{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:7Z3SZXGBZ4I36QN4DLVQAEF2Q3","short_pith_number":"pith:7Z3SZXGB","canonical_record":{"source":{"id":"2506.14304","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2025-06-17T08:30:45Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"c81606f594ecbd4c8be6a86b8e295566aa3d26c67cb9719fc384b9d9c50afd05","abstract_canon_sha256":"62ea258124e8da8a1d3ebcb4f2d55735e2f9725923c2f00855c9144e1d202334"},"schema_version":"1.0"},"canonical_sha256":"fe772cdcc1cf11bf41bc1aeb0010ba86e2f0bf9d7730d7c6ceca3bbb1e0f0adc","source":{"kind":"arxiv","id":"2506.14304","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.14304","created_at":"2026-07-05T11:59:15Z"},{"alias_kind":"arxiv_version","alias_value":"2506.14304v2","created_at":"2026-07-05T11:59:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.14304","created_at":"2026-07-05T11:59:15Z"},{"alias_kind":"pith_short_12","alias_value":"7Z3SZXGBZ4I3","created_at":"2026-07-05T11:59:15Z"},{"alias_kind":"pith_short_16","alias_value":"7Z3SZXGBZ4I36QN4","created_at":"2026-07-05T11:59:15Z"},{"alias_kind":"pith_short_8","alias_value":"7Z3SZXGB","created_at":"2026-07-05T11:59:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:7Z3SZXGBZ4I36QN4DLVQAEF2Q3","target":"record","payload":{"canonical_record":{"source":{"id":"2506.14304","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2025-06-17T08:30:45Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"c81606f594ecbd4c8be6a86b8e295566aa3d26c67cb9719fc384b9d9c50afd05","abstract_canon_sha256":"62ea258124e8da8a1d3ebcb4f2d55735e2f9725923c2f00855c9144e1d202334"},"schema_version":"1.0"},"canonical_sha256":"fe772cdcc1cf11bf41bc1aeb0010ba86e2f0bf9d7730d7c6ceca3bbb1e0f0adc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:59:15.954191Z","signature_b64":"9cMK5YlRaJrk8j14aHg4y0I/RqbKvU/wl2i0jXPJnXqvwU2OlOPoGQu1oDltcd0tY7Y2nCJduK1l+rOyy88iDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe772cdcc1cf11bf41bc1aeb0010ba86e2f0bf9d7730d7c6ceca3bbb1e0f0adc","last_reissued_at":"2026-07-05T11:59:15.953733Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:59:15.953733Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.14304","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":"2026-07-05T11:59:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4SAsN4JWrkXmHoISm5ZXgwDKlzf0ePecweR5QF5Tq0sYMyzt/hA0reEO9EJ3gfKQVmvRzx51CffCVsZuMycZBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T14:32:05.804306Z"},"content_sha256":"2f2f34e427fc6a3831ba36254200b2c66c71b290444af359f0ba4c245e21f8f6","schema_version":"1.0","event_id":"sha256:2f2f34e427fc6a3831ba36254200b2c66c71b290444af359f0ba4c245e21f8f6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:7Z3SZXGBZ4I36QN4DLVQAEF2Q3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Partial Group Symmetry in Figures I: Semidirect Products and the Six Coins","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.GR","authors_text":"Takahiro Hayashi","submitted_at":"2025-06-17T08:30:45Z","abstract_excerpt":"In this paper, we construct a partial group \\(\\mathcal{P}(F)\\) that represents the \"partial symmetry\" inherent in a subset \\(F\\) of \\(d\\)-dimensional Euclidean space. In cases where \\(F\\) is not connected, \\(\\mathcal{P}(F)\\) captures more detailed information than the conventional symmetry group \\(G(F)\\). To establish a stronger connection between \\(\\mathcal{P}(F)\\) and \\(F\\), we introduce a novel definition of partial group action. Furthermore, to characterize \\(\\mathcal{P}(F)\\) in specific cases, we define partial group actions on other partial groups and present a construction of the corres"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.14304","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.14304/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":"2026-07-05T11:59:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"G6PPwsgAnhbxdQJUudRZcF9pQpb3zQrbrA6Nkbu1Iw/x3/6mXuLoJLQZfCiw1nWrO4eWC3dj1uqbCdSAiO5iBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-14T14:32:05.804828Z"},"content_sha256":"8e9851cfda7fb937c731ee5a404d13cefa443aba828524d2dbe54b787106fcf7","schema_version":"1.0","event_id":"sha256:8e9851cfda7fb937c731ee5a404d13cefa443aba828524d2dbe54b787106fcf7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7Z3SZXGBZ4I36QN4DLVQAEF2Q3/bundle.json","state_url":"https://pith.science/pith/7Z3SZXGBZ4I36QN4DLVQAEF2Q3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7Z3SZXGBZ4I36QN4DLVQAEF2Q3/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":"2026-08-14T14:32:05Z","links":{"resolver":"https://pith.science/pith/7Z3SZXGBZ4I36QN4DLVQAEF2Q3","bundle":"https://pith.science/pith/7Z3SZXGBZ4I36QN4DLVQAEF2Q3/bundle.json","state":"https://pith.science/pith/7Z3SZXGBZ4I36QN4DLVQAEF2Q3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7Z3SZXGBZ4I36QN4DLVQAEF2Q3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:7Z3SZXGBZ4I36QN4DLVQAEF2Q3","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":"62ea258124e8da8a1d3ebcb4f2d55735e2f9725923c2f00855c9144e1d202334","cross_cats_sorted":["math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2025-06-17T08:30:45Z","title_canon_sha256":"c81606f594ecbd4c8be6a86b8e295566aa3d26c67cb9719fc384b9d9c50afd05"},"schema_version":"1.0","source":{"id":"2506.14304","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.14304","created_at":"2026-07-05T11:59:15Z"},{"alias_kind":"arxiv_version","alias_value":"2506.14304v2","created_at":"2026-07-05T11:59:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.14304","created_at":"2026-07-05T11:59:15Z"},{"alias_kind":"pith_short_12","alias_value":"7Z3SZXGBZ4I3","created_at":"2026-07-05T11:59:15Z"},{"alias_kind":"pith_short_16","alias_value":"7Z3SZXGBZ4I36QN4","created_at":"2026-07-05T11:59:15Z"},{"alias_kind":"pith_short_8","alias_value":"7Z3SZXGB","created_at":"2026-07-05T11:59:15Z"}],"graph_snapshots":[{"event_id":"sha256:8e9851cfda7fb937c731ee5a404d13cefa443aba828524d2dbe54b787106fcf7","target":"graph","created_at":"2026-07-05T11:59:15Z","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/2506.14304/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we construct a partial group \\(\\mathcal{P}(F)\\) that represents the \"partial symmetry\" inherent in a subset \\(F\\) of \\(d\\)-dimensional Euclidean space. In cases where \\(F\\) is not connected, \\(\\mathcal{P}(F)\\) captures more detailed information than the conventional symmetry group \\(G(F)\\). To establish a stronger connection between \\(\\mathcal{P}(F)\\) and \\(F\\), we introduce a novel definition of partial group action. Furthermore, to characterize \\(\\mathcal{P}(F)\\) in specific cases, we define partial group actions on other partial groups and present a construction of the corres","authors_text":"Takahiro Hayashi","cross_cats":["math.QA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2025-06-17T08:30:45Z","title":"Partial Group Symmetry in Figures I: Semidirect Products and the Six Coins"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.14304","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":"sha256:2f2f34e427fc6a3831ba36254200b2c66c71b290444af359f0ba4c245e21f8f6","target":"record","created_at":"2026-07-05T11:59:15Z","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":"62ea258124e8da8a1d3ebcb4f2d55735e2f9725923c2f00855c9144e1d202334","cross_cats_sorted":["math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2025-06-17T08:30:45Z","title_canon_sha256":"c81606f594ecbd4c8be6a86b8e295566aa3d26c67cb9719fc384b9d9c50afd05"},"schema_version":"1.0","source":{"id":"2506.14304","kind":"arxiv","version":2}},"canonical_sha256":"fe772cdcc1cf11bf41bc1aeb0010ba86e2f0bf9d7730d7c6ceca3bbb1e0f0adc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fe772cdcc1cf11bf41bc1aeb0010ba86e2f0bf9d7730d7c6ceca3bbb1e0f0adc","first_computed_at":"2026-07-05T11:59:15.953733Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:59:15.953733Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9cMK5YlRaJrk8j14aHg4y0I/RqbKvU/wl2i0jXPJnXqvwU2OlOPoGQu1oDltcd0tY7Y2nCJduK1l+rOyy88iDw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:59:15.954191Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.14304","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2f2f34e427fc6a3831ba36254200b2c66c71b290444af359f0ba4c245e21f8f6","sha256:8e9851cfda7fb937c731ee5a404d13cefa443aba828524d2dbe54b787106fcf7"],"state_sha256":"21354bce632e5ab7c2c8585a07038671beca4ec56485ff60f81a06b213fcce5a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xTDPzpbZkms7GXVZ6Wd2jR99b/MP8Uscr+3uNjD4J47tWNtMwF4TA0qLFPo2kipAjv4Iu2D/QKwzuXKysuk6Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-14T14:32:05.810121Z","bundle_sha256":"304afaad8f35ef483dadf923b8997c40319eaeb6e14fa668c70baf7893b7519a"}}