{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:TC7LOJSTMGN75KBJ676GNSWFY3","short_pith_number":"pith:TC7LOJST","canonical_record":{"source":{"id":"1010.1077","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2010-10-06T07:42:37Z","cross_cats_sorted":["math.CO","math.MG"],"title_canon_sha256":"e3801e13072705637b11fbec79fabad72887e55d693056fde4d822e1b5a5bfcf","abstract_canon_sha256":"c8865bda55db8c5b4e3c6cbe99717a753fce9050bcd47bc35aab4e6bd22ac70e"},"schema_version":"1.0"},"canonical_sha256":"98beb72653619bfea829f7fc66cac5c6ef9047cae4125382adc42df76ea477c7","source":{"kind":"arxiv","id":"1010.1077","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.1077","created_at":"2026-05-18T04:25:49Z"},{"alias_kind":"arxiv_version","alias_value":"1010.1077v2","created_at":"2026-05-18T04:25:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.1077","created_at":"2026-05-18T04:25:49Z"},{"alias_kind":"pith_short_12","alias_value":"TC7LOJSTMGN7","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"TC7LOJSTMGN75KBJ","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"TC7LOJST","created_at":"2026-05-18T12:26:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:TC7LOJSTMGN75KBJ676GNSWFY3","target":"record","payload":{"canonical_record":{"source":{"id":"1010.1077","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2010-10-06T07:42:37Z","cross_cats_sorted":["math.CO","math.MG"],"title_canon_sha256":"e3801e13072705637b11fbec79fabad72887e55d693056fde4d822e1b5a5bfcf","abstract_canon_sha256":"c8865bda55db8c5b4e3c6cbe99717a753fce9050bcd47bc35aab4e6bd22ac70e"},"schema_version":"1.0"},"canonical_sha256":"98beb72653619bfea829f7fc66cac5c6ef9047cae4125382adc42df76ea477c7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:25:49.223731Z","signature_b64":"ppC+e7FBly/tze8CUEoQStWKNqZI8OziQmiasOCbKF1E3y8T8ynneOubgbhKjJ6fIysAy2c+HSyuf/1LinWXAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98beb72653619bfea829f7fc66cac5c6ef9047cae4125382adc42df76ea477c7","last_reissued_at":"2026-05-18T04:25:49.223001Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:25:49.223001Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1010.1077","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":"2026-05-18T04:25:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/t2q8oF2sFJTWS2zpcfpNk6Cm6DJvlGTS/yWGH/TQ7JS124VDYFFuGPEKMOVBtZ/LccpDK4S7IhRf1fdMWzkBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T17:50:36.910770Z"},"content_sha256":"7033d4240942d50f9bd28d6441f66cfac7099b3ff215fa2806de862d6de05aa5","schema_version":"1.0","event_id":"sha256:7033d4240942d50f9bd28d6441f66cfac7099b3ff215fa2806de862d6de05aa5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:TC7LOJSTMGN75KBJ676GNSWFY3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Maximal lattice-free polyhedra: finiteness and an explicit description in dimension three","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.OC","authors_text":"Christian Wagner, Gennadiy Averkov, Robert Weismantel","submitted_at":"2010-10-06T07:42:37Z","abstract_excerpt":"A convex set with nonempty interior is maximal lattice-free if it is inclusion-maximal with respect to the property of not containing integer points in its interior. Maximal lattice-free convex sets are known to be polyhedra. The precision of a rational polyhedron $P$ in $\\mathbb{R}^d$ is the smallest integer $s$ such that $sP$ is an integral polyhedron. In this paper we show that, up to affine mappings preserving $\\mathbb{Z}^d$, the number of maximal lattice-free rational polyhedra of a given precision $s$ is finite. Furthermore, we present the complete list of all maximal lattice-free integr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.1077","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":"2026-05-18T04:25:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u+MZn/VcZYOr1np3H3kNqSV5gAI4qLQgfkaGYTo292gmfT6A1OWbw6RFkF+NN1WAjvKO+ZlhOo5GBDnAC8wQCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T17:50:36.911519Z"},"content_sha256":"85a3fea795fdc42bd47a857e0b281ab1caa043ed1a1cfb6b425a355db700b1d6","schema_version":"1.0","event_id":"sha256:85a3fea795fdc42bd47a857e0b281ab1caa043ed1a1cfb6b425a355db700b1d6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TC7LOJSTMGN75KBJ676GNSWFY3/bundle.json","state_url":"https://pith.science/pith/TC7LOJSTMGN75KBJ676GNSWFY3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TC7LOJSTMGN75KBJ676GNSWFY3/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":"2026-05-31T17:50:36Z","links":{"resolver":"https://pith.science/pith/TC7LOJSTMGN75KBJ676GNSWFY3","bundle":"https://pith.science/pith/TC7LOJSTMGN75KBJ676GNSWFY3/bundle.json","state":"https://pith.science/pith/TC7LOJSTMGN75KBJ676GNSWFY3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TC7LOJSTMGN75KBJ676GNSWFY3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:TC7LOJSTMGN75KBJ676GNSWFY3","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":"c8865bda55db8c5b4e3c6cbe99717a753fce9050bcd47bc35aab4e6bd22ac70e","cross_cats_sorted":["math.CO","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2010-10-06T07:42:37Z","title_canon_sha256":"e3801e13072705637b11fbec79fabad72887e55d693056fde4d822e1b5a5bfcf"},"schema_version":"1.0","source":{"id":"1010.1077","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.1077","created_at":"2026-05-18T04:25:49Z"},{"alias_kind":"arxiv_version","alias_value":"1010.1077v2","created_at":"2026-05-18T04:25:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.1077","created_at":"2026-05-18T04:25:49Z"},{"alias_kind":"pith_short_12","alias_value":"TC7LOJSTMGN7","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"TC7LOJSTMGN75KBJ","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"TC7LOJST","created_at":"2026-05-18T12:26:13Z"}],"graph_snapshots":[{"event_id":"sha256:85a3fea795fdc42bd47a857e0b281ab1caa043ed1a1cfb6b425a355db700b1d6","target":"graph","created_at":"2026-05-18T04:25:49Z","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 convex set with nonempty interior is maximal lattice-free if it is inclusion-maximal with respect to the property of not containing integer points in its interior. Maximal lattice-free convex sets are known to be polyhedra. The precision of a rational polyhedron $P$ in $\\mathbb{R}^d$ is the smallest integer $s$ such that $sP$ is an integral polyhedron. In this paper we show that, up to affine mappings preserving $\\mathbb{Z}^d$, the number of maximal lattice-free rational polyhedra of a given precision $s$ is finite. Furthermore, we present the complete list of all maximal lattice-free integr","authors_text":"Christian Wagner, Gennadiy Averkov, Robert Weismantel","cross_cats":["math.CO","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2010-10-06T07:42:37Z","title":"Maximal lattice-free polyhedra: finiteness and an explicit description in dimension three"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.1077","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":"sha256:7033d4240942d50f9bd28d6441f66cfac7099b3ff215fa2806de862d6de05aa5","target":"record","created_at":"2026-05-18T04:25:49Z","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":"c8865bda55db8c5b4e3c6cbe99717a753fce9050bcd47bc35aab4e6bd22ac70e","cross_cats_sorted":["math.CO","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2010-10-06T07:42:37Z","title_canon_sha256":"e3801e13072705637b11fbec79fabad72887e55d693056fde4d822e1b5a5bfcf"},"schema_version":"1.0","source":{"id":"1010.1077","kind":"arxiv","version":2}},"canonical_sha256":"98beb72653619bfea829f7fc66cac5c6ef9047cae4125382adc42df76ea477c7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"98beb72653619bfea829f7fc66cac5c6ef9047cae4125382adc42df76ea477c7","first_computed_at":"2026-05-18T04:25:49.223001Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:25:49.223001Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ppC+e7FBly/tze8CUEoQStWKNqZI8OziQmiasOCbKF1E3y8T8ynneOubgbhKjJ6fIysAy2c+HSyuf/1LinWXAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:25:49.223731Z","signed_message":"canonical_sha256_bytes"},"source_id":"1010.1077","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7033d4240942d50f9bd28d6441f66cfac7099b3ff215fa2806de862d6de05aa5","sha256:85a3fea795fdc42bd47a857e0b281ab1caa043ed1a1cfb6b425a355db700b1d6"],"state_sha256":"e16019524e845cc78be5ddcf68205c2a2f0b0bac3827f01a3003658d13527922"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EYy1RTQPYUE+b3AvHFsq+0KoNqkcfDAUBh3wrXMvfMv+MqSssHCe5Dvlq+UzXWJz11kSx/xl3bfhTuxhrA0GDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T17:50:36.915494Z","bundle_sha256":"988e058b2c2ead06a7fe0682e9bbe351d8dc20aa5b04cc4a75692fc8dda45e36"}}