{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:FCP76U5KSGKSD3OIDTGKCOYLHF","short_pith_number":"pith:FCP76U5K","canonical_record":{"source":{"id":"1010.4968","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-10-24T14:54:50Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ce5975c74a48c878be1712bb5b4ef08b8bb367b44727eef8a7704f7a2881f18f","abstract_canon_sha256":"67d00712e66efaa0d7af8bf46fe917d5c5554a86057ed9a53d71768f3e37c049"},"schema_version":"1.0"},"canonical_sha256":"289fff53aa919521edc81ccca13b0b394c2f24536318764536901ac5f280c95d","source":{"kind":"arxiv","id":"1010.4968","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.4968","created_at":"2026-05-18T03:48:23Z"},{"alias_kind":"arxiv_version","alias_value":"1010.4968v3","created_at":"2026-05-18T03:48:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.4968","created_at":"2026-05-18T03:48:23Z"},{"alias_kind":"pith_short_12","alias_value":"FCP76U5KSGKS","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"FCP76U5KSGKSD3OI","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"FCP76U5K","created_at":"2026-05-18T12:26:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:FCP76U5KSGKSD3OIDTGKCOYLHF","target":"record","payload":{"canonical_record":{"source":{"id":"1010.4968","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-10-24T14:54:50Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ce5975c74a48c878be1712bb5b4ef08b8bb367b44727eef8a7704f7a2881f18f","abstract_canon_sha256":"67d00712e66efaa0d7af8bf46fe917d5c5554a86057ed9a53d71768f3e37c049"},"schema_version":"1.0"},"canonical_sha256":"289fff53aa919521edc81ccca13b0b394c2f24536318764536901ac5f280c95d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:48:23.588507Z","signature_b64":"Bfwo2j3TiVo6MA3YexOQ21IhiczfVsfv26Nt2Nz+h3oQfvU1oRu0pBGwyKwrr0u5+P+TuLTe69Wp0VuiHYIeAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"289fff53aa919521edc81ccca13b0b394c2f24536318764536901ac5f280c95d","last_reissued_at":"2026-05-18T03:48:23.588076Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:48:23.588076Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1010.4968","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":"2026-05-18T03:48:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QzhHPhwUguAG/D2e35EBo81HfJQOh54PeJfHBlMO73lx840hXZlRLXxlTIU+w5EaVzvOM2zuorb0ZeC5A+irDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T21:30:50.252345Z"},"content_sha256":"564333cacd5c452d58f33664e9d5817e8293ae09e1e0133d6e52828cc8315265","schema_version":"1.0","event_id":"sha256:564333cacd5c452d58f33664e9d5817e8293ae09e1e0133d6e52828cc8315265"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:FCP76U5KSGKSD3OIDTGKCOYLHF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Applications of Three Dimensional Extremal Length, I: Tiling of a Topological Cube","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GT","authors_text":"Sa'ar Hersonsky","submitted_at":"2010-10-24T14:54:50Z","abstract_excerpt":"Given a triangulation of a closed topological cube, we show that (under some technical condition) there is an essentially unique tiling of a rectangular parallelepiped by cubes, indexed by the vertices of the triangulation. Moreover, i - the combinatorics is preserved, and ii- the boundary is preserved: vertices corresponding to the cubes at the corners of the rectangular parallelepiped are at the corners of the topological cube. Also, the sizes of the cubes are obtained as a solution of a variational problem which is a discrete version of the notion of extremal length in three dimensional Euc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.4968","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":"2026-05-18T03:48:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1cdbJzkDfzNO2yQ0ckSP1db4DuFD8vZx+MO19vQp/yqo1Yklk8dJkCxjRnx+8xpnQRISNQ/8cUpV0yLLUAZDBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T21:30:50.252693Z"},"content_sha256":"83f41ad23cb2b503e90787060f1d7d29ff084827fe65bc673b332d153a56ec36","schema_version":"1.0","event_id":"sha256:83f41ad23cb2b503e90787060f1d7d29ff084827fe65bc673b332d153a56ec36"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FCP76U5KSGKSD3OIDTGKCOYLHF/bundle.json","state_url":"https://pith.science/pith/FCP76U5KSGKSD3OIDTGKCOYLHF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FCP76U5KSGKSD3OIDTGKCOYLHF/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":"2026-06-24T21:30:50Z","links":{"resolver":"https://pith.science/pith/FCP76U5KSGKSD3OIDTGKCOYLHF","bundle":"https://pith.science/pith/FCP76U5KSGKSD3OIDTGKCOYLHF/bundle.json","state":"https://pith.science/pith/FCP76U5KSGKSD3OIDTGKCOYLHF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FCP76U5KSGKSD3OIDTGKCOYLHF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:FCP76U5KSGKSD3OIDTGKCOYLHF","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":"67d00712e66efaa0d7af8bf46fe917d5c5554a86057ed9a53d71768f3e37c049","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-10-24T14:54:50Z","title_canon_sha256":"ce5975c74a48c878be1712bb5b4ef08b8bb367b44727eef8a7704f7a2881f18f"},"schema_version":"1.0","source":{"id":"1010.4968","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.4968","created_at":"2026-05-18T03:48:23Z"},{"alias_kind":"arxiv_version","alias_value":"1010.4968v3","created_at":"2026-05-18T03:48:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.4968","created_at":"2026-05-18T03:48:23Z"},{"alias_kind":"pith_short_12","alias_value":"FCP76U5KSGKS","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"FCP76U5KSGKSD3OI","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"FCP76U5K","created_at":"2026-05-18T12:26:06Z"}],"graph_snapshots":[{"event_id":"sha256:83f41ad23cb2b503e90787060f1d7d29ff084827fe65bc673b332d153a56ec36","target":"graph","created_at":"2026-05-18T03:48:23Z","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":"Given a triangulation of a closed topological cube, we show that (under some technical condition) there is an essentially unique tiling of a rectangular parallelepiped by cubes, indexed by the vertices of the triangulation. Moreover, i - the combinatorics is preserved, and ii- the boundary is preserved: vertices corresponding to the cubes at the corners of the rectangular parallelepiped are at the corners of the topological cube. Also, the sizes of the cubes are obtained as a solution of a variational problem which is a discrete version of the notion of extremal length in three dimensional Euc","authors_text":"Sa'ar Hersonsky","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-10-24T14:54:50Z","title":"Applications of Three Dimensional Extremal Length, I: Tiling of a Topological Cube"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.4968","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":"sha256:564333cacd5c452d58f33664e9d5817e8293ae09e1e0133d6e52828cc8315265","target":"record","created_at":"2026-05-18T03:48:23Z","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":"67d00712e66efaa0d7af8bf46fe917d5c5554a86057ed9a53d71768f3e37c049","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-10-24T14:54:50Z","title_canon_sha256":"ce5975c74a48c878be1712bb5b4ef08b8bb367b44727eef8a7704f7a2881f18f"},"schema_version":"1.0","source":{"id":"1010.4968","kind":"arxiv","version":3}},"canonical_sha256":"289fff53aa919521edc81ccca13b0b394c2f24536318764536901ac5f280c95d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"289fff53aa919521edc81ccca13b0b394c2f24536318764536901ac5f280c95d","first_computed_at":"2026-05-18T03:48:23.588076Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:48:23.588076Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Bfwo2j3TiVo6MA3YexOQ21IhiczfVsfv26Nt2Nz+h3oQfvU1oRu0pBGwyKwrr0u5+P+TuLTe69Wp0VuiHYIeAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:48:23.588507Z","signed_message":"canonical_sha256_bytes"},"source_id":"1010.4968","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:564333cacd5c452d58f33664e9d5817e8293ae09e1e0133d6e52828cc8315265","sha256:83f41ad23cb2b503e90787060f1d7d29ff084827fe65bc673b332d153a56ec36"],"state_sha256":"7a763149335ee8d29673f4d1d77ac331d193a0ad4f41dbe6b803fda93e0b13ee"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d6CeYYO2EBiSLjXs0oT+Xt/dmb0ZXm08rgK5pa1hP0NrRbECdbDbgRXzP2lqRiy6DcAM9wuSmgY6UNuApWNsBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T21:30:50.254623Z","bundle_sha256":"d807f2876f7dde45d83d12332f54cae3b30c34257ff6f986c6f3f611a48e8c9c"}}