{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:TCTNDLH7QVCTYQ3HOAEAHQCSQ6","short_pith_number":"pith:TCTNDLH7","canonical_record":{"source":{"id":"1009.0487","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-09-02T17:49:18Z","cross_cats_sorted":[],"title_canon_sha256":"6e726b86eb5a3ef34b24c0cdee5e6cd1855d740620fc67554f813c7e8b94c16e","abstract_canon_sha256":"59b4f91cb4e7680c3135a20e61152c137a46cd00216e86314d393db62490658e"},"schema_version":"1.0"},"canonical_sha256":"98a6d1acff85453c4367700803c05287bbb22233315ad9b6429b93f42c7101ed","source":{"kind":"arxiv","id":"1009.0487","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.0487","created_at":"2026-05-18T04:41:26Z"},{"alias_kind":"arxiv_version","alias_value":"1009.0487v1","created_at":"2026-05-18T04:41:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.0487","created_at":"2026-05-18T04:41:26Z"},{"alias_kind":"pith_short_12","alias_value":"TCTNDLH7QVCT","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"TCTNDLH7QVCTYQ3H","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"TCTNDLH7","created_at":"2026-05-18T12:26:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:TCTNDLH7QVCTYQ3HOAEAHQCSQ6","target":"record","payload":{"canonical_record":{"source":{"id":"1009.0487","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-09-02T17:49:18Z","cross_cats_sorted":[],"title_canon_sha256":"6e726b86eb5a3ef34b24c0cdee5e6cd1855d740620fc67554f813c7e8b94c16e","abstract_canon_sha256":"59b4f91cb4e7680c3135a20e61152c137a46cd00216e86314d393db62490658e"},"schema_version":"1.0"},"canonical_sha256":"98a6d1acff85453c4367700803c05287bbb22233315ad9b6429b93f42c7101ed","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:26.519623Z","signature_b64":"7FlkI6Dh+2S/XsvZ1NMB2GfYMNcPmW2LTIPpLTYYURndtJWgQ6kz8b63TrbbBgFEOhFhqVuASzK4up5eLR/IAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98a6d1acff85453c4367700803c05287bbb22233315ad9b6429b93f42c7101ed","last_reissued_at":"2026-05-18T04:41:26.518915Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:26.518915Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1009.0487","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":"2026-05-18T04:41:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MV8MQP7xO+WL2aDiusiJl7nO+qr8XW8tmNTjncO4r2LNcujbWcrZCI89OQ3tUBvfpUlsPkKvNjtwa8KK0AU7Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T09:25:16.446698Z"},"content_sha256":"92596b5099e8dd9eabd495e49e411757f3441ace2f3fcfd99cc31a80e66703da","schema_version":"1.0","event_id":"sha256:92596b5099e8dd9eabd495e49e411757f3441ace2f3fcfd99cc31a80e66703da"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:TCTNDLH7QVCTYQ3HOAEAHQCSQ6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Unbreakable Loops","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Louis Marchand, Martin Beaudry","submitted_at":"2010-09-02T17:49:18Z","abstract_excerpt":"We say that a loop is unbreakable when it does not have nontrivial subloops. While the cyclic groups of prime order are the only unbreakable finite groups, we show that nonassociative unbreakable loops exist for every order n >= 5. We describe two families of commutative unbreakable loops of odd order, n >= 7, one where the loop's multiplication group is isomorphic to the alternating group An and another where the multiplication group is isomorphic to the symmetric group Sn. We also prove for each even n >= 6 that there exist unbreakable loops of order n whose multiplication group is isomorphi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.0487","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":"2026-05-18T04:41:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3dJ94dVb56n8432LaXIjsdq9ESEwFrVKiigv16mwEhySn9P1aSskLTduXSzw9gZYlWehVvQcAjpeonmwgJUhBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T09:25:16.447042Z"},"content_sha256":"80b8b136dfdad348712ee0795dd5dddcb8700439220eec8c99711bd0498ae805","schema_version":"1.0","event_id":"sha256:80b8b136dfdad348712ee0795dd5dddcb8700439220eec8c99711bd0498ae805"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TCTNDLH7QVCTYQ3HOAEAHQCSQ6/bundle.json","state_url":"https://pith.science/pith/TCTNDLH7QVCTYQ3HOAEAHQCSQ6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TCTNDLH7QVCTYQ3HOAEAHQCSQ6/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":"2026-06-29T09:25:16Z","links":{"resolver":"https://pith.science/pith/TCTNDLH7QVCTYQ3HOAEAHQCSQ6","bundle":"https://pith.science/pith/TCTNDLH7QVCTYQ3HOAEAHQCSQ6/bundle.json","state":"https://pith.science/pith/TCTNDLH7QVCTYQ3HOAEAHQCSQ6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TCTNDLH7QVCTYQ3HOAEAHQCSQ6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:TCTNDLH7QVCTYQ3HOAEAHQCSQ6","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":"59b4f91cb4e7680c3135a20e61152c137a46cd00216e86314d393db62490658e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-09-02T17:49:18Z","title_canon_sha256":"6e726b86eb5a3ef34b24c0cdee5e6cd1855d740620fc67554f813c7e8b94c16e"},"schema_version":"1.0","source":{"id":"1009.0487","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.0487","created_at":"2026-05-18T04:41:26Z"},{"alias_kind":"arxiv_version","alias_value":"1009.0487v1","created_at":"2026-05-18T04:41:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.0487","created_at":"2026-05-18T04:41:26Z"},{"alias_kind":"pith_short_12","alias_value":"TCTNDLH7QVCT","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"TCTNDLH7QVCTYQ3H","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"TCTNDLH7","created_at":"2026-05-18T12:26:13Z"}],"graph_snapshots":[{"event_id":"sha256:80b8b136dfdad348712ee0795dd5dddcb8700439220eec8c99711bd0498ae805","target":"graph","created_at":"2026-05-18T04:41:26Z","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 say that a loop is unbreakable when it does not have nontrivial subloops. While the cyclic groups of prime order are the only unbreakable finite groups, we show that nonassociative unbreakable loops exist for every order n >= 5. We describe two families of commutative unbreakable loops of odd order, n >= 7, one where the loop's multiplication group is isomorphic to the alternating group An and another where the multiplication group is isomorphic to the symmetric group Sn. We also prove for each even n >= 6 that there exist unbreakable loops of order n whose multiplication group is isomorphi","authors_text":"Louis Marchand, Martin Beaudry","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-09-02T17:49:18Z","title":"Unbreakable Loops"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.0487","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":"sha256:92596b5099e8dd9eabd495e49e411757f3441ace2f3fcfd99cc31a80e66703da","target":"record","created_at":"2026-05-18T04:41:26Z","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":"59b4f91cb4e7680c3135a20e61152c137a46cd00216e86314d393db62490658e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-09-02T17:49:18Z","title_canon_sha256":"6e726b86eb5a3ef34b24c0cdee5e6cd1855d740620fc67554f813c7e8b94c16e"},"schema_version":"1.0","source":{"id":"1009.0487","kind":"arxiv","version":1}},"canonical_sha256":"98a6d1acff85453c4367700803c05287bbb22233315ad9b6429b93f42c7101ed","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"98a6d1acff85453c4367700803c05287bbb22233315ad9b6429b93f42c7101ed","first_computed_at":"2026-05-18T04:41:26.518915Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:41:26.518915Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7FlkI6Dh+2S/XsvZ1NMB2GfYMNcPmW2LTIPpLTYYURndtJWgQ6kz8b63TrbbBgFEOhFhqVuASzK4up5eLR/IAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:41:26.519623Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.0487","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:92596b5099e8dd9eabd495e49e411757f3441ace2f3fcfd99cc31a80e66703da","sha256:80b8b136dfdad348712ee0795dd5dddcb8700439220eec8c99711bd0498ae805"],"state_sha256":"169e6a4d99888d4a9392abb27a3a88941fc56ba5887d6abe0b3f60f7cdee905d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l7jaYhwmC420ucKqwavQlC7s89nd34Nq7OkpR3pwI8IJ9gOZy+ZPSR/baMZhzJhm20eM9/4jg/3tVq/2C4HYDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T09:25:16.448908Z","bundle_sha256":"8bd711bcae91fae3ef74837a72ad606145c60a68ca5b2db79ac069266fa2c2a5"}}