{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:PUSMGASUKTVK7RP6HVG7AQAZDI","short_pith_number":"pith:PUSMGASU","canonical_record":{"source":{"id":"2412.01128","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2024-12-02T05:08:53Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"af4fa96375ec41261b9d7bf4fc267bc2c0fb1cce83f2ee082260c0f73ec533e7","abstract_canon_sha256":"80bb836b6c7cc81ea906736e504ec47945d875850821503b99164ec53847cab7"},"schema_version":"1.0"},"canonical_sha256":"7d24c3025454eaafc5fe3d4df040191a232e0e866e6870f29fe9604991f4a2cb","source":{"kind":"arxiv","id":"2412.01128","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.01128","created_at":"2026-07-05T09:42:54Z"},{"alias_kind":"arxiv_version","alias_value":"2412.01128v1","created_at":"2026-07-05T09:42:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.01128","created_at":"2026-07-05T09:42:54Z"},{"alias_kind":"pith_short_12","alias_value":"PUSMGASUKTVK","created_at":"2026-07-05T09:42:54Z"},{"alias_kind":"pith_short_16","alias_value":"PUSMGASUKTVK7RP6","created_at":"2026-07-05T09:42:54Z"},{"alias_kind":"pith_short_8","alias_value":"PUSMGASU","created_at":"2026-07-05T09:42:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:PUSMGASUKTVK7RP6HVG7AQAZDI","target":"record","payload":{"canonical_record":{"source":{"id":"2412.01128","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2024-12-02T05:08:53Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"af4fa96375ec41261b9d7bf4fc267bc2c0fb1cce83f2ee082260c0f73ec533e7","abstract_canon_sha256":"80bb836b6c7cc81ea906736e504ec47945d875850821503b99164ec53847cab7"},"schema_version":"1.0"},"canonical_sha256":"7d24c3025454eaafc5fe3d4df040191a232e0e866e6870f29fe9604991f4a2cb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:42:54.477290Z","signature_b64":"6ji3y3hwaYTwWP0chChCE0eor19lp4mDeKaHZo6LhH1hOv6jmPh9APE2tyFQU1kuBZ1Fwo6U8uFepbqmXUm6BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7d24c3025454eaafc5fe3d4df040191a232e0e866e6870f29fe9604991f4a2cb","last_reissued_at":"2026-07-05T09:42:54.476839Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:42:54.476839Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.01128","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":"2026-07-05T09:42:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"noWNxbf5PB/Ndz4ShZcNtfOS/IObO9d8IqGl5TIYTbeXIGA75Db8m045qaKgYhij6PmAvP9yNPCozkESbi/MAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-16T02:12:57.401242Z"},"content_sha256":"a848bd8de9c4caf9a88b4f464a5dc3d21a83a3c9926f08bda8efbcd5e9e42842","schema_version":"1.0","event_id":"sha256:a848bd8de9c4caf9a88b4f464a5dc3d21a83a3c9926f08bda8efbcd5e9e42842"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:PUSMGASUKTVK7RP6HVG7AQAZDI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Representation stability in the (co)homology of vertical configuration spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AT","authors_text":"Chenglu Wang, Chunye Yang, David Baron, Jennifer C. H. Wilson, Urshita Pal","submitted_at":"2024-12-02T05:08:53Z","abstract_excerpt":"In this paper, we study sequences of topological spaces called \"vertical configuration spaces\" of points in Euclidean space. We apply the theory of FI$_G$-modules, and results of Bianchi-Kranhold, to show that their (co)homology groups are \"representation stable\" with respect to natural actions of wreath products $S_k \\wr S_n$. In particular, we show that in each (co)homological degree, the (co)homology groups (viewed as $S_k \\wr S_n$-representations) can be expressed as induced representations of a specific form. Consequently, the characters of their rational (co)homology groups, and the patt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.01128","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/2412.01128/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":"2026-07-05T09:42:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HYrEwljCZwH6DF2BTR6ePOtkj4GKutAwk0+25I47ttnxKISSw3PYqgIcGCbDhinvzYvwhA+195G6OBgw2EG1AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-16T02:12:57.401635Z"},"content_sha256":"d2311754180402d7936b3cecdd13fa5c6cc688ecab02908f4a306436d2c60115","schema_version":"1.0","event_id":"sha256:d2311754180402d7936b3cecdd13fa5c6cc688ecab02908f4a306436d2c60115"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PUSMGASUKTVK7RP6HVG7AQAZDI/bundle.json","state_url":"https://pith.science/pith/PUSMGASUKTVK7RP6HVG7AQAZDI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PUSMGASUKTVK7RP6HVG7AQAZDI/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":"2026-07-16T02:12:57Z","links":{"resolver":"https://pith.science/pith/PUSMGASUKTVK7RP6HVG7AQAZDI","bundle":"https://pith.science/pith/PUSMGASUKTVK7RP6HVG7AQAZDI/bundle.json","state":"https://pith.science/pith/PUSMGASUKTVK7RP6HVG7AQAZDI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PUSMGASUKTVK7RP6HVG7AQAZDI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:PUSMGASUKTVK7RP6HVG7AQAZDI","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":"80bb836b6c7cc81ea906736e504ec47945d875850821503b99164ec53847cab7","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2024-12-02T05:08:53Z","title_canon_sha256":"af4fa96375ec41261b9d7bf4fc267bc2c0fb1cce83f2ee082260c0f73ec533e7"},"schema_version":"1.0","source":{"id":"2412.01128","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.01128","created_at":"2026-07-05T09:42:54Z"},{"alias_kind":"arxiv_version","alias_value":"2412.01128v1","created_at":"2026-07-05T09:42:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.01128","created_at":"2026-07-05T09:42:54Z"},{"alias_kind":"pith_short_12","alias_value":"PUSMGASUKTVK","created_at":"2026-07-05T09:42:54Z"},{"alias_kind":"pith_short_16","alias_value":"PUSMGASUKTVK7RP6","created_at":"2026-07-05T09:42:54Z"},{"alias_kind":"pith_short_8","alias_value":"PUSMGASU","created_at":"2026-07-05T09:42:54Z"}],"graph_snapshots":[{"event_id":"sha256:d2311754180402d7936b3cecdd13fa5c6cc688ecab02908f4a306436d2c60115","target":"graph","created_at":"2026-07-05T09:42:54Z","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/2412.01128/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we study sequences of topological spaces called \"vertical configuration spaces\" of points in Euclidean space. We apply the theory of FI$_G$-modules, and results of Bianchi-Kranhold, to show that their (co)homology groups are \"representation stable\" with respect to natural actions of wreath products $S_k \\wr S_n$. In particular, we show that in each (co)homological degree, the (co)homology groups (viewed as $S_k \\wr S_n$-representations) can be expressed as induced representations of a specific form. Consequently, the characters of their rational (co)homology groups, and the patt","authors_text":"Chenglu Wang, Chunye Yang, David Baron, Jennifer C. H. Wilson, Urshita Pal","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2024-12-02T05:08:53Z","title":"Representation stability in the (co)homology of vertical configuration spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.01128","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:a848bd8de9c4caf9a88b4f464a5dc3d21a83a3c9926f08bda8efbcd5e9e42842","target":"record","created_at":"2026-07-05T09:42:54Z","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":"80bb836b6c7cc81ea906736e504ec47945d875850821503b99164ec53847cab7","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2024-12-02T05:08:53Z","title_canon_sha256":"af4fa96375ec41261b9d7bf4fc267bc2c0fb1cce83f2ee082260c0f73ec533e7"},"schema_version":"1.0","source":{"id":"2412.01128","kind":"arxiv","version":1}},"canonical_sha256":"7d24c3025454eaafc5fe3d4df040191a232e0e866e6870f29fe9604991f4a2cb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7d24c3025454eaafc5fe3d4df040191a232e0e866e6870f29fe9604991f4a2cb","first_computed_at":"2026-07-05T09:42:54.476839Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:42:54.476839Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6ji3y3hwaYTwWP0chChCE0eor19lp4mDeKaHZo6LhH1hOv6jmPh9APE2tyFQU1kuBZ1Fwo6U8uFepbqmXUm6BQ==","signature_status":"signed_v1","signed_at":"2026-07-05T09:42:54.477290Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.01128","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a848bd8de9c4caf9a88b4f464a5dc3d21a83a3c9926f08bda8efbcd5e9e42842","sha256:d2311754180402d7936b3cecdd13fa5c6cc688ecab02908f4a306436d2c60115"],"state_sha256":"f0aa696f2737374ef1facc94f12af0e71ce9ce82063bac544346f3bb6fc83783"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MV1zrX0eSLd5RFWyl3XnPdEFolEA5nJfnDjYjdpvYOk9JmrsE2kD9dKE4bm3sDMIpGCNMZisNaRSsBEP7c8JBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-16T02:12:57.403840Z","bundle_sha256":"daa87d49d580d774c96fd515411fde7a2dc33691104dc51d13cc88bd89e58caa"}}