{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:3VEWXGWXCM7HBASYF3245SJV6B","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":"1bf6496cf891ad04736e120c88fec17738548eb36c8af24d0f8bb82ab1090164","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2026-06-05T13:58:40Z","title_canon_sha256":"520860295e741921c9476d6a726f4470cac9a46146f299216251c94d26364c19"},"schema_version":"1.0","source":{"id":"2606.07286","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.07286","created_at":"2026-06-08T01:05:17Z"},{"alias_kind":"arxiv_version","alias_value":"2606.07286v1","created_at":"2026-06-08T01:05:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.07286","created_at":"2026-06-08T01:05:17Z"},{"alias_kind":"pith_short_12","alias_value":"3VEWXGWXCM7H","created_at":"2026-06-08T01:05:17Z"},{"alias_kind":"pith_short_16","alias_value":"3VEWXGWXCM7HBASY","created_at":"2026-06-08T01:05:17Z"},{"alias_kind":"pith_short_8","alias_value":"3VEWXGWX","created_at":"2026-06-08T01:05:17Z"}],"graph_snapshots":[{"event_id":"sha256:33737e15c42add8a82d49a89af2e0d6d35909aa47ad0679ef61809268595a4d2","target":"graph","created_at":"2026-06-08T01:05:17Z","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/2606.07286/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove Fourier positivity for spherical functions on a semisimple linear algebraic group $G$ over a local field restricted to its split tori $A$ for unitary principal series parameters of $G$. For ${\\rm SL}_n(F)$, where $F$ is a local field, we obtain an explicit recursive formula for the Fourier transform on the diagonal split torus in terms of local Rankin--Selberg factors for ${\\rm GL}_n\\times {\\rm GL}_{n-1}$, together with uniform exponential lower bounds in the spectral parameters.\n  The main input is a Plancherel expansion for the restriction of a ${\\rm GL}_n(F)$-spherical function to ","authors_text":"Dongwen Liu, Genkai Zhang, Jun Yu, Michael Bj\\\"orklund","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2026-06-05T13:58:40Z","title":"Fourier positivity for spherical functions I: split tori and spherical principal series"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.07286","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":"sha256:086b31fc1f2611f088d8785f3bed90622e6525161d94faedadbf28736065782c","target":"record","created_at":"2026-06-08T01:05:17Z","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":"1bf6496cf891ad04736e120c88fec17738548eb36c8af24d0f8bb82ab1090164","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2026-06-05T13:58:40Z","title_canon_sha256":"520860295e741921c9476d6a726f4470cac9a46146f299216251c94d26364c19"},"schema_version":"1.0","source":{"id":"2606.07286","kind":"arxiv","version":1}},"canonical_sha256":"dd496b9ad7133e7082582ef5cec935f046caaacff28f426f6620bc4c57dbae7c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dd496b9ad7133e7082582ef5cec935f046caaacff28f426f6620bc4c57dbae7c","first_computed_at":"2026-06-08T01:05:17.487127Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-08T01:05:17.487127Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lwHzxbWr/3tbAn9mXlzyVmhtLGTHEbplbehddMNaO98K5mYFI3M4XtW0NgtSeCJjtQJSd/INvbJekDPQFnrQBw==","signature_status":"signed_v1","signed_at":"2026-06-08T01:05:17.487910Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.07286","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:086b31fc1f2611f088d8785f3bed90622e6525161d94faedadbf28736065782c","sha256:33737e15c42add8a82d49a89af2e0d6d35909aa47ad0679ef61809268595a4d2"],"state_sha256":"c15b8c85e207ef456bade9fb0313c1f8d185e84e4a5326a1273a8aca532bdc46"}