{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:FIHIZPUJLCKM6D4XDWUSMGR7EQ","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":"6507b034799867e315494450cd79c340bfd39f880e8b8c3166d7d4e90fe9b1f5","cross_cats_sorted":["math.NT","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-02-10T04:19:37Z","title_canon_sha256":"f3d1f5af4340ff2ada2fe23c76eecd4ec2e8004a9e4805ca7cf5ffa5bf2be86b"},"schema_version":"1.0","source":{"id":"2502.06143","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.06143","created_at":"2026-07-05T10:11:48Z"},{"alias_kind":"arxiv_version","alias_value":"2502.06143v1","created_at":"2026-07-05T10:11:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.06143","created_at":"2026-07-05T10:11:48Z"},{"alias_kind":"pith_short_12","alias_value":"FIHIZPUJLCKM","created_at":"2026-07-05T10:11:48Z"},{"alias_kind":"pith_short_16","alias_value":"FIHIZPUJLCKM6D4X","created_at":"2026-07-05T10:11:48Z"},{"alias_kind":"pith_short_8","alias_value":"FIHIZPUJ","created_at":"2026-07-05T10:11:48Z"}],"graph_snapshots":[{"event_id":"sha256:590a273d391e741a0ac050d849d38cb0a208c68d918d6efd52d04b9e1de18677","target":"graph","created_at":"2026-07-05T10:11:48Z","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/2502.06143/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We establish that the singular numbers (arising from Cartan decomposition) and corners (emerging from Iwasawa decomposition) in split reductive groups over non-archimedean fields are fundamentally determined by Hall-Littlewood polynomials. Through applications of the Satake isomorphism, we extend Van Peski's results (arXiv:2011.09356, Theorem 1.3) to encompass arbitrary root systems. Leveraging this theoretical foundation, we further develop Shen's work (arXiv:2411.01104, Theorem 1.1) to demonstrate that both singular numbers and corners of such products exhibit minimal separation. This charac","authors_text":"Jiahe Shen","cross_cats":["math.NT","math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-02-10T04:19:37Z","title":"Gaussian Universality of Products Over Split Reductive Groups and the Satake Isomorphism"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.06143","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":"sha256:3bc55d17add463bc5d113bf1157aa782744f1005adc0c53cd666ae8c390b3acf","target":"record","created_at":"2026-07-05T10:11:48Z","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":"6507b034799867e315494450cd79c340bfd39f880e8b8c3166d7d4e90fe9b1f5","cross_cats_sorted":["math.NT","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-02-10T04:19:37Z","title_canon_sha256":"f3d1f5af4340ff2ada2fe23c76eecd4ec2e8004a9e4805ca7cf5ffa5bf2be86b"},"schema_version":"1.0","source":{"id":"2502.06143","kind":"arxiv","version":1}},"canonical_sha256":"2a0e8cbe895894cf0f971da9261a3f24004494b274d10586ae2d41474da51205","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2a0e8cbe895894cf0f971da9261a3f24004494b274d10586ae2d41474da51205","first_computed_at":"2026-07-05T10:11:48.271173Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:11:48.271173Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ng+3qHkz69GnlUPCJ7Pz0vlLI1sL67yubsjOWNHyOcUtJXisynr6Llp5kG/WggKwFVQFUAXdqbdnoPAuOalGAA==","signature_status":"signed_v1","signed_at":"2026-07-05T10:11:48.271700Z","signed_message":"canonical_sha256_bytes"},"source_id":"2502.06143","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3bc55d17add463bc5d113bf1157aa782744f1005adc0c53cd666ae8c390b3acf","sha256:590a273d391e741a0ac050d849d38cb0a208c68d918d6efd52d04b9e1de18677"],"state_sha256":"d4928b15946d7969658df0e656a40db72cf780ce2a0590c01b2d2f15f53b81a0"}