{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:QOEYSZQAY76GHIVHSFPVXYQRS6","short_pith_number":"pith:QOEYSZQA","canonical_record":{"source":{"id":"1410.6128","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-22T18:25:05Z","cross_cats_sorted":["math.MG","math.OC"],"title_canon_sha256":"6659e8b670e48b611e63bc6c2d2badd860f5e92ef70152daca5a30c49325dbff","abstract_canon_sha256":"293e3caf65a9bcc7dd83adaec26019acb60aabcd5b4a30812dfda8b36e66bfc9"},"schema_version":"1.0"},"canonical_sha256":"8389896600c7fc63a2a7915f5be21197bd0aafadcca975735e77313f6a6e62cb","source":{"kind":"arxiv","id":"1410.6128","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.6128","created_at":"2026-05-18T02:39:32Z"},{"alias_kind":"arxiv_version","alias_value":"1410.6128v1","created_at":"2026-05-18T02:39:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.6128","created_at":"2026-05-18T02:39:32Z"},{"alias_kind":"pith_short_12","alias_value":"QOEYSZQAY76G","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"QOEYSZQAY76GHIVH","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"QOEYSZQA","created_at":"2026-05-18T12:28:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:QOEYSZQAY76GHIVHSFPVXYQRS6","target":"record","payload":{"canonical_record":{"source":{"id":"1410.6128","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-22T18:25:05Z","cross_cats_sorted":["math.MG","math.OC"],"title_canon_sha256":"6659e8b670e48b611e63bc6c2d2badd860f5e92ef70152daca5a30c49325dbff","abstract_canon_sha256":"293e3caf65a9bcc7dd83adaec26019acb60aabcd5b4a30812dfda8b36e66bfc9"},"schema_version":"1.0"},"canonical_sha256":"8389896600c7fc63a2a7915f5be21197bd0aafadcca975735e77313f6a6e62cb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:39:32.547990Z","signature_b64":"RKPvhMxzrdlNSWQlj7dkXMlInVH1QHRUCAYXpD62vq7TyDf7/J4YhuZG27tM6jMh7o7qaAT+H5Z3B/DwIHhjDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8389896600c7fc63a2a7915f5be21197bd0aafadcca975735e77313f6a6e62cb","last_reissued_at":"2026-05-18T02:39:32.547422Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:39:32.547422Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1410.6128","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":"2026-05-18T02:39:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QCBJ5wiNPMe/CjX3uCDteJO6UVuMTnA//z37tYKfC24htfe6JzBZTr9g1UK0TuLYdw76KEtBYUIgtYBcAxTBBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T08:07:14.281602Z"},"content_sha256":"f8e7b9dddb6932441124e8f14f97d64b5b87be79a2730e0164f5152fe72a8c67","schema_version":"1.0","event_id":"sha256:f8e7b9dddb6932441124e8f14f97d64b5b87be79a2730e0164f5152fe72a8c67"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:QOEYSZQAY76GHIVHSFPVXYQRS6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A sharp quantitative version of Hales' isoperimetric honeycomb theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.OC"],"primary_cat":"math.AP","authors_text":"Francesco Maggi, Marco Caroccia","submitted_at":"2014-10-22T18:25:05Z","abstract_excerpt":"We prove a sharp quantitative version of Hales' isoperimetric honeycomb theorem by exploiting a quantitative isoperimetric inequality for polygons and an improved convergence theorem for planar bubble clusters. Further applications include the description of isoperimetric tilings of the torus with respect to almost unit-area constraints or with respect to almost flat Riemannian metrics."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.6128","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":"2026-05-18T02:39:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3etD7XbtCyA8Rb4nEUYULjQFvEFnyG+RrKGMasLRhpF+fkCSDSNqxR/y/Gxenm72qO9/JeuM2Z4+dL+gUrNkCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T08:07:14.281943Z"},"content_sha256":"fd73e63187e830a76cd15fde7963991423eeaf09e66afc17362af23f09e8bae3","schema_version":"1.0","event_id":"sha256:fd73e63187e830a76cd15fde7963991423eeaf09e66afc17362af23f09e8bae3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QOEYSZQAY76GHIVHSFPVXYQRS6/bundle.json","state_url":"https://pith.science/pith/QOEYSZQAY76GHIVHSFPVXYQRS6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QOEYSZQAY76GHIVHSFPVXYQRS6/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":"2026-06-20T08:07:14Z","links":{"resolver":"https://pith.science/pith/QOEYSZQAY76GHIVHSFPVXYQRS6","bundle":"https://pith.science/pith/QOEYSZQAY76GHIVHSFPVXYQRS6/bundle.json","state":"https://pith.science/pith/QOEYSZQAY76GHIVHSFPVXYQRS6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QOEYSZQAY76GHIVHSFPVXYQRS6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:QOEYSZQAY76GHIVHSFPVXYQRS6","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":"293e3caf65a9bcc7dd83adaec26019acb60aabcd5b4a30812dfda8b36e66bfc9","cross_cats_sorted":["math.MG","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-22T18:25:05Z","title_canon_sha256":"6659e8b670e48b611e63bc6c2d2badd860f5e92ef70152daca5a30c49325dbff"},"schema_version":"1.0","source":{"id":"1410.6128","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1410.6128","created_at":"2026-05-18T02:39:32Z"},{"alias_kind":"arxiv_version","alias_value":"1410.6128v1","created_at":"2026-05-18T02:39:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.6128","created_at":"2026-05-18T02:39:32Z"},{"alias_kind":"pith_short_12","alias_value":"QOEYSZQAY76G","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"QOEYSZQAY76GHIVH","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"QOEYSZQA","created_at":"2026-05-18T12:28:46Z"}],"graph_snapshots":[{"event_id":"sha256:fd73e63187e830a76cd15fde7963991423eeaf09e66afc17362af23f09e8bae3","target":"graph","created_at":"2026-05-18T02:39:32Z","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 prove a sharp quantitative version of Hales' isoperimetric honeycomb theorem by exploiting a quantitative isoperimetric inequality for polygons and an improved convergence theorem for planar bubble clusters. Further applications include the description of isoperimetric tilings of the torus with respect to almost unit-area constraints or with respect to almost flat Riemannian metrics.","authors_text":"Francesco Maggi, Marco Caroccia","cross_cats":["math.MG","math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-22T18:25:05Z","title":"A sharp quantitative version of Hales' isoperimetric honeycomb theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.6128","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:f8e7b9dddb6932441124e8f14f97d64b5b87be79a2730e0164f5152fe72a8c67","target":"record","created_at":"2026-05-18T02:39:32Z","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":"293e3caf65a9bcc7dd83adaec26019acb60aabcd5b4a30812dfda8b36e66bfc9","cross_cats_sorted":["math.MG","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-22T18:25:05Z","title_canon_sha256":"6659e8b670e48b611e63bc6c2d2badd860f5e92ef70152daca5a30c49325dbff"},"schema_version":"1.0","source":{"id":"1410.6128","kind":"arxiv","version":1}},"canonical_sha256":"8389896600c7fc63a2a7915f5be21197bd0aafadcca975735e77313f6a6e62cb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8389896600c7fc63a2a7915f5be21197bd0aafadcca975735e77313f6a6e62cb","first_computed_at":"2026-05-18T02:39:32.547422Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:39:32.547422Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RKPvhMxzrdlNSWQlj7dkXMlInVH1QHRUCAYXpD62vq7TyDf7/J4YhuZG27tM6jMh7o7qaAT+H5Z3B/DwIHhjDg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:39:32.547990Z","signed_message":"canonical_sha256_bytes"},"source_id":"1410.6128","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f8e7b9dddb6932441124e8f14f97d64b5b87be79a2730e0164f5152fe72a8c67","sha256:fd73e63187e830a76cd15fde7963991423eeaf09e66afc17362af23f09e8bae3"],"state_sha256":"e90f832a32b6b29a6533fdc44744a1d5e013a947ee0ab84f407dd2a98decf446"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kEr7ElS6islRO1oZaYLbxVFU9fSnGNG69IOP0xcA3cD3WaQMAe4jHu3aWoyTe6XneNauq5A7Sx9+8pjZZpyjCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T08:07:14.283877Z","bundle_sha256":"97ff349475c27ec1cc489c9849802ea9bbcac5693392ec91fd9156b180692bef"}}