{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:7RTTKOHKEPJIWJBLD6FVDXLJH3","short_pith_number":"pith:7RTTKOHK","canonical_record":{"source":{"id":"1607.07559","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-07-26T06:59:53Z","cross_cats_sorted":[],"title_canon_sha256":"3d5ba418a58cf9cf5165026dc31fd12c5eb86a3e8071e1a15ff1e794ce4defe0","abstract_canon_sha256":"ca92530b11d03e4569da3405c1fdf4ec09abe8ce1365ebbe09afd1996d59d255"},"schema_version":"1.0"},"canonical_sha256":"fc673538ea23d28b242b1f8b51dd693ee35ba791830559f7dbd66bb76d707678","source":{"kind":"arxiv","id":"1607.07559","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.07559","created_at":"2026-05-18T00:16:50Z"},{"alias_kind":"arxiv_version","alias_value":"1607.07559v2","created_at":"2026-05-18T00:16:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.07559","created_at":"2026-05-18T00:16:50Z"},{"alias_kind":"pith_short_12","alias_value":"7RTTKOHKEPJI","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7RTTKOHKEPJIWJBL","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7RTTKOHK","created_at":"2026-05-18T12:30:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:7RTTKOHKEPJIWJBLD6FVDXLJH3","target":"record","payload":{"canonical_record":{"source":{"id":"1607.07559","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-07-26T06:59:53Z","cross_cats_sorted":[],"title_canon_sha256":"3d5ba418a58cf9cf5165026dc31fd12c5eb86a3e8071e1a15ff1e794ce4defe0","abstract_canon_sha256":"ca92530b11d03e4569da3405c1fdf4ec09abe8ce1365ebbe09afd1996d59d255"},"schema_version":"1.0"},"canonical_sha256":"fc673538ea23d28b242b1f8b51dd693ee35ba791830559f7dbd66bb76d707678","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:50.914452Z","signature_b64":"hpRgibyNkYiS2CODCwSVb6LtFsaSjqfsjF0Hs2OGbGUqISSlPuheLPaxWX76zmFlNIL7iR0wynJ5c2jzmN1vCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc673538ea23d28b242b1f8b51dd693ee35ba791830559f7dbd66bb76d707678","last_reissued_at":"2026-05-18T00:16:50.913542Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:50.913542Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1607.07559","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":"2026-05-18T00:16:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IeIUQQb6tGZ/yw4vX1gqfjzuSGVHCSoluTc4brWYmmZOFB0a1BraHzIS5vzvmdbjQf0evo9KwDkyh6s/QLLyBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T15:07:52.414622Z"},"content_sha256":"e5db2b06c60ebba5642f04f647d7954ffbbb45faf55206fb4631b936ef1bcef1","schema_version":"1.0","event_id":"sha256:e5db2b06c60ebba5642f04f647d7954ffbbb45faf55206fb4631b936ef1bcef1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:7RTTKOHKEPJIWJBLD6FVDXLJH3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Symmetric products and subgroup lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AT","authors_text":"Markus Hausmann","submitted_at":"2016-07-26T06:59:53Z","abstract_excerpt":"Let G be a finite group. We show that the rational homotopy groups of symmetric products of the G-equivariant sphere spectrum are naturally isomorphic to the rational homology groups of certain subcomplexes of the subgroup lattice of G."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.07559","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":"2026-05-18T00:16:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FrKogNHS5ER+WoWpCold+XhrRyegfzyv0mAfgqO5HMaNrOCGhYKnIAU0IuzT73qwBSARRyo23BQneT/DgtneAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T15:07:52.415242Z"},"content_sha256":"ec29f8530274bd1da9720197c15761fd863484ca6451734e9a0164be573784dc","schema_version":"1.0","event_id":"sha256:ec29f8530274bd1da9720197c15761fd863484ca6451734e9a0164be573784dc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7RTTKOHKEPJIWJBLD6FVDXLJH3/bundle.json","state_url":"https://pith.science/pith/7RTTKOHKEPJIWJBLD6FVDXLJH3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7RTTKOHKEPJIWJBLD6FVDXLJH3/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":"2026-05-30T15:07:52Z","links":{"resolver":"https://pith.science/pith/7RTTKOHKEPJIWJBLD6FVDXLJH3","bundle":"https://pith.science/pith/7RTTKOHKEPJIWJBLD6FVDXLJH3/bundle.json","state":"https://pith.science/pith/7RTTKOHKEPJIWJBLD6FVDXLJH3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7RTTKOHKEPJIWJBLD6FVDXLJH3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:7RTTKOHKEPJIWJBLD6FVDXLJH3","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":"ca92530b11d03e4569da3405c1fdf4ec09abe8ce1365ebbe09afd1996d59d255","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-07-26T06:59:53Z","title_canon_sha256":"3d5ba418a58cf9cf5165026dc31fd12c5eb86a3e8071e1a15ff1e794ce4defe0"},"schema_version":"1.0","source":{"id":"1607.07559","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.07559","created_at":"2026-05-18T00:16:50Z"},{"alias_kind":"arxiv_version","alias_value":"1607.07559v2","created_at":"2026-05-18T00:16:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.07559","created_at":"2026-05-18T00:16:50Z"},{"alias_kind":"pith_short_12","alias_value":"7RTTKOHKEPJI","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7RTTKOHKEPJIWJBL","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7RTTKOHK","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:ec29f8530274bd1da9720197c15761fd863484ca6451734e9a0164be573784dc","target":"graph","created_at":"2026-05-18T00:16:50Z","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 G be a finite group. We show that the rational homotopy groups of symmetric products of the G-equivariant sphere spectrum are naturally isomorphic to the rational homology groups of certain subcomplexes of the subgroup lattice of G.","authors_text":"Markus Hausmann","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-07-26T06:59:53Z","title":"Symmetric products and subgroup lattices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.07559","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:e5db2b06c60ebba5642f04f647d7954ffbbb45faf55206fb4631b936ef1bcef1","target":"record","created_at":"2026-05-18T00:16:50Z","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":"ca92530b11d03e4569da3405c1fdf4ec09abe8ce1365ebbe09afd1996d59d255","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2016-07-26T06:59:53Z","title_canon_sha256":"3d5ba418a58cf9cf5165026dc31fd12c5eb86a3e8071e1a15ff1e794ce4defe0"},"schema_version":"1.0","source":{"id":"1607.07559","kind":"arxiv","version":2}},"canonical_sha256":"fc673538ea23d28b242b1f8b51dd693ee35ba791830559f7dbd66bb76d707678","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fc673538ea23d28b242b1f8b51dd693ee35ba791830559f7dbd66bb76d707678","first_computed_at":"2026-05-18T00:16:50.913542Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:16:50.913542Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hpRgibyNkYiS2CODCwSVb6LtFsaSjqfsjF0Hs2OGbGUqISSlPuheLPaxWX76zmFlNIL7iR0wynJ5c2jzmN1vCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:16:50.914452Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.07559","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e5db2b06c60ebba5642f04f647d7954ffbbb45faf55206fb4631b936ef1bcef1","sha256:ec29f8530274bd1da9720197c15761fd863484ca6451734e9a0164be573784dc"],"state_sha256":"4fc8b1d1b60e62289bf4194e6c4e8e7f0150c0ef7da6c06e81b1892bc85e8298"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Xy2t4yTTO3PnysyN/6tOtLNz0dLMJ4ufy0GvX1Xy2jKu/UxBa6gV+WpJ+boU7wRWdmrMEwJvfE+YZDH4iIjeAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T15:07:52.418422Z","bundle_sha256":"f73c5ef331e111d17c7fdc10770448562a40c9ede5e98f16d53593475a6749c7"}}