-05-17T23:38:55Z"},{"alias_kind":"pith_short_12","alias_value":"JFWAVOV5FZEC","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"JFWAVOV5FZECYTY4","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"JFWAVOV5","created_at":"2026-05-18T12:33:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:JFWAVOV5FZECYTY4AT7JEQ3JJ5","target":"record","payload":{"canonical_record":{"source":{"id":"2605.14902","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T14:40:16Z","cross_cats_sorted":["cs.DM","cs.DS"],"title_canon_sha256":"0dca85ea299f3aa0a5529a204a7bf682c5dcea2a3fe315ce7c6821c7674ca00d","abstract_canon_sha256":"3dcc84423f0fb47e079146fde560dfea9f7c0436889d6f1b8dd20f0f09f6154e"},"schema_version":"1.0"},"canonical_sha256":"496c0ababd2e482c4f1c04fe9243694f72acf186b003d99bfeb68096225e6fb3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:38:55.846652Z","signature_b64":"7ilCF6ckAKQg2jB4YsOe/FUXr7/HMvZ2RH0k6edaPjbyx5IO2MyFUS7RI0T6c6NES2Sc3wPaCz6IMb3zWRULAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"496c0ababd2e482c4f1c04fe9243694f72acf186b003d99bfeb68096225e6fb3","last_reissued_at":"2026-05-17T23:38:55.845922Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:38:55.845922Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.14902","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":"2026-05-17T23:38:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"59YdXQovAXmGYNwsGRPO3jzP3z1m9aXAiIDcdxrVbai9X/RYEJ2B8i+tjGIv6wEAb8xnlBm8SCq7PsrurtlMCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T07:30:54.909479Z"},"content_sha256":"c41b70ff42bebe1e5ec95391894bc4ed1531e958716af4d22748e6d2546cf0c5","schema_version":"1.0","event_id":"sha256:c41b70ff42bebe1e5ec95391894bc4ed1531e958716af4d22748e6d2546cf0c5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:JFWAVOV5FZECYTY4AT7JEQ3JJ5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Optimal Bounds for the k-Disjoint Paths Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.DS"],"primary_cat":"math.CO","authors_text":"Dario Cavallaro, Dimitrios M. Thilikos, Maximilian Gorsky, Sebastian Wiederrecht, Stephan Kreutzer","submitted_at":"2026-05-14T14:40:16Z","abstract_excerpt":"The Graph Minors Series of Robertson and Seymour forms the foundation of algorithmic structural graph theory, yielding fixed-parameter algorithms for problems such as Disjoint Paths, Rooted Minor Checking, and Folio. A key ingredient behind the fixed-parameter tractability of the $k$-Disjoint Paths problem is the irrelevant-vertex technique. This machinery is governed by the Vital Linkage Theorem and the so-called Linkage Function $\\ell$. However, despite its foundational role, the best known bounds on the Linkage Function are enormous and are only implicitly understood. The quantitative bound"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.14902","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":"2026-05-17T23:38:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0Aa0WReu6wFU9sQtOHiMc0JZZlrsGhFMknAW5xWw5mGNvMhnzHswv/O7wk8XZrM7kf9MM8j3jsQBY0o1VUMEBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T07:30:54.910082Z"},"content_sha256":"813b441084e5f31a4934bce1721721f5bec61345112516439082bf7d8aa47a25","schema_version":"1.0","event_id":"sha256:813b441084e5f31a4934bce1721721f5bec61345112516439082bf7d8aa47a25"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JFWAVOV5FZECYTY4AT7JEQ3JJ5/bundle.json","state_url":"https://pith.science/pith/JFWAVOV5FZECYTY4AT7JEQ3JJ5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JFWAVOV5FZECYTY4AT7JEQ3JJ5/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":"2026-05-26T07:30:54Z","links":{"resolver":"https://pith.science/pith/JFWAVOV5FZECYTY4AT7JEQ3JJ5","bundle":"https://pith.science/pith/JFWAVOV5FZECYTY4AT7JEQ3JJ5/bundle.json","state":"https://pith.science/pith/JFWAVOV5FZECYTY4AT7JEQ3JJ5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JFWAVOV5FZECYTY4AT7JEQ3JJ5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:JFWAVOV5FZECYTY4AT7JEQ3JJ5","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":"3dcc84423f0fb47e079146fde560dfea9f7c0436889d6f1b8dd20f0f09f6154e","cross_cats_sorted":["cs.DM","cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T14:40:16Z","title_canon_sha256":"0dca85ea299f3aa0a5529a204a7bf682c5dcea2a3fe315ce7c6821c7674ca00d"},"schema_version":"1.0","source":{"id":"2605.14902","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.14902","created_at":"2026-05-17T23:38:55Z"},{"alias_kind":"arxiv_version","alias_value":"2605.14902v1","created_at":"2026-05-17T23:38:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.14902","created_at":"2026-05-17T23:38:55Z"},{"alias_kind":"pith_short_12","alias_value":"JFWAVOV5FZEC","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"JFWAVOV5FZECYTY4","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"JFWAVOV5","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:813b441084e5f31a4934bce1721721f5bec61345112516439082bf7d8aa47a25","target":"graph","created_at":"2026-05-17T23:38:55Z","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":"The Graph Minors Series of Robertson and Seymour forms the foundation of algorithmic structural graph theory, yielding fixed-parameter algorithms for problems such as Disjoint Paths, Rooted Minor Checking, and Folio. A key ingredient behind the fixed-parameter tractability of the $k$-Disjoint Paths problem is the irrelevant-vertex technique. This machinery is governed by the Vital Linkage Theorem and the so-called Linkage Function $\\ell$. However, despite its foundational role, the best known bounds on the Linkage Function are enormous and are only implicitly understood. The quantitative bound","authors_text":"Dario Cavallaro, Dimitrios M. Thilikos, Maximilian Gorsky, Sebastian Wiederrecht, Stephan Kreutzer","cross_cats":["cs.DM","cs.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T14:40:16Z","title":"Optimal Bounds for the k-Disjoint Paths Problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.14902","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:c41b70ff42bebe1e5ec95391894bc4ed1531e958716af4d22748e6d2546cf0c5","target":"record","created_at":"2026-05-17T23:38:55Z","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":"3dcc84423f0fb47e079146fde560dfea9f7c0436889d6f1b8dd20f0f09f6154e","cross_cats_sorted":["cs.DM","cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T14:40:16Z","title_canon_sha256":"0dca85ea299f3aa0a5529a204a7bf682c5dcea2a3fe315ce7c6821c7674ca00d"},"schema_version":"1.0","source":{"id":"2605.14902","kind":"arxiv","version":1}},"canonical_sha256":"496c0ababd2e482c4f1c04fe9243694f72acf186b003d99bfeb68096225e6fb3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"496c0ababd2e482c4f1c04fe9243694f72acf186b003d99bfeb68096225e6fb3","first_computed_at":"2026-05-17T23:38:55.845922Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:38:55.845922Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7ilCF6ckAKQg2jB4YsOe/FUXr7/HMvZ2RH0k6edaPjbyx5IO2MyFUS7RI0T6c6NES2Sc3wPaCz6IMb3zWRULAA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:38:55.846652Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.14902","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c41b70ff42bebe1e5ec95391894bc4ed1531e958716af4d22748e6d2546cf0c5","sha256:813b441084e5f31a4934bce1721721f5bec61345112516439082bf7d8aa47a25"],"state_sha256":"b6fb96e9c7dd4c8ef62402e933d374e24e2c015288811ba74ffcc52a0a049e3c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UAGats1OLjRu89a9sz+TAkTtxMgXKCO388MS3MBF3LswqUvKWk2cOP6u8SfpyM42mxQtzsvGNXzRLT+DAc35Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T07:30:54.913875Z","bundle_sha256":"eea033a9e78767d72a5cad954cf4d94feb83d893eec6fad452d450c7cf55b9a0"}}