{"paper":{"title":"The Fourier extension conjecture for the paraboloid","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The Fourier extension conjecture for the paraboloid holds in every dimension greater than 2.","cross_cats":[],"primary_cat":"math.CA","authors_text":"Cristian Rios, Eric T. Sawyer","submitted_at":"2025-12-31T17:36:38Z","abstract_excerpt":"We give a proof of Fourier extension conjecture on the paraboloid in all dimensions bigger than 2 that begins with a decomposition suggested in Sawyer [Saw8] of writing a smooth Alpert projection as a sum of pieces whose Fourier extensions are localized. This is then used to establish a local inequality that is well known to be equivalent to the Fourier extension conjecture, and is accomplished by using a variant of the bilinear equivalence of the Fourier extension conjecture given by Tao, Vargas and Vega in [TaVaVe]. A key aspect of our proof is that the bilinear inequality, when taken over s"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We give a proof of Fourier extension conjecture on the paraboloid in all dimensions bigger than 2 that begins with a decomposition suggested in Sawyer [Saw8] of writing a smooth Alpert projection as a sum of pieces whose Fourier extensions are localized.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The bilinear inequality, when taken over smooth Alpert projections, only requires an averaging over grids of functions mollified by discrete multipliers which converts a difficult exponential sum into an oscillatory integral with periodic amplitude.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A proof is given that the Fourier extension conjecture holds for the paraboloid in dimensions d greater than 2.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Fourier extension conjecture for the paraboloid holds in every dimension greater than 2.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"ee60e4b1aef7d4c6d17d465a75fef91d04bd7b97e77750bd5970f978253fa055"},"source":{"id":"2512.24990","kind":"arxiv","version":7},"verdict":{"id":"18d72f74-f65e-4c62-955e-7dfb01efc0de","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T18:43:26.098871Z","strongest_claim":"We give a proof of Fourier extension conjecture on the paraboloid in all dimensions bigger than 2 that begins with a decomposition suggested in Sawyer [Saw8] of writing a smooth Alpert projection as a sum of pieces whose Fourier extensions are localized.","one_line_summary":"A proof is given that the Fourier extension conjecture holds for the paraboloid in dimensions d greater than 2.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The bilinear inequality, when taken over smooth Alpert projections, only requires an averaging over grids of functions mollified by discrete multipliers which converts a difficult exponential sum into an oscillatory integral with periodic amplitude.","pith_extraction_headline":"The Fourier extension conjecture for the paraboloid holds in every dimension greater than 2."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2512.24990/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"}