{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:QAYZRT4CYDN33OQXAR63KMSB3D","short_pith_number":"pith:QAYZRT4C","canonical_record":{"source":{"id":"2605.25833","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-05-25T13:29:54Z","cross_cats_sorted":[],"title_canon_sha256":"71ed4e355e2a0199528eb58b6422f793b8d5cad2406239cf3b14a1872b61073f","abstract_canon_sha256":"4e5bb6c3bfda30933fb5df096103dc8c6a4b479cd42751d33d297456e47215bd"},"schema_version":"1.0"},"canonical_sha256":"803198cf82c0dbbdba17047db53241d8c7eb9b0c637591376d74861b71b0ce15","source":{"kind":"arxiv","id":"2605.25833","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.25833","created_at":"2026-05-26T02:05:14Z"},{"alias_kind":"arxiv_version","alias_value":"2605.25833v1","created_at":"2026-05-26T02:05:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.25833","created_at":"2026-05-26T02:05:14Z"},{"alias_kind":"pith_short_12","alias_value":"QAYZRT4CYDN3","created_at":"2026-05-26T02:05:14Z"},{"alias_kind":"pith_short_16","alias_value":"QAYZRT4CYDN33OQX","created_at":"2026-05-26T02:05:14Z"},{"alias_kind":"pith_short_8","alias_value":"QAYZRT4C","created_at":"2026-05-26T02:05:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:QAYZRT4CYDN33OQXAR63KMSB3D","target":"record","payload":{"canonical_record":{"source":{"id":"2605.25833","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-05-25T13:29:54Z","cross_cats_sorted":[],"title_canon_sha256":"71ed4e355e2a0199528eb58b6422f793b8d5cad2406239cf3b14a1872b61073f","abstract_canon_sha256":"4e5bb6c3bfda30933fb5df096103dc8c6a4b479cd42751d33d297456e47215bd"},"schema_version":"1.0"},"canonical_sha256":"803198cf82c0dbbdba17047db53241d8c7eb9b0c637591376d74861b71b0ce15","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T02:05:14.370638Z","signature_b64":"1xUhwcE+UieI+MHTXuXxmC9t+URJ3oo0tkG5ubFQ3osNB3baxab6fqvYqYloIANf64nLhBaMctFlfPLaK188DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"803198cf82c0dbbdba17047db53241d8c7eb9b0c637591376d74861b71b0ce15","last_reissued_at":"2026-05-26T02:05:14.369869Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T02:05:14.369869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.25833","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":"2026-05-26T02:05:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QyNbwwpJqBmXsy/iQZxiCmabN/ylzK2zb4HUVkgPtaSr40KXAdADfBuLWDUcIvMLaCdbIw0hLgScpnfAU6dqCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T23:02:32.102423Z"},"content_sha256":"f54eaebb8ecd83a1b54629b4befe8c4193a9355d82aeaf1ac924f0b3786849e9","schema_version":"1.0","event_id":"sha256:f54eaebb8ecd83a1b54629b4befe8c4193a9355d82aeaf1ac924f0b3786849e9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:QAYZRT4CYDN33OQXAR63KMSB3D","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Pointwise Convergence of Schr\\\"{o}dinger Operators in Bessel Potential Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jiheng Tan, Wenchang Sun, Yucheng Pan","submitted_at":"2026-05-25T13:29:54Z","abstract_excerpt":"We study the pointwise convergence of solutions to the free Schr\\\"{o}dinger equation with initial data in the Bessel potential spaces $L_s^p(\\mathbb{R}^n)$. We establish new sufficient regularity indices for pointwise convergence across the full range $1 \\leq p < \\infty$, and demonstrate via counterexamples that these indices are sharp for all $1 \\leq p \\leq 2$ in one dimension, as well as at the endpoint $p=1$ and $p>2+p_n$ in higher dimensions. The proofs rely on the high-dimensional stationary phase method."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.25833","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/2605.25833/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":"2026-05-26T02:05:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w15aMHFDqYtUQYXjU4DwEQQn+B2wRM5I2fQgE4LFabA3WPng/7eT96ETeMzkzIM/EwQ8xpGgJqmE+ztczE67BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T23:02:32.103170Z"},"content_sha256":"bb343d1f8e952e1a0ee5aade30a43c01eb8d379096c2880bd656ec3bda9e8833","schema_version":"1.0","event_id":"sha256:bb343d1f8e952e1a0ee5aade30a43c01eb8d379096c2880bd656ec3bda9e8833"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QAYZRT4CYDN33OQXAR63KMSB3D/bundle.json","state_url":"https://pith.science/pith/QAYZRT4CYDN33OQXAR63KMSB3D/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QAYZRT4CYDN33OQXAR63KMSB3D/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":"2026-06-11T23:02:32Z","links":{"resolver":"https://pith.science/pith/QAYZRT4CYDN33OQXAR63KMSB3D","bundle":"https://pith.science/pith/QAYZRT4CYDN33OQXAR63KMSB3D/bundle.json","state":"https://pith.science/pith/QAYZRT4CYDN33OQXAR63KMSB3D/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QAYZRT4CYDN33OQXAR63KMSB3D/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:QAYZRT4CYDN33OQXAR63KMSB3D","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":"4e5bb6c3bfda30933fb5df096103dc8c6a4b479cd42751d33d297456e47215bd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-05-25T13:29:54Z","title_canon_sha256":"71ed4e355e2a0199528eb58b6422f793b8d5cad2406239cf3b14a1872b61073f"},"schema_version":"1.0","source":{"id":"2605.25833","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.25833","created_at":"2026-05-26T02:05:14Z"},{"alias_kind":"arxiv_version","alias_value":"2605.25833v1","created_at":"2026-05-26T02:05:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.25833","created_at":"2026-05-26T02:05:14Z"},{"alias_kind":"pith_short_12","alias_value":"QAYZRT4CYDN3","created_at":"2026-05-26T02:05:14Z"},{"alias_kind":"pith_short_16","alias_value":"QAYZRT4CYDN33OQX","created_at":"2026-05-26T02:05:14Z"},{"alias_kind":"pith_short_8","alias_value":"QAYZRT4C","created_at":"2026-05-26T02:05:14Z"}],"graph_snapshots":[{"event_id":"sha256:bb343d1f8e952e1a0ee5aade30a43c01eb8d379096c2880bd656ec3bda9e8833","target":"graph","created_at":"2026-05-26T02:05:14Z","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/2605.25833/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the pointwise convergence of solutions to the free Schr\\\"{o}dinger equation with initial data in the Bessel potential spaces $L_s^p(\\mathbb{R}^n)$. We establish new sufficient regularity indices for pointwise convergence across the full range $1 \\leq p < \\infty$, and demonstrate via counterexamples that these indices are sharp for all $1 \\leq p \\leq 2$ in one dimension, as well as at the endpoint $p=1$ and $p>2+p_n$ in higher dimensions. The proofs rely on the high-dimensional stationary phase method.","authors_text":"Jiheng Tan, Wenchang Sun, Yucheng Pan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-05-25T13:29:54Z","title":"Pointwise Convergence of Schr\\\"{o}dinger Operators in Bessel Potential Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.25833","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:f54eaebb8ecd83a1b54629b4befe8c4193a9355d82aeaf1ac924f0b3786849e9","target":"record","created_at":"2026-05-26T02:05:14Z","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":"4e5bb6c3bfda30933fb5df096103dc8c6a4b479cd42751d33d297456e47215bd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-05-25T13:29:54Z","title_canon_sha256":"71ed4e355e2a0199528eb58b6422f793b8d5cad2406239cf3b14a1872b61073f"},"schema_version":"1.0","source":{"id":"2605.25833","kind":"arxiv","version":1}},"canonical_sha256":"803198cf82c0dbbdba17047db53241d8c7eb9b0c637591376d74861b71b0ce15","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"803198cf82c0dbbdba17047db53241d8c7eb9b0c637591376d74861b71b0ce15","first_computed_at":"2026-05-26T02:05:14.369869Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-26T02:05:14.369869Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1xUhwcE+UieI+MHTXuXxmC9t+URJ3oo0tkG5ubFQ3osNB3baxab6fqvYqYloIANf64nLhBaMctFlfPLaK188DA==","signature_status":"signed_v1","signed_at":"2026-05-26T02:05:14.370638Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.25833","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f54eaebb8ecd83a1b54629b4befe8c4193a9355d82aeaf1ac924f0b3786849e9","sha256:bb343d1f8e952e1a0ee5aade30a43c01eb8d379096c2880bd656ec3bda9e8833"],"state_sha256":"2dbd8fe8a811ee51685643b66db51c4e473cb88938bb54aa19c482f9bf8c62ef"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nGzh78iL7xHhSJlrB1paR4mvGScxehFMByj+bHYGngT6iGXkFPvbKDb/JYmdDXXCljyssCgzZnfpRjvZYUOcCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T23:02:32.106747Z","bundle_sha256":"4b93f6da49b21f5f7a658e536167878101e26bdb9b7b544bff454ea39afed9e5"}}