{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:CTXP3VS4KHQ5FBWQMCBCYF3KAI","short_pith_number":"pith:CTXP3VS4","canonical_record":{"source":{"id":"1209.4652","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-09-20T20:00:59Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"3cfc951fc95c4c7b87e961dc86a819887e557686e20fcc7f9355d37fa3b2bfe0","abstract_canon_sha256":"cc957eb4933adc45a30985c35ed197a045ecbeefe4489b3b824b1b95292293da"},"schema_version":"1.0"},"canonical_sha256":"14eefdd65c51e1d286d060822c176a0228f28b17d019f537868e2ab847da8350","source":{"kind":"arxiv","id":"1209.4652","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.4652","created_at":"2026-05-18T02:55:27Z"},{"alias_kind":"arxiv_version","alias_value":"1209.4652v2","created_at":"2026-05-18T02:55:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.4652","created_at":"2026-05-18T02:55:27Z"},{"alias_kind":"pith_short_12","alias_value":"CTXP3VS4KHQ5","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"CTXP3VS4KHQ5FBWQ","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"CTXP3VS4","created_at":"2026-05-18T12:27:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:CTXP3VS4KHQ5FBWQMCBCYF3KAI","target":"record","payload":{"canonical_record":{"source":{"id":"1209.4652","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-09-20T20:00:59Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"3cfc951fc95c4c7b87e961dc86a819887e557686e20fcc7f9355d37fa3b2bfe0","abstract_canon_sha256":"cc957eb4933adc45a30985c35ed197a045ecbeefe4489b3b824b1b95292293da"},"schema_version":"1.0"},"canonical_sha256":"14eefdd65c51e1d286d060822c176a0228f28b17d019f537868e2ab847da8350","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:55:27.800099Z","signature_b64":"DXTu+a4Dk1TOFJud4VJxBLZsS4mHQmQz34jrTyNQElPgnNrOU5I3LCrdTj0EbdDgIEL4DdeiA/INioXKH+CdBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"14eefdd65c51e1d286d060822c176a0228f28b17d019f537868e2ab847da8350","last_reissued_at":"2026-05-18T02:55:27.799627Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:55:27.799627Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.4652","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":"2026-05-18T02:55:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"15Adqh1N6Cn0BJVeJkOB2yZ0feXbcHNmKY2cDJM0X0oQestl/CZKyhUcygw5drut4ANgqKBLaBn/K/InA5HsDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T06:44:02.776023Z"},"content_sha256":"3216468f4887ba380b7e076bbe76e3b0a88d3c78ca7cbff73e414cfccff0b055","schema_version":"1.0","event_id":"sha256:3216468f4887ba380b7e076bbe76e3b0a88d3c78ca7cbff73e414cfccff0b055"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:CTXP3VS4KHQ5FBWQMCBCYF3KAI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the sum of the Voronoi polytope of a lattice with a zonotope","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"Alexander Magazinov, Mathieu Dutour Sikiric, Viatcheslav Grishukhin","submitted_at":"2012-09-20T20:00:59Z","abstract_excerpt":"A parallelotope $P$ is a polytope that admits a facet-to-facet tiling of space by translation copies of $P$ along a lattice. The Voronoi cell $P_V(L)$ of a lattice $L$ is an example of a parallelotope. A parallelotope can be uniquely decomposed as the Minkowski sum of a zone closed parallelotope $P$ and a zonotope $Z(U)$, where $U$ is the set of vectors used to generate the zonotope. In this paper we consider the related question: When is the Minkowski sum of a general parallelotope and a zonotope $P+Z(U)$ a parallelotope? We give two necessary conditions and show that the vectors $U$ have to "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.4652","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":"2026-05-18T02:55:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zjHrlmV9FiQTk1MPPN2S7NGXHjiUz8JOc2LzCYdGzIn81efURrDW8elP5EPliIVteRt6bf04r/KT/CYFzYRpDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T06:44:02.776382Z"},"content_sha256":"7420c525abd6056d7a4fb68f00551cff6bc96fd651c1bfabd470ac2d21f26df9","schema_version":"1.0","event_id":"sha256:7420c525abd6056d7a4fb68f00551cff6bc96fd651c1bfabd470ac2d21f26df9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CTXP3VS4KHQ5FBWQMCBCYF3KAI/bundle.json","state_url":"https://pith.science/pith/CTXP3VS4KHQ5FBWQMCBCYF3KAI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CTXP3VS4KHQ5FBWQMCBCYF3KAI/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":"2026-06-25T06:44:02Z","links":{"resolver":"https://pith.science/pith/CTXP3VS4KHQ5FBWQMCBCYF3KAI","bundle":"https://pith.science/pith/CTXP3VS4KHQ5FBWQMCBCYF3KAI/bundle.json","state":"https://pith.science/pith/CTXP3VS4KHQ5FBWQMCBCYF3KAI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CTXP3VS4KHQ5FBWQMCBCYF3KAI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:CTXP3VS4KHQ5FBWQMCBCYF3KAI","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":"cc957eb4933adc45a30985c35ed197a045ecbeefe4489b3b824b1b95292293da","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-09-20T20:00:59Z","title_canon_sha256":"3cfc951fc95c4c7b87e961dc86a819887e557686e20fcc7f9355d37fa3b2bfe0"},"schema_version":"1.0","source":{"id":"1209.4652","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.4652","created_at":"2026-05-18T02:55:27Z"},{"alias_kind":"arxiv_version","alias_value":"1209.4652v2","created_at":"2026-05-18T02:55:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.4652","created_at":"2026-05-18T02:55:27Z"},{"alias_kind":"pith_short_12","alias_value":"CTXP3VS4KHQ5","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"CTXP3VS4KHQ5FBWQ","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"CTXP3VS4","created_at":"2026-05-18T12:27:01Z"}],"graph_snapshots":[{"event_id":"sha256:7420c525abd6056d7a4fb68f00551cff6bc96fd651c1bfabd470ac2d21f26df9","target":"graph","created_at":"2026-05-18T02:55:27Z","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 parallelotope $P$ is a polytope that admits a facet-to-facet tiling of space by translation copies of $P$ along a lattice. The Voronoi cell $P_V(L)$ of a lattice $L$ is an example of a parallelotope. A parallelotope can be uniquely decomposed as the Minkowski sum of a zone closed parallelotope $P$ and a zonotope $Z(U)$, where $U$ is the set of vectors used to generate the zonotope. In this paper we consider the related question: When is the Minkowski sum of a general parallelotope and a zonotope $P+Z(U)$ a parallelotope? We give two necessary conditions and show that the vectors $U$ have to ","authors_text":"Alexander Magazinov, Mathieu Dutour Sikiric, Viatcheslav Grishukhin","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-09-20T20:00:59Z","title":"On the sum of the Voronoi polytope of a lattice with a zonotope"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.4652","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":"sha256:3216468f4887ba380b7e076bbe76e3b0a88d3c78ca7cbff73e414cfccff0b055","target":"record","created_at":"2026-05-18T02:55:27Z","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":"cc957eb4933adc45a30985c35ed197a045ecbeefe4489b3b824b1b95292293da","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2012-09-20T20:00:59Z","title_canon_sha256":"3cfc951fc95c4c7b87e961dc86a819887e557686e20fcc7f9355d37fa3b2bfe0"},"schema_version":"1.0","source":{"id":"1209.4652","kind":"arxiv","version":2}},"canonical_sha256":"14eefdd65c51e1d286d060822c176a0228f28b17d019f537868e2ab847da8350","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"14eefdd65c51e1d286d060822c176a0228f28b17d019f537868e2ab847da8350","first_computed_at":"2026-05-18T02:55:27.799627Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:55:27.799627Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DXTu+a4Dk1TOFJud4VJxBLZsS4mHQmQz34jrTyNQElPgnNrOU5I3LCrdTj0EbdDgIEL4DdeiA/INioXKH+CdBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:55:27.800099Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.4652","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3216468f4887ba380b7e076bbe76e3b0a88d3c78ca7cbff73e414cfccff0b055","sha256:7420c525abd6056d7a4fb68f00551cff6bc96fd651c1bfabd470ac2d21f26df9"],"state_sha256":"d2bfd63473c435b0acf5d3c99103f48d4a5a931ecfbb8a6bb5abff45d2c97c51"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sHRoqqpeMkr3LydZHQysdDTG6J45rZnKwb2501DeHVPVve6vJwnH1VARS/19uLkbtlsJY2HhpPSBQfUvDmT5Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T06:44:02.778272Z","bundle_sha256":"f76bd44767edef1a6eebadc8c0e74d79434de119a9b9b7dd77923c2ecb78c989"}}