{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:I2PBSMLN7YU33M2DST74RGJ6XP","short_pith_number":"pith:I2PBSMLN","schema_version":"1.0","canonical_sha256":"469e19316dfe29bdb34394ffc8993ebbd9446a93bf1c3d6198dc44ac4def9c9f","source":{"kind":"arxiv","id":"2506.08831","version":1},"attestation_state":"computed","paper":{"title":"Dispersive estimates for Dirac Operators in dimension four with obstructions at threshold energies","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Benjamin Lyons, Connor Lane, William R. Green","submitted_at":"2025-06-10T14:19:47Z","abstract_excerpt":"We investigate $L^1\\to L^\\infty$ dispersive estimates for the Dirac equation with a potential in four spatial dimensions. We classify the structure of the obstructions at the thresholds as being composed of an at most two dimensional space of resonances per threshold, and finitely many eigenfunctions. Similar to the Schr\\\"odinger evolution, we prove the natural $t^{-2}$ decay rate when the thresholds are regular. When there is a threshold resonance or eigenvalue, we show that there is a time dependent, finite rank operator satisfying $\\|F_t\\|_{L^1\\to L^\\infty}\\lesssim (\\log t)^{-1}$ for $t>2$ "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2506.08831","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-10T14:19:47Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"5d7e0615feb5dbc1c3b3c9c888176329785a2359fd8ad3d16357763535c61018","abstract_canon_sha256":"ea12b9acaf82f94a15649721cb06bd28e6ca9517aa65bb358e5d204d7dec8b28"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:19:13.903604Z","signature_b64":"NOM4ob7f3pVVcmC6vc7F8Ac9OJGwNWrNL3SqlxzF9Y5ZF7HnZ0gr31DOR112YX+b191mS9doOk2QV3iRrvV1AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"469e19316dfe29bdb34394ffc8993ebbd9446a93bf1c3d6198dc44ac4def9c9f","last_reissued_at":"2026-07-05T11:19:13.903141Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:19:13.903141Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dispersive estimates for Dirac Operators in dimension four with obstructions at threshold energies","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Benjamin Lyons, Connor Lane, William R. Green","submitted_at":"2025-06-10T14:19:47Z","abstract_excerpt":"We investigate $L^1\\to L^\\infty$ dispersive estimates for the Dirac equation with a potential in four spatial dimensions. We classify the structure of the obstructions at the thresholds as being composed of an at most two dimensional space of resonances per threshold, and finitely many eigenfunctions. Similar to the Schr\\\"odinger evolution, we prove the natural $t^{-2}$ decay rate when the thresholds are regular. When there is a threshold resonance or eigenvalue, we show that there is a time dependent, finite rank operator satisfying $\\|F_t\\|_{L^1\\to L^\\infty}\\lesssim (\\log t)^{-1}$ for $t>2$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.08831","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/2506.08831/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2506.08831","created_at":"2026-07-05T11:19:13.903198+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.08831v1","created_at":"2026-07-05T11:19:13.903198+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.08831","created_at":"2026-07-05T11:19:13.903198+00:00"},{"alias_kind":"pith_short_12","alias_value":"I2PBSMLN7YU3","created_at":"2026-07-05T11:19:13.903198+00:00"},{"alias_kind":"pith_short_16","alias_value":"I2PBSMLN7YU33M2D","created_at":"2026-07-05T11:19:13.903198+00:00"},{"alias_kind":"pith_short_8","alias_value":"I2PBSMLN","created_at":"2026-07-05T11:19:13.903198+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/I2PBSMLN7YU33M2DST74RGJ6XP","json":"https://pith.science/pith/I2PBSMLN7YU33M2DST74RGJ6XP.json","graph_json":"https://pith.science/api/pith-number/I2PBSMLN7YU33M2DST74RGJ6XP/graph.json","events_json":"https://pith.science/api/pith-number/I2PBSMLN7YU33M2DST74RGJ6XP/events.json","paper":"https://pith.science/paper/I2PBSMLN"},"agent_actions":{"view_html":"https://pith.science/pith/I2PBSMLN7YU33M2DST74RGJ6XP","download_json":"https://pith.science/pith/I2PBSMLN7YU33M2DST74RGJ6XP.json","view_paper":"https://pith.science/paper/I2PBSMLN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.08831&json=true","fetch_graph":"https://pith.science/api/pith-number/I2PBSMLN7YU33M2DST74RGJ6XP/graph.json","fetch_events":"https://pith.science/api/pith-number/I2PBSMLN7YU33M2DST74RGJ6XP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I2PBSMLN7YU33M2DST74RGJ6XP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I2PBSMLN7YU33M2DST74RGJ6XP/action/storage_attestation","attest_author":"https://pith.science/pith/I2PBSMLN7YU33M2DST74RGJ6XP/action/author_attestation","sign_citation":"https://pith.science/pith/I2PBSMLN7YU33M2DST74RGJ6XP/action/citation_signature","submit_replication":"https://pith.science/pith/I2PBSMLN7YU33M2DST74RGJ6XP/action/replication_record"}},"created_at":"2026-07-05T11:19:13.903198+00:00","updated_at":"2026-07-05T11:19:13.903198+00:00"}