{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:QVNMRRESZGSARNUWA23MH42BMN","short_pith_number":"pith:QVNMRRES","canonical_record":{"source":{"id":"2307.02816","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-07-06T07:20:44Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"3d283c9bc5cf9ff55ce8572d215c7856dcccb11368ec1944263af5ddd4282ba7","abstract_canon_sha256":"5b5b16e542166afc276b639909462ed1986c21e62c9ac57073f2c24359b282e6"},"schema_version":"1.0"},"canonical_sha256":"855ac8c492c9a408b69606b6c3f341635f045b73ce14aa874aef5c3e889fbfa5","source":{"kind":"arxiv","id":"2307.02816","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2307.02816","created_at":"2026-06-02T02:04:44Z"},{"alias_kind":"arxiv_version","alias_value":"2307.02816v2","created_at":"2026-06-02T02:04:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2307.02816","created_at":"2026-06-02T02:04:44Z"},{"alias_kind":"pith_short_12","alias_value":"QVNMRRESZGSA","created_at":"2026-06-02T02:04:44Z"},{"alias_kind":"pith_short_16","alias_value":"QVNMRRESZGSARNUW","created_at":"2026-06-02T02:04:44Z"},{"alias_kind":"pith_short_8","alias_value":"QVNMRRES","created_at":"2026-06-02T02:04:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:QVNMRRESZGSARNUWA23MH42BMN","target":"record","payload":{"canonical_record":{"source":{"id":"2307.02816","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-07-06T07:20:44Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"3d283c9bc5cf9ff55ce8572d215c7856dcccb11368ec1944263af5ddd4282ba7","abstract_canon_sha256":"5b5b16e542166afc276b639909462ed1986c21e62c9ac57073f2c24359b282e6"},"schema_version":"1.0"},"canonical_sha256":"855ac8c492c9a408b69606b6c3f341635f045b73ce14aa874aef5c3e889fbfa5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T02:04:44.620714Z","signature_b64":"d9ncDdYKGapTOosMcRx0S5+SLdkT2WVM/Pc+FROtffv8zN9K6qQuQ21TUHDqpRJdWL4SPju4XByLEdnhk2geAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"855ac8c492c9a408b69606b6c3f341635f045b73ce14aa874aef5c3e889fbfa5","last_reissued_at":"2026-06-02T02:04:44.620233Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T02:04:44.620233Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2307.02816","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-02T02:04:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0rgHBdfyA31K0NWMdhu/ijT4/NGH8LLp/lbV7feqUhw9CvnDIqR/EVFFqE3sEUsa/QXSuCYFI6siSAfksEQCDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T18:27:33.894691Z"},"content_sha256":"b447f254ffcb8bf1f8e6e4034498b6dd3bc3aac03627e3e6be4a30af49f72063","schema_version":"1.0","event_id":"sha256:b447f254ffcb8bf1f8e6e4034498b6dd3bc3aac03627e3e6be4a30af49f72063"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:QVNMRRESZGSARNUWA23MH42BMN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The grid-minor theorem revisited","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Cl\\'ement Rambaud, David R. Wood, Gwena\\\"el Joret, Hoang La, J\\k{e}drzej Hodor, Pat Morin, Piotr Micek, Robert Hickingbotham, Vida Dujmovi\\'c","submitted_at":"2023-07-06T07:20:44Z","abstract_excerpt":"We prove that for every planar graph $X$ of treedepth $h$, there exists a positive integer $c$ such that for every $X$-minor-free graph $G$, there exists a graph $H$ of treewidth at most $f(h)$ such that $G$ is isomorphic to a subgraph of $H\\boxtimes K_c$. This is a qualitative strengthening of the Grid-Minor Theorem of Robertson and Seymour (JCTB 1986), and treedepth is the optimal parameter in such a result. As an example application, we use this result to improve the upper bound for weak coloring numbers of graphs excluding a fixed graph as a minor."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.02816","kind":"arxiv","version":2},"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/2307.02816/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":"2026-06-02T02:04:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xvornM3PSyniLzH2j6/rpACK/Yg2tCqI4k3H6jU5bsSHxPxPUkO/S1D3wu1nTiehlDLDsqTvkFbcqw9guRw/Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T18:27:33.895285Z"},"content_sha256":"3e806b874ac235ebddaf297aa02e4789c284eb58cd5f2af418aa0e0f82edffb8","schema_version":"1.0","event_id":"sha256:3e806b874ac235ebddaf297aa02e4789c284eb58cd5f2af418aa0e0f82edffb8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QVNMRRESZGSARNUWA23MH42BMN/bundle.json","state_url":"https://pith.science/pith/QVNMRRESZGSARNUWA23MH42BMN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QVNMRRESZGSARNUWA23MH42BMN/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":"2026-06-05T18:27:33Z","links":{"resolver":"https://pith.science/pith/QVNMRRESZGSARNUWA23MH42BMN","bundle":"https://pith.science/pith/QVNMRRESZGSARNUWA23MH42BMN/bundle.json","state":"https://pith.science/pith/QVNMRRESZGSARNUWA23MH42BMN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QVNMRRESZGSARNUWA23MH42BMN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:QVNMRRESZGSARNUWA23MH42BMN","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":"5b5b16e542166afc276b639909462ed1986c21e62c9ac57073f2c24359b282e6","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-07-06T07:20:44Z","title_canon_sha256":"3d283c9bc5cf9ff55ce8572d215c7856dcccb11368ec1944263af5ddd4282ba7"},"schema_version":"1.0","source":{"id":"2307.02816","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2307.02816","created_at":"2026-06-02T02:04:44Z"},{"alias_kind":"arxiv_version","alias_value":"2307.02816v2","created_at":"2026-06-02T02:04:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2307.02816","created_at":"2026-06-02T02:04:44Z"},{"alias_kind":"pith_short_12","alias_value":"QVNMRRESZGSA","created_at":"2026-06-02T02:04:44Z"},{"alias_kind":"pith_short_16","alias_value":"QVNMRRESZGSARNUW","created_at":"2026-06-02T02:04:44Z"},{"alias_kind":"pith_short_8","alias_value":"QVNMRRES","created_at":"2026-06-02T02:04:44Z"}],"graph_snapshots":[{"event_id":"sha256:3e806b874ac235ebddaf297aa02e4789c284eb58cd5f2af418aa0e0f82edffb8","target":"graph","created_at":"2026-06-02T02:04:44Z","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/2307.02816/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove that for every planar graph $X$ of treedepth $h$, there exists a positive integer $c$ such that for every $X$-minor-free graph $G$, there exists a graph $H$ of treewidth at most $f(h)$ such that $G$ is isomorphic to a subgraph of $H\\boxtimes K_c$. This is a qualitative strengthening of the Grid-Minor Theorem of Robertson and Seymour (JCTB 1986), and treedepth is the optimal parameter in such a result. As an example application, we use this result to improve the upper bound for weak coloring numbers of graphs excluding a fixed graph as a minor.","authors_text":"Cl\\'ement Rambaud, David R. Wood, Gwena\\\"el Joret, Hoang La, J\\k{e}drzej Hodor, Pat Morin, Piotr Micek, Robert Hickingbotham, Vida Dujmovi\\'c","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-07-06T07:20:44Z","title":"The grid-minor theorem revisited"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2307.02816","kind":"arxiv","version":2},"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:b447f254ffcb8bf1f8e6e4034498b6dd3bc3aac03627e3e6be4a30af49f72063","target":"record","created_at":"2026-06-02T02:04:44Z","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":"5b5b16e542166afc276b639909462ed1986c21e62c9ac57073f2c24359b282e6","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2023-07-06T07:20:44Z","title_canon_sha256":"3d283c9bc5cf9ff55ce8572d215c7856dcccb11368ec1944263af5ddd4282ba7"},"schema_version":"1.0","source":{"id":"2307.02816","kind":"arxiv","version":2}},"canonical_sha256":"855ac8c492c9a408b69606b6c3f341635f045b73ce14aa874aef5c3e889fbfa5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"855ac8c492c9a408b69606b6c3f341635f045b73ce14aa874aef5c3e889fbfa5","first_computed_at":"2026-06-02T02:04:44.620233Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T02:04:44.620233Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"d9ncDdYKGapTOosMcRx0S5+SLdkT2WVM/Pc+FROtffv8zN9K6qQuQ21TUHDqpRJdWL4SPju4XByLEdnhk2geAQ==","signature_status":"signed_v1","signed_at":"2026-06-02T02:04:44.620714Z","signed_message":"canonical_sha256_bytes"},"source_id":"2307.02816","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b447f254ffcb8bf1f8e6e4034498b6dd3bc3aac03627e3e6be4a30af49f72063","sha256:3e806b874ac235ebddaf297aa02e4789c284eb58cd5f2af418aa0e0f82edffb8"],"state_sha256":"1aaeb9bb88eeb06163fe5bde5b4f8d595886ed69e27475f9b5c51de1bbf9b5aa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Gdb+G4U6GhgScpbr+8JQ2sDjJh5AGTAUVE9mVG38Uc06pwfD5nlViAMXJimtUt1dYifP0PLlPvyMcFYaz0dlAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T18:27:33.898471Z","bundle_sha256":"10e7f14d8bf9dbf47ad6c10a72bda9d48747397887ce5a76b3ae7e0e5f882c0d"}}