{"paper":{"title":"The Grimmer--Shu--Wang Certificate and the Drori--Teboulle Minimax Constant-Stepsize Bound for $N\\ge 3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For every number of steps N at least 3, positive vectors exist that satisfy the equations of the strengthened low-rank certificate for the worst-case analysis of gradient descent with constant stepsize.","cross_cats":[],"primary_cat":"math.OC","authors_text":"Lixing Zhang","submitted_at":"2026-05-12T02:15:33Z","abstract_excerpt":"We prove, for every horizon \\(N\\ge 3\\), the existence of the strengthened low-rank performance-estimation certificate proposed by Grimmer, Shu, and Wang for the Drori--Teboulle constant-step gradient-descent bound. For each \\(N\\ge 3\\), let \\(\\rho_N\\in(0,1)\\) be determined by \\(\\rho_N^{2N}(2N\\rho_N+2N+1)=1\\). We show that the GSW certificate equations admit positive vectors \\(a,b,c,d\\) satisfying all residual equations. The proof proceeds through a reduced residual system in the variables \\(d\\), a simplex existence argument for a positive reduced zero, a terminal residual completion identity, a"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove, for every horizon N >= 3, the existence of the strengthened low-rank performance-estimation certificate proposed by Grimmer, Shu, and Wang for the Drori-Teboulle minimax nonnegative constant-stepsize problem for gradient descent.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The GSW certificate equations admit positive vectors a, b, c, d satisfying all residual equations, shown via a reduced residual system, simplex existence argument, terminal residual completion identity, and tail-square convolution argument proving cumulative margins.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The Grimmer-Shu-Wang low-rank PEP certificate exists for every horizon N >= 3 and establishes the exact Drori-Teboulle minimax nonnegative constant-stepsize bound for gradient descent.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For every number of steps N at least 3, positive vectors exist that satisfy the equations of the strengthened low-rank certificate for the worst-case analysis of gradient descent with constant stepsize.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"fef0149551dfe6c4e7f82ba85fc61b0dda72e64013e4411b21ce5aae646f9ea3"},"source":{"id":"2605.11421","kind":"arxiv","version":2},"verdict":{"id":"6a642e79-12b1-445f-9d6c-44b421c37a7d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-13T02:37:04.909515Z","strongest_claim":"We prove, for every horizon N >= 3, the existence of the strengthened low-rank performance-estimation certificate proposed by Grimmer, Shu, and Wang for the Drori-Teboulle minimax nonnegative constant-stepsize problem for gradient descent.","one_line_summary":"The Grimmer-Shu-Wang low-rank PEP certificate exists for every horizon N >= 3 and establishes the exact Drori-Teboulle minimax nonnegative constant-stepsize bound for gradient descent.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The GSW certificate equations admit positive vectors a, b, c, d satisfying all residual equations, shown via a reduced residual system, simplex existence argument, terminal residual completion identity, and tail-square convolution argument proving cumulative margins.","pith_extraction_headline":"For every number of steps N at least 3, positive vectors exist that satisfy the equations of the strengthened low-rank certificate for the worst-case analysis of gradient descent with constant stepsize."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.11421/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T04:22:00.458781Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T12:36:22.419663Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T10:01:16.556914Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T08:28:42.511585Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"6316e6376e763f65d698e16e053cb43dcf3197c056b7afeb6140b2ad9c540444"},"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"}