{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:UUJN7SKTNRLS5IK5F3P2H7RA5M","short_pith_number":"pith:UUJN7SKT","canonical_record":{"source":{"id":"1504.04821","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-19T11:05:08Z","cross_cats_sorted":[],"title_canon_sha256":"552127ad72400d9a606cbebe1bd403e692091e66bd3c855c3e34746ee64a0690","abstract_canon_sha256":"a21a9112c91c18acc0e381dc9d0e5f353ebc59d3d9df75d6a8e4887758fe5f2b"},"schema_version":"1.0"},"canonical_sha256":"a512dfc9536c572ea15d2edfa3fe20eb22db92e18984c24578412633f519a059","source":{"kind":"arxiv","id":"1504.04821","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.04821","created_at":"2026-05-18T02:18:24Z"},{"alias_kind":"arxiv_version","alias_value":"1504.04821v1","created_at":"2026-05-18T02:18:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.04821","created_at":"2026-05-18T02:18:24Z"},{"alias_kind":"pith_short_12","alias_value":"UUJN7SKTNRLS","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"UUJN7SKTNRLS5IK5","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"UUJN7SKT","created_at":"2026-05-18T12:29:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:UUJN7SKTNRLS5IK5F3P2H7RA5M","target":"record","payload":{"canonical_record":{"source":{"id":"1504.04821","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-19T11:05:08Z","cross_cats_sorted":[],"title_canon_sha256":"552127ad72400d9a606cbebe1bd403e692091e66bd3c855c3e34746ee64a0690","abstract_canon_sha256":"a21a9112c91c18acc0e381dc9d0e5f353ebc59d3d9df75d6a8e4887758fe5f2b"},"schema_version":"1.0"},"canonical_sha256":"a512dfc9536c572ea15d2edfa3fe20eb22db92e18984c24578412633f519a059","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:24.658508Z","signature_b64":"qV+6h/J8JFP180+DpaooZAmr8PjkD9OMP0aMWEZQe82rM5CzygM4zYaUDSzs13sGhYE84Whn9txC6MX3/QHzBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a512dfc9536c572ea15d2edfa3fe20eb22db92e18984c24578412633f519a059","last_reissued_at":"2026-05-18T02:18:24.657821Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:24.657821Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.04821","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":"2026-05-18T02:18:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+g/WOIH/xCLEwt/D7fD34UCjldd/yqnUzFf0+hhpsph4J5hEecyYME6ni9INucMaW4yx3d6T/5DL/1gBWVGBAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T23:41:52.115699Z"},"content_sha256":"5618ff7f7683adfb3fa2b6ec4f52e0c087a0371a79c0ebcca646b33290214164","schema_version":"1.0","event_id":"sha256:5618ff7f7683adfb3fa2b6ec4f52e0c087a0371a79c0ebcca646b33290214164"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:UUJN7SKTNRLS5IK5F3P2H7RA5M","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Strongly sublinear separators and polynomial expansion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sergey Norin, Zdenek Dvorak","submitted_at":"2015-04-19T11:05:08Z","abstract_excerpt":"A result of Plotkin, Rao, and Smith implies that graphs with polynomial expansion have strongly sublinear separators. We prove a converse of this result showing that hereditary classes of graphs with strongly sublinear separators have polynomial expansion. This confirms a conjecture of the first author."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04821","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":"2026-05-18T02:18:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"y5PklF5jT+OZxQAxjq64fLb+ind1BlNLN+ImhU2izMGtX1Ty3juV8ve6UiLwL/Jun24u5KZbDiXz3y2nSmoHDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T23:41:52.116166Z"},"content_sha256":"c7bab99ab062d06b3bd29f67cf7c3eda3b0e4cbe90fb7b8e137d10e507d26197","schema_version":"1.0","event_id":"sha256:c7bab99ab062d06b3bd29f67cf7c3eda3b0e4cbe90fb7b8e137d10e507d26197"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UUJN7SKTNRLS5IK5F3P2H7RA5M/bundle.json","state_url":"https://pith.science/pith/UUJN7SKTNRLS5IK5F3P2H7RA5M/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UUJN7SKTNRLS5IK5F3P2H7RA5M/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":"2026-05-25T23:41:52Z","links":{"resolver":"https://pith.science/pith/UUJN7SKTNRLS5IK5F3P2H7RA5M","bundle":"https://pith.science/pith/UUJN7SKTNRLS5IK5F3P2H7RA5M/bundle.json","state":"https://pith.science/pith/UUJN7SKTNRLS5IK5F3P2H7RA5M/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UUJN7SKTNRLS5IK5F3P2H7RA5M/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:UUJN7SKTNRLS5IK5F3P2H7RA5M","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":"a21a9112c91c18acc0e381dc9d0e5f353ebc59d3d9df75d6a8e4887758fe5f2b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-19T11:05:08Z","title_canon_sha256":"552127ad72400d9a606cbebe1bd403e692091e66bd3c855c3e34746ee64a0690"},"schema_version":"1.0","source":{"id":"1504.04821","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.04821","created_at":"2026-05-18T02:18:24Z"},{"alias_kind":"arxiv_version","alias_value":"1504.04821v1","created_at":"2026-05-18T02:18:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.04821","created_at":"2026-05-18T02:18:24Z"},{"alias_kind":"pith_short_12","alias_value":"UUJN7SKTNRLS","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"UUJN7SKTNRLS5IK5","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"UUJN7SKT","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:c7bab99ab062d06b3bd29f67cf7c3eda3b0e4cbe90fb7b8e137d10e507d26197","target":"graph","created_at":"2026-05-18T02:18:24Z","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":"A result of Plotkin, Rao, and Smith implies that graphs with polynomial expansion have strongly sublinear separators. We prove a converse of this result showing that hereditary classes of graphs with strongly sublinear separators have polynomial expansion. This confirms a conjecture of the first author.","authors_text":"Sergey Norin, Zdenek Dvorak","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-19T11:05:08Z","title":"Strongly sublinear separators and polynomial expansion"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.04821","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":"sha256:5618ff7f7683adfb3fa2b6ec4f52e0c087a0371a79c0ebcca646b33290214164","target":"record","created_at":"2026-05-18T02:18:24Z","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":"a21a9112c91c18acc0e381dc9d0e5f353ebc59d3d9df75d6a8e4887758fe5f2b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-19T11:05:08Z","title_canon_sha256":"552127ad72400d9a606cbebe1bd403e692091e66bd3c855c3e34746ee64a0690"},"schema_version":"1.0","source":{"id":"1504.04821","kind":"arxiv","version":1}},"canonical_sha256":"a512dfc9536c572ea15d2edfa3fe20eb22db92e18984c24578412633f519a059","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a512dfc9536c572ea15d2edfa3fe20eb22db92e18984c24578412633f519a059","first_computed_at":"2026-05-18T02:18:24.657821Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:18:24.657821Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qV+6h/J8JFP180+DpaooZAmr8PjkD9OMP0aMWEZQe82rM5CzygM4zYaUDSzs13sGhYE84Whn9txC6MX3/QHzBg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:18:24.658508Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.04821","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5618ff7f7683adfb3fa2b6ec4f52e0c087a0371a79c0ebcca646b33290214164","sha256:c7bab99ab062d06b3bd29f67cf7c3eda3b0e4cbe90fb7b8e137d10e507d26197"],"state_sha256":"203b4e91fff49383ff2609ee837145b1ca087e40ea62fd8518ffe11397f2e02d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c2JX52c1O+JOdgldq0kExKpuDwD025T7PhAHcg8Eq7997NJI4TwktPQ1RigGrHkp1+Wq1MSXl6QYdPIvLuMWDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T23:41:52.119030Z","bundle_sha256":"1f5ffb8fb9697b6ed470539579ec27e88cec6d559faceef9177f419b1e2089aa"}}