{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:N73CGXFBRTLZ6SY2LBH5PUNFRI","short_pith_number":"pith:N73CGXFB","canonical_record":{"source":{"id":"1207.6927","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-30T13:03:28Z","cross_cats_sorted":[],"title_canon_sha256":"7f246e62fb43a7dbc87da838c470abb223afebce17cefbadb2f18248b93d73ac","abstract_canon_sha256":"bd02e9385179152cc2dc42b5e97f08db0be250189e8536b73f89ee457035a4de"},"schema_version":"1.0"},"canonical_sha256":"6ff6235ca18cd79f4b1a584fd7d1a58a2d36cb5964dd015b58018b985913659d","source":{"kind":"arxiv","id":"1207.6927","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.6927","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"arxiv_version","alias_value":"1207.6927v3","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.6927","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"pith_short_12","alias_value":"N73CGXFBRTLZ","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"N73CGXFBRTLZ6SY2","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"N73CGXFB","created_at":"2026-05-18T12:27:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:N73CGXFBRTLZ6SY2LBH5PUNFRI","target":"record","payload":{"canonical_record":{"source":{"id":"1207.6927","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-30T13:03:28Z","cross_cats_sorted":[],"title_canon_sha256":"7f246e62fb43a7dbc87da838c470abb223afebce17cefbadb2f18248b93d73ac","abstract_canon_sha256":"bd02e9385179152cc2dc42b5e97f08db0be250189e8536b73f89ee457035a4de"},"schema_version":"1.0"},"canonical_sha256":"6ff6235ca18cd79f4b1a584fd7d1a58a2d36cb5964dd015b58018b985913659d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:57.178151Z","signature_b64":"FadKdjj/07vvCyR1jCCSci2fwaLA6lpNgN8Qy0Wkobqp9yfTdGBrW1cSfRxxM/Qv9pHdTsW6AwzmPCu9XPZ2AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6ff6235ca18cd79f4b1a584fd7d1a58a2d36cb5964dd015b58018b985913659d","last_reissued_at":"2026-05-18T01:07:57.177718Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:57.177718Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.6927","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:07:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eCwiA7n54Bu8f2bG1BPHsscOiHnJdZPZEwczRcClw1kHq3dumu/HlP2LIio/4v7Re52pQWuZT+jfqKgVqU1/Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T06:18:39.088988Z"},"content_sha256":"d2f3129d450a5df0f0e27f8ec61ef7a0986629f28e4a2eb977c8aaf740d6a9fd","schema_version":"1.0","event_id":"sha256:d2f3129d450a5df0f0e27f8ec61ef7a0986629f28e4a2eb977c8aaf740d6a9fd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:N73CGXFBRTLZ6SY2LBH5PUNFRI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A New Proof of the Flat Wall Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ken-ichi Kawarabayashi, Paul Wollan, Robin Thomas","submitted_at":"2012-07-30T13:03:28Z","abstract_excerpt":"We give an elementary and self-contained proof, and a numerical improvement, of a weaker form of the excluded clique minor theorem of Robertson and Seymour, the following. Let t,r>0 be integers, and let R=49152t^{24}(40t^2+r). An r-wall is obtained from a (2r x r)-grid by deleting every odd vertical edge in every odd row and every even vertical edge in every even row, then deleting the two resulting vertices of degree one, and finally subdividing edges arbitrarily. The vertices of degree two that existed before the subdivision are called the pegs of the r-wall. Let G be a graph with no K_t min"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.6927","kind":"arxiv","version":3},"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":"2026-05-18T01:07:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u0QKzg/ZeV21BwTPrlMoS2C0oBilUAQqMPX/zDiDAuZLJPwLu7dohu/wlOahFk8QIndllEVt+mXdt4D7cjPNDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T06:18:39.089333Z"},"content_sha256":"0073e5437db6ad15668f88632e2857d084c2d4e33e33006c4160c16f0ea60a46","schema_version":"1.0","event_id":"sha256:0073e5437db6ad15668f88632e2857d084c2d4e33e33006c4160c16f0ea60a46"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/N73CGXFBRTLZ6SY2LBH5PUNFRI/bundle.json","state_url":"https://pith.science/pith/N73CGXFBRTLZ6SY2LBH5PUNFRI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/N73CGXFBRTLZ6SY2LBH5PUNFRI/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":"2026-06-03T06:18:39Z","links":{"resolver":"https://pith.science/pith/N73CGXFBRTLZ6SY2LBH5PUNFRI","bundle":"https://pith.science/pith/N73CGXFBRTLZ6SY2LBH5PUNFRI/bundle.json","state":"https://pith.science/pith/N73CGXFBRTLZ6SY2LBH5PUNFRI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/N73CGXFBRTLZ6SY2LBH5PUNFRI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:N73CGXFBRTLZ6SY2LBH5PUNFRI","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":"bd02e9385179152cc2dc42b5e97f08db0be250189e8536b73f89ee457035a4de","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-30T13:03:28Z","title_canon_sha256":"7f246e62fb43a7dbc87da838c470abb223afebce17cefbadb2f18248b93d73ac"},"schema_version":"1.0","source":{"id":"1207.6927","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.6927","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"arxiv_version","alias_value":"1207.6927v3","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.6927","created_at":"2026-05-18T01:07:57Z"},{"alias_kind":"pith_short_12","alias_value":"N73CGXFBRTLZ","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"N73CGXFBRTLZ6SY2","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"N73CGXFB","created_at":"2026-05-18T12:27:16Z"}],"graph_snapshots":[{"event_id":"sha256:0073e5437db6ad15668f88632e2857d084c2d4e33e33006c4160c16f0ea60a46","target":"graph","created_at":"2026-05-18T01:07:57Z","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":"We give an elementary and self-contained proof, and a numerical improvement, of a weaker form of the excluded clique minor theorem of Robertson and Seymour, the following. Let t,r>0 be integers, and let R=49152t^{24}(40t^2+r). An r-wall is obtained from a (2r x r)-grid by deleting every odd vertical edge in every odd row and every even vertical edge in every even row, then deleting the two resulting vertices of degree one, and finally subdividing edges arbitrarily. The vertices of degree two that existed before the subdivision are called the pegs of the r-wall. Let G be a graph with no K_t min","authors_text":"Ken-ichi Kawarabayashi, Paul Wollan, Robin Thomas","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-30T13:03:28Z","title":"A New Proof of the Flat Wall Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.6927","kind":"arxiv","version":3},"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:d2f3129d450a5df0f0e27f8ec61ef7a0986629f28e4a2eb977c8aaf740d6a9fd","target":"record","created_at":"2026-05-18T01:07:57Z","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":"bd02e9385179152cc2dc42b5e97f08db0be250189e8536b73f89ee457035a4de","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-07-30T13:03:28Z","title_canon_sha256":"7f246e62fb43a7dbc87da838c470abb223afebce17cefbadb2f18248b93d73ac"},"schema_version":"1.0","source":{"id":"1207.6927","kind":"arxiv","version":3}},"canonical_sha256":"6ff6235ca18cd79f4b1a584fd7d1a58a2d36cb5964dd015b58018b985913659d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6ff6235ca18cd79f4b1a584fd7d1a58a2d36cb5964dd015b58018b985913659d","first_computed_at":"2026-05-18T01:07:57.177718Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:07:57.177718Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FadKdjj/07vvCyR1jCCSci2fwaLA6lpNgN8Qy0Wkobqp9yfTdGBrW1cSfRxxM/Qv9pHdTsW6AwzmPCu9XPZ2AA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:07:57.178151Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.6927","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d2f3129d450a5df0f0e27f8ec61ef7a0986629f28e4a2eb977c8aaf740d6a9fd","sha256:0073e5437db6ad15668f88632e2857d084c2d4e33e33006c4160c16f0ea60a46"],"state_sha256":"e42baed4f9fa365325e34f9726288a0108f9605aac837ec69f742e15f05aa1e6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"faUimx41LVYpz7EpE5l9AZFYX1aTeK7E1n1Js8uFlnjesGqfPdpkHlnMwPLvk341BJuZAYjruY2h/EREE9xcAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T06:18:39.091153Z","bundle_sha256":"7cc46b19b95907f11710c6682c66de87e5fc845faf682b4d6c344bb13788459d"}}