{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:KGG6MCMEJRZ4KN7JRYCU47DYNC","short_pith_number":"pith:KGG6MCME","canonical_record":{"source":{"id":"0908.1830","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-08-13T05:09:35Z","cross_cats_sorted":["math.CO","math.PR"],"title_canon_sha256":"5fde2193e694498d2480e85c5883ac7ab7cf24cb94b1232430486d6de231d517","abstract_canon_sha256":"ad3d861e3b86e83fb0af27e5918000169ba25993df2f65af3e6cfccdb8a7c6d8"},"schema_version":"1.0"},"canonical_sha256":"518de609844c73c537e98e054e7c786897fa82e2b6094118300b03cef6d3e4e9","source":{"kind":"arxiv","id":"0908.1830","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0908.1830","created_at":"2026-05-18T04:38:14Z"},{"alias_kind":"arxiv_version","alias_value":"0908.1830v3","created_at":"2026-05-18T04:38:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.1830","created_at":"2026-05-18T04:38:14Z"},{"alias_kind":"pith_short_12","alias_value":"KGG6MCMEJRZ4","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"KGG6MCMEJRZ4KN7J","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"KGG6MCME","created_at":"2026-05-18T12:26:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:KGG6MCMEJRZ4KN7JRYCU47DYNC","target":"record","payload":{"canonical_record":{"source":{"id":"0908.1830","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-08-13T05:09:35Z","cross_cats_sorted":["math.CO","math.PR"],"title_canon_sha256":"5fde2193e694498d2480e85c5883ac7ab7cf24cb94b1232430486d6de231d517","abstract_canon_sha256":"ad3d861e3b86e83fb0af27e5918000169ba25993df2f65af3e6cfccdb8a7c6d8"},"schema_version":"1.0"},"canonical_sha256":"518de609844c73c537e98e054e7c786897fa82e2b6094118300b03cef6d3e4e9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:38:14.402056Z","signature_b64":"y+2QRpuvMxlFNKHpb4tmdZx7HpRhEku0nbf9cStljvG/PGisGzMClJJDBRWjCVrmHO0bWBGKuBNhgFXfbnA1BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"518de609844c73c537e98e054e7c786897fa82e2b6094118300b03cef6d3e4e9","last_reissued_at":"2026-05-18T04:38:14.401581Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:38:14.401581Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0908.1830","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":"2026-05-18T04:38:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FyfcqG378fo0F9xoyQ14f77gXadJu9EjvvyF5H6MMEO80Xg4OqJYs7tx6YW/yadt77ClpNgizwzG6joVMtuLBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T07:32:21.598748Z"},"content_sha256":"5e30a4aad0b5784c132860fbb89533bcf7dd42e03c95d8ad72cd6a079121b3ff","schema_version":"1.0","event_id":"sha256:5e30a4aad0b5784c132860fbb89533bcf7dd42e03c95d8ad72cd6a079121b3ff"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:KGG6MCMEJRZ4KN7JRYCU47DYNC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sparse stable configurations of hard discs in a box","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.PR"],"primary_cat":"math.MG","authors_text":"Matthew Kahle","submitted_at":"2009-08-13T05:09:35Z","abstract_excerpt":"We construct stable configurations of n overlapping discs of radius r in a unit square, with r = O(1/n). By a result of Diaconis, Lebeau, and Michel, this result is best possible, up to a constant factor. A consequence is that the Metropolis algorithm, a well-studied Markov chain on the hardcore model, is not irreducible in this range of parameters."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.1830","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":"2026-05-18T04:38:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IFIyoj+LJfnyKFmIganbRKV+6LYUeXR2y07+lOKEa4jRqE8Lza2J+6nsGTSUP+Y5Dl2R5ckDWQEkciQP3y7tBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T07:32:21.599160Z"},"content_sha256":"951e594dc9ddf777de8c080874534b445dca8a9b7137f059bd2f63a2a20dedef","schema_version":"1.0","event_id":"sha256:951e594dc9ddf777de8c080874534b445dca8a9b7137f059bd2f63a2a20dedef"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KGG6MCMEJRZ4KN7JRYCU47DYNC/bundle.json","state_url":"https://pith.science/pith/KGG6MCMEJRZ4KN7JRYCU47DYNC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KGG6MCMEJRZ4KN7JRYCU47DYNC/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":"2026-06-02T07:32:21Z","links":{"resolver":"https://pith.science/pith/KGG6MCMEJRZ4KN7JRYCU47DYNC","bundle":"https://pith.science/pith/KGG6MCMEJRZ4KN7JRYCU47DYNC/bundle.json","state":"https://pith.science/pith/KGG6MCMEJRZ4KN7JRYCU47DYNC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KGG6MCMEJRZ4KN7JRYCU47DYNC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:KGG6MCMEJRZ4KN7JRYCU47DYNC","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":"ad3d861e3b86e83fb0af27e5918000169ba25993df2f65af3e6cfccdb8a7c6d8","cross_cats_sorted":["math.CO","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-08-13T05:09:35Z","title_canon_sha256":"5fde2193e694498d2480e85c5883ac7ab7cf24cb94b1232430486d6de231d517"},"schema_version":"1.0","source":{"id":"0908.1830","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0908.1830","created_at":"2026-05-18T04:38:14Z"},{"alias_kind":"arxiv_version","alias_value":"0908.1830v3","created_at":"2026-05-18T04:38:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.1830","created_at":"2026-05-18T04:38:14Z"},{"alias_kind":"pith_short_12","alias_value":"KGG6MCMEJRZ4","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"KGG6MCMEJRZ4KN7J","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"KGG6MCME","created_at":"2026-05-18T12:26:00Z"}],"graph_snapshots":[{"event_id":"sha256:951e594dc9ddf777de8c080874534b445dca8a9b7137f059bd2f63a2a20dedef","target":"graph","created_at":"2026-05-18T04:38:14Z","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 construct stable configurations of n overlapping discs of radius r in a unit square, with r = O(1/n). By a result of Diaconis, Lebeau, and Michel, this result is best possible, up to a constant factor. A consequence is that the Metropolis algorithm, a well-studied Markov chain on the hardcore model, is not irreducible in this range of parameters.","authors_text":"Matthew Kahle","cross_cats":["math.CO","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-08-13T05:09:35Z","title":"Sparse stable configurations of hard discs in a box"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.1830","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":"sha256:5e30a4aad0b5784c132860fbb89533bcf7dd42e03c95d8ad72cd6a079121b3ff","target":"record","created_at":"2026-05-18T04:38:14Z","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":"ad3d861e3b86e83fb0af27e5918000169ba25993df2f65af3e6cfccdb8a7c6d8","cross_cats_sorted":["math.CO","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2009-08-13T05:09:35Z","title_canon_sha256":"5fde2193e694498d2480e85c5883ac7ab7cf24cb94b1232430486d6de231d517"},"schema_version":"1.0","source":{"id":"0908.1830","kind":"arxiv","version":3}},"canonical_sha256":"518de609844c73c537e98e054e7c786897fa82e2b6094118300b03cef6d3e4e9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"518de609844c73c537e98e054e7c786897fa82e2b6094118300b03cef6d3e4e9","first_computed_at":"2026-05-18T04:38:14.401581Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:38:14.401581Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"y+2QRpuvMxlFNKHpb4tmdZx7HpRhEku0nbf9cStljvG/PGisGzMClJJDBRWjCVrmHO0bWBGKuBNhgFXfbnA1BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:38:14.402056Z","signed_message":"canonical_sha256_bytes"},"source_id":"0908.1830","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5e30a4aad0b5784c132860fbb89533bcf7dd42e03c95d8ad72cd6a079121b3ff","sha256:951e594dc9ddf777de8c080874534b445dca8a9b7137f059bd2f63a2a20dedef"],"state_sha256":"26f56846df809905d69bd9d930922ce2c5de879177808062293908510ad9510e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mcf6BJDC4qLhb1H+kB+tjyzSIPw8Q8iOIuxESmp0bsW40KbfgG85/BSerHZJBvo+MiyJVvCpAfuRdl54gMY8AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T07:32:21.601455Z","bundle_sha256":"28e50967d2bd7747e47e380765f0c12f9bea5cedbe8ce50cbd29ce1dc7417c50"}}