{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:PGG26LY6BO2BOVQW3OTX2OYJ2R","short_pith_number":"pith:PGG26LY6","canonical_record":{"source":{"id":"1601.07887","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-01-28T20:26:20Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"e5df71017b905545f81512aa9ef80f2b49524f45daee5c6f32a142b3a40efee6","abstract_canon_sha256":"c7b7293bfd5c4a8ba2dabfa0ae698f7c8d6362c1d4b6e1acb9c2f34bf4878030"},"schema_version":"1.0"},"canonical_sha256":"798daf2f1e0bb4175616dba77d3b09d447bfb8dca438a8b1d2cd14a07e462180","source":{"kind":"arxiv","id":"1601.07887","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.07887","created_at":"2026-05-18T01:07:56Z"},{"alias_kind":"arxiv_version","alias_value":"1601.07887v2","created_at":"2026-05-18T01:07:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.07887","created_at":"2026-05-18T01:07:56Z"},{"alias_kind":"pith_short_12","alias_value":"PGG26LY6BO2B","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_16","alias_value":"PGG26LY6BO2BOVQW","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_8","alias_value":"PGG26LY6","created_at":"2026-05-18T12:30:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:PGG26LY6BO2BOVQW3OTX2OYJ2R","target":"record","payload":{"canonical_record":{"source":{"id":"1601.07887","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-01-28T20:26:20Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"e5df71017b905545f81512aa9ef80f2b49524f45daee5c6f32a142b3a40efee6","abstract_canon_sha256":"c7b7293bfd5c4a8ba2dabfa0ae698f7c8d6362c1d4b6e1acb9c2f34bf4878030"},"schema_version":"1.0"},"canonical_sha256":"798daf2f1e0bb4175616dba77d3b09d447bfb8dca438a8b1d2cd14a07e462180","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:56.860185Z","signature_b64":"4JfGCj1iQTXRG3eQNLCQkCLtcaFP+RfhsZd2sj6b2IN5QS1/u967O0avN75bQz4Ds/XBwH26nqYJeI3y3BNvBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"798daf2f1e0bb4175616dba77d3b09d447bfb8dca438a8b1d2cd14a07e462180","last_reissued_at":"2026-05-18T01:07:56.859661Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:56.859661Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.07887","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":"2026-05-18T01:07:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bF41Cmsn0rLtQTnjLfcC77qhIkmAQ4RB1IF7Idms+gpi9vT8mbLUuSjyzJskoQV919ixNYySXWD/Zy8fQDbmDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T07:42:13.128645Z"},"content_sha256":"51a4e11dee6ad93c23d164c9a219569077712b4c770d22bdd3779926d45013c9","schema_version":"1.0","event_id":"sha256:51a4e11dee6ad93c23d164c9a219569077712b4c770d22bdd3779926d45013c9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:PGG26LY6BO2BOVQW3OTX2OYJ2R","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weighted stationary phase of higher orders","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CA","authors_text":"Haiwei Sun, Mark McKee, Yangbo Ye","submitted_at":"2016-01-28T20:26:20Z","abstract_excerpt":"An $n$th-order first derivative test for oscillatoric integrals is established. When the phase has a single stationary point, an $n$th-order asymptotic expansion of a weighted stationary phase integral is proved for arbitrary $n\\geq1$. This asymptotic expansion sharpened the classical result for $n=1$ by Huxley. Possible applications include analysis and analytic number theory."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.07887","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":"2026-05-18T01:07:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LMx6lDC1gamwWUwT6B6+Qc8OkIXE4RD8YZXx9KwpUZYgSlXZndEvVWjeRjImc1/+LgXVE6+LFdkj6Kk4dz4rDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T07:42:13.129297Z"},"content_sha256":"d5ee9b2f1322945046fca05e74222ed6225fdae33fdb04ed51c6d14fd6be3a4e","schema_version":"1.0","event_id":"sha256:d5ee9b2f1322945046fca05e74222ed6225fdae33fdb04ed51c6d14fd6be3a4e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PGG26LY6BO2BOVQW3OTX2OYJ2R/bundle.json","state_url":"https://pith.science/pith/PGG26LY6BO2BOVQW3OTX2OYJ2R/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PGG26LY6BO2BOVQW3OTX2OYJ2R/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":"2026-06-05T07:42:13Z","links":{"resolver":"https://pith.science/pith/PGG26LY6BO2BOVQW3OTX2OYJ2R","bundle":"https://pith.science/pith/PGG26LY6BO2BOVQW3OTX2OYJ2R/bundle.json","state":"https://pith.science/pith/PGG26LY6BO2BOVQW3OTX2OYJ2R/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PGG26LY6BO2BOVQW3OTX2OYJ2R/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:PGG26LY6BO2BOVQW3OTX2OYJ2R","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":"c7b7293bfd5c4a8ba2dabfa0ae698f7c8d6362c1d4b6e1acb9c2f34bf4878030","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-01-28T20:26:20Z","title_canon_sha256":"e5df71017b905545f81512aa9ef80f2b49524f45daee5c6f32a142b3a40efee6"},"schema_version":"1.0","source":{"id":"1601.07887","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.07887","created_at":"2026-05-18T01:07:56Z"},{"alias_kind":"arxiv_version","alias_value":"1601.07887v2","created_at":"2026-05-18T01:07:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.07887","created_at":"2026-05-18T01:07:56Z"},{"alias_kind":"pith_short_12","alias_value":"PGG26LY6BO2B","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_16","alias_value":"PGG26LY6BO2BOVQW","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_8","alias_value":"PGG26LY6","created_at":"2026-05-18T12:30:39Z"}],"graph_snapshots":[{"event_id":"sha256:d5ee9b2f1322945046fca05e74222ed6225fdae33fdb04ed51c6d14fd6be3a4e","target":"graph","created_at":"2026-05-18T01:07:56Z","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":"An $n$th-order first derivative test for oscillatoric integrals is established. When the phase has a single stationary point, an $n$th-order asymptotic expansion of a weighted stationary phase integral is proved for arbitrary $n\\geq1$. This asymptotic expansion sharpened the classical result for $n=1$ by Huxley. Possible applications include analysis and analytic number theory.","authors_text":"Haiwei Sun, Mark McKee, Yangbo Ye","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-01-28T20:26:20Z","title":"Weighted stationary phase of higher orders"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.07887","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":"sha256:51a4e11dee6ad93c23d164c9a219569077712b4c770d22bdd3779926d45013c9","target":"record","created_at":"2026-05-18T01:07:56Z","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":"c7b7293bfd5c4a8ba2dabfa0ae698f7c8d6362c1d4b6e1acb9c2f34bf4878030","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-01-28T20:26:20Z","title_canon_sha256":"e5df71017b905545f81512aa9ef80f2b49524f45daee5c6f32a142b3a40efee6"},"schema_version":"1.0","source":{"id":"1601.07887","kind":"arxiv","version":2}},"canonical_sha256":"798daf2f1e0bb4175616dba77d3b09d447bfb8dca438a8b1d2cd14a07e462180","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"798daf2f1e0bb4175616dba77d3b09d447bfb8dca438a8b1d2cd14a07e462180","first_computed_at":"2026-05-18T01:07:56.859661Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:07:56.859661Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4JfGCj1iQTXRG3eQNLCQkCLtcaFP+RfhsZd2sj6b2IN5QS1/u967O0avN75bQz4Ds/XBwH26nqYJeI3y3BNvBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:07:56.860185Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.07887","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:51a4e11dee6ad93c23d164c9a219569077712b4c770d22bdd3779926d45013c9","sha256:d5ee9b2f1322945046fca05e74222ed6225fdae33fdb04ed51c6d14fd6be3a4e"],"state_sha256":"5b57255581a00bf06be8c2d4bac55004d0dfb8ef53635862280690802a80a0e5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E5pgFEhGbK1aeHWXWC/CJiSZGh0OUazxJSv0ulOJNmLO12dQx7la0P0dUBwtKO9J/7QwWDDMdNowpZfQqKRfDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T07:42:13.132870Z","bundle_sha256":"096fb7663260daad28569b1ad53698cf120a13a061357f8c288c71f4bc0d3a16"}}