{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:QMM5BN3XBEVHTFX6YFNQTQGA63","short_pith_number":"pith:QMM5BN3X","canonical_record":{"source":{"id":"1411.2142","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-11-08T17:41:09Z","cross_cats_sorted":[],"title_canon_sha256":"9463865240f8f9a1eb631fb757f1afb9e856ddfb38268dfd670fe4daa361e3eb","abstract_canon_sha256":"0b8533e80b723d43e79fb35a854216257e1defde3bb4d568a952df1ed3f558d8"},"schema_version":"1.0"},"canonical_sha256":"8319d0b777092a7996fec15b09c0c0f6d3672c9f5577d767b075ddd05ae8584f","source":{"kind":"arxiv","id":"1411.2142","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.2142","created_at":"2026-05-18T02:38:08Z"},{"alias_kind":"arxiv_version","alias_value":"1411.2142v1","created_at":"2026-05-18T02:38:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.2142","created_at":"2026-05-18T02:38:08Z"},{"alias_kind":"pith_short_12","alias_value":"QMM5BN3XBEVH","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"QMM5BN3XBEVHTFX6","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"QMM5BN3X","created_at":"2026-05-18T12:28:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:QMM5BN3XBEVHTFX6YFNQTQGA63","target":"record","payload":{"canonical_record":{"source":{"id":"1411.2142","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-11-08T17:41:09Z","cross_cats_sorted":[],"title_canon_sha256":"9463865240f8f9a1eb631fb757f1afb9e856ddfb38268dfd670fe4daa361e3eb","abstract_canon_sha256":"0b8533e80b723d43e79fb35a854216257e1defde3bb4d568a952df1ed3f558d8"},"schema_version":"1.0"},"canonical_sha256":"8319d0b777092a7996fec15b09c0c0f6d3672c9f5577d767b075ddd05ae8584f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:38:08.466882Z","signature_b64":"g9olqeFsO+ecU4Z8002rvtpAudMmrcmWJSg03krRGOkccm4KYC3MZwD8LirKC/f1oOKL8qonPDvIL3+DkwYVBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8319d0b777092a7996fec15b09c0c0f6d3672c9f5577d767b075ddd05ae8584f","last_reissued_at":"2026-05-18T02:38:08.466341Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:38:08.466341Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1411.2142","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":"2026-05-18T02:38:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HhznMG9wcOjqDm3dyx6Q+OakQr3RXEGSCh8uj5HA3qWJ8gUdJBuaMTK/u9RQ6/XsV4DpHXp4+/X4PeCxlrUaAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T07:13:10.818739Z"},"content_sha256":"cfdfba01e449f9dc9fd4f983d893539fd0f43375801d31c55fd4d43770926c65","schema_version":"1.0","event_id":"sha256:cfdfba01e449f9dc9fd4f983d893539fd0f43375801d31c55fd4d43770926c65"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:QMM5BN3XBEVHTFX6YFNQTQGA63","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Isodualit\\'e des r\\'eseaux euclidiens en petite dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Christophe Bavard","submitted_at":"2014-11-08T17:41:09Z","abstract_excerpt":"We propose an algebraic and a geometric classification of euclidean isodual lattices of fixed rank. First, we prove that these lattices are distribued according to a finite number of algebraic types. Second, we show that they are parametrized by a finite number of symmetric spaces associated to the classical groups ${\\bf SO}_0(p,q)$, ${\\bf Sp}(2g,{\\bf R})$ and ${\\bf SU}(p,q)$. We obtain a complete discription of algebraic types and Gram matrices of isodual lattices up to rank 7. The maximal density problem is also discussed."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.2142","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":"2026-05-18T02:38:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2kqX3HocvhBG4sMxe460WbBkHAzbI0+3berCWweYIraxfSu+IVALIH2LjiXW9Sp6jpRFtW0gTSfp8dFC4t8xCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T07:13:10.819238Z"},"content_sha256":"6de33ee20ede94a6e1d3e8e516b0bb83920adc3f9e38bd3906182938b013d20b","schema_version":"1.0","event_id":"sha256:6de33ee20ede94a6e1d3e8e516b0bb83920adc3f9e38bd3906182938b013d20b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QMM5BN3XBEVHTFX6YFNQTQGA63/bundle.json","state_url":"https://pith.science/pith/QMM5BN3XBEVHTFX6YFNQTQGA63/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QMM5BN3XBEVHTFX6YFNQTQGA63/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":"2026-06-11T07:13:10Z","links":{"resolver":"https://pith.science/pith/QMM5BN3XBEVHTFX6YFNQTQGA63","bundle":"https://pith.science/pith/QMM5BN3XBEVHTFX6YFNQTQGA63/bundle.json","state":"https://pith.science/pith/QMM5BN3XBEVHTFX6YFNQTQGA63/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QMM5BN3XBEVHTFX6YFNQTQGA63/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:QMM5BN3XBEVHTFX6YFNQTQGA63","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":"0b8533e80b723d43e79fb35a854216257e1defde3bb4d568a952df1ed3f558d8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-11-08T17:41:09Z","title_canon_sha256":"9463865240f8f9a1eb631fb757f1afb9e856ddfb38268dfd670fe4daa361e3eb"},"schema_version":"1.0","source":{"id":"1411.2142","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1411.2142","created_at":"2026-05-18T02:38:08Z"},{"alias_kind":"arxiv_version","alias_value":"1411.2142v1","created_at":"2026-05-18T02:38:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.2142","created_at":"2026-05-18T02:38:08Z"},{"alias_kind":"pith_short_12","alias_value":"QMM5BN3XBEVH","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"QMM5BN3XBEVHTFX6","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"QMM5BN3X","created_at":"2026-05-18T12:28:46Z"}],"graph_snapshots":[{"event_id":"sha256:6de33ee20ede94a6e1d3e8e516b0bb83920adc3f9e38bd3906182938b013d20b","target":"graph","created_at":"2026-05-18T02:38:08Z","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 propose an algebraic and a geometric classification of euclidean isodual lattices of fixed rank. First, we prove that these lattices are distribued according to a finite number of algebraic types. Second, we show that they are parametrized by a finite number of symmetric spaces associated to the classical groups ${\\bf SO}_0(p,q)$, ${\\bf Sp}(2g,{\\bf R})$ and ${\\bf SU}(p,q)$. We obtain a complete discription of algebraic types and Gram matrices of isodual lattices up to rank 7. The maximal density problem is also discussed.","authors_text":"Christophe Bavard","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-11-08T17:41:09Z","title":"Isodualit\\'e des r\\'eseaux euclidiens en petite dimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.2142","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:cfdfba01e449f9dc9fd4f983d893539fd0f43375801d31c55fd4d43770926c65","target":"record","created_at":"2026-05-18T02:38:08Z","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":"0b8533e80b723d43e79fb35a854216257e1defde3bb4d568a952df1ed3f558d8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-11-08T17:41:09Z","title_canon_sha256":"9463865240f8f9a1eb631fb757f1afb9e856ddfb38268dfd670fe4daa361e3eb"},"schema_version":"1.0","source":{"id":"1411.2142","kind":"arxiv","version":1}},"canonical_sha256":"8319d0b777092a7996fec15b09c0c0f6d3672c9f5577d767b075ddd05ae8584f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8319d0b777092a7996fec15b09c0c0f6d3672c9f5577d767b075ddd05ae8584f","first_computed_at":"2026-05-18T02:38:08.466341Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:38:08.466341Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g9olqeFsO+ecU4Z8002rvtpAudMmrcmWJSg03krRGOkccm4KYC3MZwD8LirKC/f1oOKL8qonPDvIL3+DkwYVBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:38:08.466882Z","signed_message":"canonical_sha256_bytes"},"source_id":"1411.2142","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cfdfba01e449f9dc9fd4f983d893539fd0f43375801d31c55fd4d43770926c65","sha256:6de33ee20ede94a6e1d3e8e516b0bb83920adc3f9e38bd3906182938b013d20b"],"state_sha256":"da803b4916960bb517f873d3cebd8ec2034744664c48f6e72e2728c28eba1e74"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xl76f2FuM5UhTCduI5EfjySwAAsoHF6C226YXyWvMEaGm97QbtwUbDhGAeIaTDcj6hFaeJ1G+xs9v41nL/KrDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T07:13:10.822923Z","bundle_sha256":"743cabaad80248d7810f7be9640e37eb7c8b691c1567e61999b014c94399aac5"}}