{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:XNIJ4SW3EGJEAFWVL2324QJBGY","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":"bc12fb476db282c8f8f0e49a553bc5bae66bc1e572229242497aa450a6b56fd2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2024-10-23T07:51:05Z","title_canon_sha256":"92ce8e661914111e8dd078329a2fedd2649d8af5430500b56137b9e35d6ceb42"},"schema_version":"1.0","source":{"id":"2410.17634","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2410.17634","created_at":"2026-07-05T09:24:41Z"},{"alias_kind":"arxiv_version","alias_value":"2410.17634v1","created_at":"2026-07-05T09:24:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.17634","created_at":"2026-07-05T09:24:41Z"},{"alias_kind":"pith_short_12","alias_value":"XNIJ4SW3EGJE","created_at":"2026-07-05T09:24:41Z"},{"alias_kind":"pith_short_16","alias_value":"XNIJ4SW3EGJEAFWV","created_at":"2026-07-05T09:24:41Z"},{"alias_kind":"pith_short_8","alias_value":"XNIJ4SW3","created_at":"2026-07-05T09:24:41Z"}],"graph_snapshots":[{"event_id":"sha256:ccb110d84bb4bbc59d5bf8ba57c1ff564c2d8e43b5be8f9e51de3dab7fe1f3b3","target":"graph","created_at":"2026-07-05T09:24:41Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2410.17634/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We investigate the problem of defining group or loop structures on spheres, where by ''sphere'' we mean the level set q(x) = c of a general K-valued quadratic form q, for an invertible scalar c. When K is a field and q non-degenerate, then this corresponds to the classical theory of composition algebras; in particular, for K = R and positive definite forms, we obtain the sequence of the four real division algebras R, C, H (quaternions), O (octonions). Our theory is more general, allowing that K is merely a ring, and the form q possibly degenerate. To achieve this goal, we give a more geometric","authors_text":"Wolfgang Bertram (IECL)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2024-10-23T07:51:05Z","title":"On group and loop spheres"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.17634","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":"sha256:201da96e876e9cdfbb6a69d08073d332f4e02cc12ce3a3d0eeb14f5c8e09cde0","target":"record","created_at":"2026-07-05T09:24:41Z","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":"bc12fb476db282c8f8f0e49a553bc5bae66bc1e572229242497aa450a6b56fd2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2024-10-23T07:51:05Z","title_canon_sha256":"92ce8e661914111e8dd078329a2fedd2649d8af5430500b56137b9e35d6ceb42"},"schema_version":"1.0","source":{"id":"2410.17634","kind":"arxiv","version":1}},"canonical_sha256":"bb509e4adb21924016d55eb7ae4121361d66d2932a3371f2c0a619cc9d6610d7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bb509e4adb21924016d55eb7ae4121361d66d2932a3371f2c0a619cc9d6610d7","first_computed_at":"2026-07-05T09:24:41.837692Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:24:41.837692Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AA6GLX2ui3uKmYfujg8Glz4O1aLLS1/qV54Kkvz8LBktlYgYNzWxfj5sSguAtmAep6hoSgLlcIz4XOyXElrXBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T09:24:41.838130Z","signed_message":"canonical_sha256_bytes"},"source_id":"2410.17634","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:201da96e876e9cdfbb6a69d08073d332f4e02cc12ce3a3d0eeb14f5c8e09cde0","sha256:ccb110d84bb4bbc59d5bf8ba57c1ff564c2d8e43b5be8f9e51de3dab7fe1f3b3"],"state_sha256":"96086f14d91558c09fc6a43768a8c31efc00851da73227c106dbdcbf31bd19dc"}