{"paper":{"title":"The Kikuchi Hierarchy is Sharp for $k$XOR","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CC","quant-ph"],"primary_cat":"cs.DS","authors_text":"Alexander Schmidhuber, Matthew B. Hastings","submitted_at":"2026-07-31T17:54:11Z","abstract_excerpt":"Planted noisy $k$XOR and the strong refutation of random $k$XOR are governed by a conjectured trade-off between signal strength and time: Level $\\ell$ of the Kikuchi hierarchy should achieve the smooth curve \\begin{equation*}\n  m\\ \\gtrsim\\ \\rho^{-2}n^{k/2}/\\ell^{k/2-1}\\ \\text{clauses}\n  \\quad\\Longleftrightarrow\\quad\n  \\text{solvable in time }n^{O(\\ell)}, \\end{equation*} where $\\rho$ is the bias of the planted signal or, for refutation, the target advantage. However, every spectral analysis of sparse $k$XOR to date loses polylogarithmic factors against this curve, a loss that enters the exponen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.29672","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/2607.29672/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"}