{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:LVL7TQJRMTNMMEAHXGPCNSH7TL","short_pith_number":"pith:LVL7TQJR","canonical_record":{"source":{"id":"2109.00878","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2021-09-02T12:41:12Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"ce5482c9bad3b5fe0d728949e55df67c535172e8fe37c80e83713e054052923c","abstract_canon_sha256":"ab14d840cfd25b76c9c3d6a84d8cb710b56416606b5a8c8cf546c24281cc29f4"},"schema_version":"1.0"},"canonical_sha256":"5d57f9c13164dac61007b99e26c8ff9aefd9c1c9c5da92266572a993c9b22ad0","source":{"kind":"arxiv","id":"2109.00878","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.00878","created_at":"2026-07-05T03:10:59Z"},{"alias_kind":"arxiv_version","alias_value":"2109.00878v1","created_at":"2026-07-05T03:10:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.00878","created_at":"2026-07-05T03:10:59Z"},{"alias_kind":"pith_short_12","alias_value":"LVL7TQJRMTNM","created_at":"2026-07-05T03:10:59Z"},{"alias_kind":"pith_short_16","alias_value":"LVL7TQJRMTNMMEAH","created_at":"2026-07-05T03:10:59Z"},{"alias_kind":"pith_short_8","alias_value":"LVL7TQJR","created_at":"2026-07-05T03:10:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:LVL7TQJRMTNMMEAHXGPCNSH7TL","target":"record","payload":{"canonical_record":{"source":{"id":"2109.00878","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2021-09-02T12:41:12Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"ce5482c9bad3b5fe0d728949e55df67c535172e8fe37c80e83713e054052923c","abstract_canon_sha256":"ab14d840cfd25b76c9c3d6a84d8cb710b56416606b5a8c8cf546c24281cc29f4"},"schema_version":"1.0"},"canonical_sha256":"5d57f9c13164dac61007b99e26c8ff9aefd9c1c9c5da92266572a993c9b22ad0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:10:59.189023Z","signature_b64":"oqiI3v5+U9G3PWrGBNEebINAmJM86IdtiAClP1bR9f6sGAZDMogJpKkbsNasy0atLOyoAvkAk+SLEJ7LOfJGBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5d57f9c13164dac61007b99e26c8ff9aefd9c1c9c5da92266572a993c9b22ad0","last_reissued_at":"2026-07-05T03:10:59.188678Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:10:59.188678Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2109.00878","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":"2026-07-05T03:10:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"74URA8RtLoiqT/B9bvEjEAoPHBOG0B22Z1AIbrqty1IQDu3nCWN+2h8sjfzMEghKUAbKqzF0nUMSRNGZSURWDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T16:20:01.301637Z"},"content_sha256":"1fe1669f12e650ba975fe66c836a6e46c32897a594339c6d621ca6c397a82f6a","schema_version":"1.0","event_id":"sha256:1fe1669f12e650ba975fe66c836a6e46c32897a594339c6d621ca6c397a82f6a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:LVL7TQJRMTNMMEAHXGPCNSH7TL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Graded sets, graded groups, and Clifford algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CT","authors_text":"Wolfgang Bertram (IECL)","submitted_at":"2021-09-02T12:41:12Z","abstract_excerpt":"We define a general notion of centrally $\\Gamma$-graded sets and groups and of their graded products, and prove some basic results about the corresponding categories: most importantly, they form braided monoidal categories. Here, $\\Gamma$ is an arbitrary (generalized) ring. The case $\\Gamma$ = Z/2Z is studied in detail: it is related to Clifford algebras and their discrete Clifford groups (also called Salingaros Vee groups)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.00878","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2109.00878/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T03:10:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CxTEzo3hBXHGeWo03gQDg9EnXiVP43VC+eIx7yiBpeLWz1Upxj64oXAsgo7yWIOV2pVIh8Fj4NbG9/0pMw2KAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-23T16:20:01.302536Z"},"content_sha256":"e5aa2c98d25264331389058baea7c251731c9a22695857f54e6d92fb821b007d","schema_version":"1.0","event_id":"sha256:e5aa2c98d25264331389058baea7c251731c9a22695857f54e6d92fb821b007d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LVL7TQJRMTNMMEAHXGPCNSH7TL/bundle.json","state_url":"https://pith.science/pith/LVL7TQJRMTNMMEAHXGPCNSH7TL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LVL7TQJRMTNMMEAHXGPCNSH7TL/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":"2026-08-23T16:20:01Z","links":{"resolver":"https://pith.science/pith/LVL7TQJRMTNMMEAHXGPCNSH7TL","bundle":"https://pith.science/pith/LVL7TQJRMTNMMEAHXGPCNSH7TL/bundle.json","state":"https://pith.science/pith/LVL7TQJRMTNMMEAHXGPCNSH7TL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LVL7TQJRMTNMMEAHXGPCNSH7TL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:LVL7TQJRMTNMMEAHXGPCNSH7TL","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":"ab14d840cfd25b76c9c3d6a84d8cb710b56416606b5a8c8cf546c24281cc29f4","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2021-09-02T12:41:12Z","title_canon_sha256":"ce5482c9bad3b5fe0d728949e55df67c535172e8fe37c80e83713e054052923c"},"schema_version":"1.0","source":{"id":"2109.00878","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.00878","created_at":"2026-07-05T03:10:59Z"},{"alias_kind":"arxiv_version","alias_value":"2109.00878v1","created_at":"2026-07-05T03:10:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.00878","created_at":"2026-07-05T03:10:59Z"},{"alias_kind":"pith_short_12","alias_value":"LVL7TQJRMTNM","created_at":"2026-07-05T03:10:59Z"},{"alias_kind":"pith_short_16","alias_value":"LVL7TQJRMTNMMEAH","created_at":"2026-07-05T03:10:59Z"},{"alias_kind":"pith_short_8","alias_value":"LVL7TQJR","created_at":"2026-07-05T03:10:59Z"}],"graph_snapshots":[{"event_id":"sha256:e5aa2c98d25264331389058baea7c251731c9a22695857f54e6d92fb821b007d","target":"graph","created_at":"2026-07-05T03:10:59Z","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/2109.00878/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We define a general notion of centrally $\\Gamma$-graded sets and groups and of their graded products, and prove some basic results about the corresponding categories: most importantly, they form braided monoidal categories. Here, $\\Gamma$ is an arbitrary (generalized) ring. The case $\\Gamma$ = Z/2Z is studied in detail: it is related to Clifford algebras and their discrete Clifford groups (also called Salingaros Vee groups).","authors_text":"Wolfgang Bertram (IECL)","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2021-09-02T12:41:12Z","title":"Graded sets, graded groups, and Clifford algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.00878","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":"sha256:1fe1669f12e650ba975fe66c836a6e46c32897a594339c6d621ca6c397a82f6a","target":"record","created_at":"2026-07-05T03:10:59Z","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":"ab14d840cfd25b76c9c3d6a84d8cb710b56416606b5a8c8cf546c24281cc29f4","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CT","submitted_at":"2021-09-02T12:41:12Z","title_canon_sha256":"ce5482c9bad3b5fe0d728949e55df67c535172e8fe37c80e83713e054052923c"},"schema_version":"1.0","source":{"id":"2109.00878","kind":"arxiv","version":1}},"canonical_sha256":"5d57f9c13164dac61007b99e26c8ff9aefd9c1c9c5da92266572a993c9b22ad0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5d57f9c13164dac61007b99e26c8ff9aefd9c1c9c5da92266572a993c9b22ad0","first_computed_at":"2026-07-05T03:10:59.188678Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:10:59.188678Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oqiI3v5+U9G3PWrGBNEebINAmJM86IdtiAClP1bR9f6sGAZDMogJpKkbsNasy0atLOyoAvkAk+SLEJ7LOfJGBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T03:10:59.189023Z","signed_message":"canonical_sha256_bytes"},"source_id":"2109.00878","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1fe1669f12e650ba975fe66c836a6e46c32897a594339c6d621ca6c397a82f6a","sha256:e5aa2c98d25264331389058baea7c251731c9a22695857f54e6d92fb821b007d"],"state_sha256":"6fa0f5f48603960ae33a69a99ee0ab9d48d7ca08df7ccff574744e7be69c7e67"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iB5+u9KcUVSpL53fAlj0LXGok3wGcl53vvRpTilb/CMD+s/3Q8vYDtnGN8PYgsPm4Olz0mLSTLI+xzn2Ld5rCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-23T16:20:01.309545Z","bundle_sha256":"6c889180cd205917c81840d0c0243332432b922719446a55bba4dfd9864934b8"}}