{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:GWVVFLPC7HCP4VE2TEWMG32NRS","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":"ee910baa07af6bdbbeddfced5252703e554df08378dcf6580de6d11fe21e6954","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-14T10:23:27Z","title_canon_sha256":"694640fe2207ca5f972b69e452c04f59701682b8814877885c31cd7220f1605d"},"schema_version":"1.0","source":{"id":"2606.15729","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.15729","created_at":"2026-07-17T01:20:52Z"},{"alias_kind":"arxiv_version","alias_value":"2606.15729v2","created_at":"2026-07-17T01:20:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.15729","created_at":"2026-07-17T01:20:52Z"},{"alias_kind":"pith_short_12","alias_value":"GWVVFLPC7HCP","created_at":"2026-07-17T01:20:52Z"},{"alias_kind":"pith_short_16","alias_value":"GWVVFLPC7HCP4VE2","created_at":"2026-07-17T01:20:52Z"},{"alias_kind":"pith_short_8","alias_value":"GWVVFLPC","created_at":"2026-07-17T01:20:52Z"}],"graph_snapshots":[{"event_id":"sha256:8a7e3a1c8db11f2598e29613db937ec27f2b9677991af5c5084bf733f6838952","target":"graph","created_at":"2026-07-17T01:20:52Z","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/2606.15729/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This paper revisits the functional equation of the symmetric-square $L$-function from the perspective of trace formulas. We show that, for level-one holomorphic cusp forms and even Hecke--Maass cusp forms, this symmetry can be recovered directly from the Petersson and Kuznetsov trace formulas. Although the functional equation itself is classical, our aim is to provide a concrete and transparent example of how the analytic structure of an $L$-function can emerge from a comparison of the spectral and geometric sides of a trace formula. The argument leads to an averaged reciprocity identity and t","authors_text":"Dongsheng Wu, Taiwang Deng","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-14T10:23:27Z","title":"A trace formula derivation of the functional equation for symmetric square L-functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.15729","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:a2af9e212ea71f1536054fbed98c15be33b84bdc0a721ec5af1d9742ac5137cf","target":"record","created_at":"2026-07-17T01:20:52Z","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":"ee910baa07af6bdbbeddfced5252703e554df08378dcf6580de6d11fe21e6954","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-06-14T10:23:27Z","title_canon_sha256":"694640fe2207ca5f972b69e452c04f59701682b8814877885c31cd7220f1605d"},"schema_version":"1.0","source":{"id":"2606.15729","kind":"arxiv","version":2}},"canonical_sha256":"35ab52ade2f9c4fe549a992cc36f4d8c808b0029c28c573f6cf4345987c0f224","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"35ab52ade2f9c4fe549a992cc36f4d8c808b0029c28c573f6cf4345987c0f224","first_computed_at":"2026-07-17T01:20:52.487886Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-17T01:20:52.487886Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3fAPrNF+O07pFPGBT8sL34jkmCMgL82xXfYv/dDLq+g4LtgrGv80zKnmJPkWEKz7HdO5+pdUGD/vpzTnre8DDg==","signature_status":"signed_v1","signed_at":"2026-07-17T01:20:52.488737Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.15729","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a2af9e212ea71f1536054fbed98c15be33b84bdc0a721ec5af1d9742ac5137cf","sha256:8a7e3a1c8db11f2598e29613db937ec27f2b9677991af5c5084bf733f6838952"],"state_sha256":"7434a584b4dc87bed138277e169b0039dbd59d4671469db5d7309456015c5b0c"}