{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:U7UEE3PSRQJQTS4PLFB7ZPNZHH","short_pith_number":"pith:U7UEE3PS","canonical_record":{"source":{"id":"1905.03008","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2019-05-08T11:16:04Z","cross_cats_sorted":["cs.DM","math.CO"],"title_canon_sha256":"bc25a41a260d9378d9fa80c03af92352cabc9c7a54e8bd3befa1d47abefc8eaf","abstract_canon_sha256":"6db49dd8a1692704ba5a11bb987541d4bafed105de4cd266af0fe303c4da6273"},"schema_version":"1.0"},"canonical_sha256":"a7e8426df28c1309cb8f5943fcbdb939e6fb8af807eb6f905ce12fb01bd3e04c","source":{"kind":"arxiv","id":"1905.03008","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1905.03008","created_at":"2026-05-17T23:46:40Z"},{"alias_kind":"arxiv_version","alias_value":"1905.03008v2","created_at":"2026-05-17T23:46:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.03008","created_at":"2026-05-17T23:46:40Z"},{"alias_kind":"pith_short_12","alias_value":"U7UEE3PSRQJQ","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"U7UEE3PSRQJQTS4P","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"U7UEE3PS","created_at":"2026-05-18T12:33:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:U7UEE3PSRQJQTS4PLFB7ZPNZHH","target":"record","payload":{"canonical_record":{"source":{"id":"1905.03008","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2019-05-08T11:16:04Z","cross_cats_sorted":["cs.DM","math.CO"],"title_canon_sha256":"bc25a41a260d9378d9fa80c03af92352cabc9c7a54e8bd3befa1d47abefc8eaf","abstract_canon_sha256":"6db49dd8a1692704ba5a11bb987541d4bafed105de4cd266af0fe303c4da6273"},"schema_version":"1.0"},"canonical_sha256":"a7e8426df28c1309cb8f5943fcbdb939e6fb8af807eb6f905ce12fb01bd3e04c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:46:40.646411Z","signature_b64":"bhAJqFlRtltVaNkfU0VOxtoB2z8mcCwgpD5sgeM+KTDxsHXkuTLfMd00lgUp2Jn2ves9fr/nnVJkWh8ipZSiDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a7e8426df28c1309cb8f5943fcbdb939e6fb8af807eb6f905ce12fb01bd3e04c","last_reissued_at":"2026-05-17T23:46:40.645632Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:46:40.645632Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1905.03008","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":"2026-05-17T23:46:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qb3SgkLu+W4XqboXwcYcQ1fLOjdwbxUzvOjUwSIMAX799TwKCG9agojj0YVCEfsXQXdu5cTV4i9FtePxrQJ0Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T19:21:29.952227Z"},"content_sha256":"15341b4d9f95cc38dfe15e82cb4b684c4810752c229d8b6ad8daf81968415958","schema_version":"1.0","event_id":"sha256:15341b4d9f95cc38dfe15e82cb4b684c4810752c229d8b6ad8daf81968415958"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:U7UEE3PSRQJQTS4PLFB7ZPNZHH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Walk refinement, walk logic, and the iteration number of the Weisfeiler-Leman algorithm","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"cs.LO","authors_text":"Ilia Ponomarenko, Moritz Lichter, Pascal Schweitzer","submitted_at":"2019-05-08T11:16:04Z","abstract_excerpt":"We show that the 2-dimensional Weisfeiler-Leman algorithm stabilizes n-vertex graphs after at most O(n log n) iterations. This implies that if such graphs are distinguishable in 3-variable first order logic with counting, then they can also be distinguished in this logic by a formula of quantifier depth at most O(n log n).\n  For this we exploit a new refinement based on counting walks and argue that its iteration number differs from the classic Weisfeiler-Leman refinement by at most a logarithmic factor. We then prove matching linear upper and lower bounds on the number of iterations of the wa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.03008","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":""},"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":"2026-05-17T23:46:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9mrh83gvffL85/Kasjxp7eKes49po71hU74+KHynBQKOHqpfSiN6Lc4ot28KcW6+0ocQqSvlouh9VF74F/DMDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T19:21:29.952592Z"},"content_sha256":"0ca564293f4df4862d0649e5cd0537693a5488cf3608a542ddd1f81026b91ca2","schema_version":"1.0","event_id":"sha256:0ca564293f4df4862d0649e5cd0537693a5488cf3608a542ddd1f81026b91ca2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U7UEE3PSRQJQTS4PLFB7ZPNZHH/bundle.json","state_url":"https://pith.science/pith/U7UEE3PSRQJQTS4PLFB7ZPNZHH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U7UEE3PSRQJQTS4PLFB7ZPNZHH/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":"2026-06-21T19:21:29Z","links":{"resolver":"https://pith.science/pith/U7UEE3PSRQJQTS4PLFB7ZPNZHH","bundle":"https://pith.science/pith/U7UEE3PSRQJQTS4PLFB7ZPNZHH/bundle.json","state":"https://pith.science/pith/U7UEE3PSRQJQTS4PLFB7ZPNZHH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U7UEE3PSRQJQTS4PLFB7ZPNZHH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:U7UEE3PSRQJQTS4PLFB7ZPNZHH","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":"6db49dd8a1692704ba5a11bb987541d4bafed105de4cd266af0fe303c4da6273","cross_cats_sorted":["cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2019-05-08T11:16:04Z","title_canon_sha256":"bc25a41a260d9378d9fa80c03af92352cabc9c7a54e8bd3befa1d47abefc8eaf"},"schema_version":"1.0","source":{"id":"1905.03008","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1905.03008","created_at":"2026-05-17T23:46:40Z"},{"alias_kind":"arxiv_version","alias_value":"1905.03008v2","created_at":"2026-05-17T23:46:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.03008","created_at":"2026-05-17T23:46:40Z"},{"alias_kind":"pith_short_12","alias_value":"U7UEE3PSRQJQ","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"U7UEE3PSRQJQTS4P","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"U7UEE3PS","created_at":"2026-05-18T12:33:30Z"}],"graph_snapshots":[{"event_id":"sha256:0ca564293f4df4862d0649e5cd0537693a5488cf3608a542ddd1f81026b91ca2","target":"graph","created_at":"2026-05-17T23:46:40Z","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 show that the 2-dimensional Weisfeiler-Leman algorithm stabilizes n-vertex graphs after at most O(n log n) iterations. This implies that if such graphs are distinguishable in 3-variable first order logic with counting, then they can also be distinguished in this logic by a formula of quantifier depth at most O(n log n).\n  For this we exploit a new refinement based on counting walks and argue that its iteration number differs from the classic Weisfeiler-Leman refinement by at most a logarithmic factor. We then prove matching linear upper and lower bounds on the number of iterations of the wa","authors_text":"Ilia Ponomarenko, Moritz Lichter, Pascal Schweitzer","cross_cats":["cs.DM","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2019-05-08T11:16:04Z","title":"Walk refinement, walk logic, and the iteration number of the Weisfeiler-Leman algorithm"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.03008","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":"sha256:15341b4d9f95cc38dfe15e82cb4b684c4810752c229d8b6ad8daf81968415958","target":"record","created_at":"2026-05-17T23:46:40Z","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":"6db49dd8a1692704ba5a11bb987541d4bafed105de4cd266af0fe303c4da6273","cross_cats_sorted":["cs.DM","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2019-05-08T11:16:04Z","title_canon_sha256":"bc25a41a260d9378d9fa80c03af92352cabc9c7a54e8bd3befa1d47abefc8eaf"},"schema_version":"1.0","source":{"id":"1905.03008","kind":"arxiv","version":2}},"canonical_sha256":"a7e8426df28c1309cb8f5943fcbdb939e6fb8af807eb6f905ce12fb01bd3e04c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a7e8426df28c1309cb8f5943fcbdb939e6fb8af807eb6f905ce12fb01bd3e04c","first_computed_at":"2026-05-17T23:46:40.645632Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:46:40.645632Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bhAJqFlRtltVaNkfU0VOxtoB2z8mcCwgpD5sgeM+KTDxsHXkuTLfMd00lgUp2Jn2ves9fr/nnVJkWh8ipZSiDA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:46:40.646411Z","signed_message":"canonical_sha256_bytes"},"source_id":"1905.03008","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:15341b4d9f95cc38dfe15e82cb4b684c4810752c229d8b6ad8daf81968415958","sha256:0ca564293f4df4862d0649e5cd0537693a5488cf3608a542ddd1f81026b91ca2"],"state_sha256":"e02d295a8df2be9550aedab3a9bcebe193761984213723e0691999d0919d9ddf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Zz3WLuWyQyOm+W75/zhjkaGLPULZVrHYtZeIhRTpfNyyDqgGFOppKvRBmktIDPrRjfT9/RRpAQGoiHKUhBHiCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T19:21:29.954553Z","bundle_sha256":"2c2dc645200a0795a6d81c4502bc1baf4b61ab3455551103dcd981b38f33eeaf"}}