{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:EX2Q25QAISX7BWOEVNT4Z4KGXB","short_pith_number":"pith:EX2Q25QA","canonical_record":{"source":{"id":"1210.0894","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-10-02T19:48:30Z","cross_cats_sorted":["math.RT","math.SP"],"title_canon_sha256":"0473a657273efe0eb0986e741d43be39d614b0376e3a13d1138f1ea68aa5195f","abstract_canon_sha256":"5abf8dad8c1cb8740b1f0b7048dcda8d67605274d02d58f61c3a677bff531087"},"schema_version":"1.0"},"canonical_sha256":"25f50d760044aff0d9c4ab67ccf146b847acfab12edd0b7a5291e30df6ece7b6","source":{"kind":"arxiv","id":"1210.0894","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.0894","created_at":"2026-05-18T02:53:54Z"},{"alias_kind":"arxiv_version","alias_value":"1210.0894v2","created_at":"2026-05-18T02:53:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.0894","created_at":"2026-05-18T02:53:54Z"},{"alias_kind":"pith_short_12","alias_value":"EX2Q25QAISX7","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"EX2Q25QAISX7BWOE","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"EX2Q25QA","created_at":"2026-05-18T12:27:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:EX2Q25QAISX7BWOEVNT4Z4KGXB","target":"record","payload":{"canonical_record":{"source":{"id":"1210.0894","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-10-02T19:48:30Z","cross_cats_sorted":["math.RT","math.SP"],"title_canon_sha256":"0473a657273efe0eb0986e741d43be39d614b0376e3a13d1138f1ea68aa5195f","abstract_canon_sha256":"5abf8dad8c1cb8740b1f0b7048dcda8d67605274d02d58f61c3a677bff531087"},"schema_version":"1.0"},"canonical_sha256":"25f50d760044aff0d9c4ab67ccf146b847acfab12edd0b7a5291e30df6ece7b6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:53:54.140383Z","signature_b64":"snI/0skm5SiAKHLVbxsaMjyu8/sZPfrTk/JGKFr3J5KFZJb3SKaNEPHkthlwRV5dVq0+gNY9Hek3RTG4TU7nDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"25f50d760044aff0d9c4ab67ccf146b847acfab12edd0b7a5291e30df6ece7b6","last_reissued_at":"2026-05-18T02:53:54.139431Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:53:54.139431Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1210.0894","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":"2026-05-18T02:53:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3vWt+I6fwmSvyo+xR8GXsMOqLRzqsyGnLRqzyYL5hlklIUhU2gmCT8rzXbAKCnryQFUblqFU+bVlSQWLDjuACg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T18:05:56.776115Z"},"content_sha256":"92aceb26ce80920d4f931460ab6d458d1e305742a96c46c36608ee1d335aab22","schema_version":"1.0","event_id":"sha256:92aceb26ce80920d4f931460ab6d458d1e305742a96c46c36608ee1d335aab22"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:EX2Q25QAISX7BWOEVNT4Z4KGXB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Representation equivalent Bieberbach groups and strongly isospectral flat manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT","math.SP"],"primary_cat":"math.DG","authors_text":"Emilio A. Lauret","submitted_at":"2012-10-02T19:48:30Z","abstract_excerpt":"Let $\\Gamma_1$ and $\\Gamma_2$ be Bieberbach groups contained in the full isometry group $G$ of $\\mathbb{R}^n$. We prove that if the compact flat manifolds $\\Gamma_1\\backslash\\mathbb{R}^n$ and $\\Gamma_2\\backslash\\mathbb{R}^n$ are strongly isospectral then the Bieberbach groups $\\Gamma_1$ and $\\Gamma_2$ are representation equivalent, that is, the right regular representations $L^2(\\Gamma_1\\backslash G)$ and $L^2(\\Gamma_2\\backslash G)$ are unitarily equivalent."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.0894","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":"2026-05-18T02:53:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"axHXCViik1y/+U5CdBY5EU/DgBjoX19+T1MqG3McToGo05zLoEeSscwKHttdOURdu205wSqsawiQeBDCAhI+Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T18:05:56.776668Z"},"content_sha256":"0faac5ce64a679a367ec2629aa846614ddaff1d9304258fa4de52b846bc1cfd1","schema_version":"1.0","event_id":"sha256:0faac5ce64a679a367ec2629aa846614ddaff1d9304258fa4de52b846bc1cfd1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EX2Q25QAISX7BWOEVNT4Z4KGXB/bundle.json","state_url":"https://pith.science/pith/EX2Q25QAISX7BWOEVNT4Z4KGXB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EX2Q25QAISX7BWOEVNT4Z4KGXB/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":"2026-06-08T18:05:56Z","links":{"resolver":"https://pith.science/pith/EX2Q25QAISX7BWOEVNT4Z4KGXB","bundle":"https://pith.science/pith/EX2Q25QAISX7BWOEVNT4Z4KGXB/bundle.json","state":"https://pith.science/pith/EX2Q25QAISX7BWOEVNT4Z4KGXB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EX2Q25QAISX7BWOEVNT4Z4KGXB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:EX2Q25QAISX7BWOEVNT4Z4KGXB","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":"5abf8dad8c1cb8740b1f0b7048dcda8d67605274d02d58f61c3a677bff531087","cross_cats_sorted":["math.RT","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-10-02T19:48:30Z","title_canon_sha256":"0473a657273efe0eb0986e741d43be39d614b0376e3a13d1138f1ea68aa5195f"},"schema_version":"1.0","source":{"id":"1210.0894","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.0894","created_at":"2026-05-18T02:53:54Z"},{"alias_kind":"arxiv_version","alias_value":"1210.0894v2","created_at":"2026-05-18T02:53:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.0894","created_at":"2026-05-18T02:53:54Z"},{"alias_kind":"pith_short_12","alias_value":"EX2Q25QAISX7","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"EX2Q25QAISX7BWOE","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"EX2Q25QA","created_at":"2026-05-18T12:27:04Z"}],"graph_snapshots":[{"event_id":"sha256:0faac5ce64a679a367ec2629aa846614ddaff1d9304258fa4de52b846bc1cfd1","target":"graph","created_at":"2026-05-18T02:53:54Z","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":"Let $\\Gamma_1$ and $\\Gamma_2$ be Bieberbach groups contained in the full isometry group $G$ of $\\mathbb{R}^n$. We prove that if the compact flat manifolds $\\Gamma_1\\backslash\\mathbb{R}^n$ and $\\Gamma_2\\backslash\\mathbb{R}^n$ are strongly isospectral then the Bieberbach groups $\\Gamma_1$ and $\\Gamma_2$ are representation equivalent, that is, the right regular representations $L^2(\\Gamma_1\\backslash G)$ and $L^2(\\Gamma_2\\backslash G)$ are unitarily equivalent.","authors_text":"Emilio A. Lauret","cross_cats":["math.RT","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-10-02T19:48:30Z","title":"Representation equivalent Bieberbach groups and strongly isospectral flat manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.0894","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":"sha256:92aceb26ce80920d4f931460ab6d458d1e305742a96c46c36608ee1d335aab22","target":"record","created_at":"2026-05-18T02:53:54Z","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":"5abf8dad8c1cb8740b1f0b7048dcda8d67605274d02d58f61c3a677bff531087","cross_cats_sorted":["math.RT","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-10-02T19:48:30Z","title_canon_sha256":"0473a657273efe0eb0986e741d43be39d614b0376e3a13d1138f1ea68aa5195f"},"schema_version":"1.0","source":{"id":"1210.0894","kind":"arxiv","version":2}},"canonical_sha256":"25f50d760044aff0d9c4ab67ccf146b847acfab12edd0b7a5291e30df6ece7b6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"25f50d760044aff0d9c4ab67ccf146b847acfab12edd0b7a5291e30df6ece7b6","first_computed_at":"2026-05-18T02:53:54.139431Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:53:54.139431Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"snI/0skm5SiAKHLVbxsaMjyu8/sZPfrTk/JGKFr3J5KFZJb3SKaNEPHkthlwRV5dVq0+gNY9Hek3RTG4TU7nDg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:53:54.140383Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.0894","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:92aceb26ce80920d4f931460ab6d458d1e305742a96c46c36608ee1d335aab22","sha256:0faac5ce64a679a367ec2629aa846614ddaff1d9304258fa4de52b846bc1cfd1"],"state_sha256":"9e96baf889eca4eb6ee1a73c9f78fa5a3d63c3db60a93907a5eaf47ef2e59547"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j3JF/TCFFW5PTQOK5JvShr1EyOWZiikFGrMo3iiujMp0RqDBNnUPkqW6lRU8o/XfwMUkVe10yJgzmaBu9uARDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T18:05:56.779517Z","bundle_sha256":"73fd4f75ea41538e459e89e319304e738f4236e4da3aff20b966a961aaaf2818"}}