{"paper":{"title":"Explicit cutoff profiles for colored top-$m$-to-random shuffles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The p-colored top-m-to-random shuffle on the wreath product has its mixing cutoff at k = floor(n/m (log n + c)), where the number of never-chosen labels converges in law to Poisson(e^{-c}).","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Ivan Z. Feng","submitted_at":"2026-04-10T22:22:55Z","abstract_excerpt":"We study $p$-colored top-$m$-to-random on the wreath product $G_{n,p}=C_p\\wr S_n$, with $m$ fixed. Using the Nakano-Sadahiro-Sakurai basis elements $B_m$, we obtain exact nested-set occupancy mixtures and reduce the likelihood ratio to the single statistic $L_p$. This yields exact formulas for separation and $L^\\infty(U)$, and exact one-dimensional formulas for total variation, $L^q(U)$ ($1\\le q<\\infty$), $\\chi^2$, and relative entropy. At $k=\\Bigl\\lfloor \\frac{n}{m}(\\log n+c)\\Bigr\\rfloor$, the number of never-chosen labels in the associated $m$-subset occupancy model converges in law to $\\mat"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"At k=⌊n/m (log n + c)⌋, the number of never-chosen labels converges in law to Poisson(e^{-c}), giving the total-variation profile f_p(c), the separation profile, and the corresponding L^q(U), L^∞(U), χ², and relative-entropy profiles.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the Nakano-Sadahiro-Sakurai basis elements B_m yield exact nested-set occupancy mixtures on the wreath product G_{n,p}, allowing the likelihood ratio to reduce to the single statistic L_p.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Explicit cutoff profiles for total variation, separation, and other distances are obtained for colored top-m-to-random shuffles, with unused labels converging to Poisson(e^{-c}) at the cutoff time.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The p-colored top-m-to-random shuffle on the wreath product has its mixing cutoff at k = floor(n/m (log n + c)), where the number of never-chosen labels converges in law to Poisson(e^{-c}).","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"957ca7042cd2c87191502f5cd621e43f6a61d7cbccccbbc1ba78074d5f4d5e33"},"source":{"id":"2604.09933","kind":"arxiv","version":2},"verdict":{"id":"438dbd1c-60ef-4cf0-a932-6915be5fc969","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T16:25:41.376796Z","strongest_claim":"At k=⌊n/m (log n + c)⌋, the number of never-chosen labels converges in law to Poisson(e^{-c}), giving the total-variation profile f_p(c), the separation profile, and the corresponding L^q(U), L^∞(U), χ², and relative-entropy profiles.","one_line_summary":"Explicit cutoff profiles for total variation, separation, and other distances are obtained for colored top-m-to-random shuffles, with unused labels converging to Poisson(e^{-c}) at the cutoff time.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the Nakano-Sadahiro-Sakurai basis elements B_m yield exact nested-set occupancy mixtures on the wreath product G_{n,p}, allowing the likelihood ratio to reduce to the single statistic L_p.","pith_extraction_headline":"The p-colored top-m-to-random shuffle on the wreath product has its mixing cutoff at k = floor(n/m (log n + c)), where the number of never-chosen labels converges in law to Poisson(e^{-c})."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.09933/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":2,"snapshot_sha256":"0f13a63a723245295920e410d6dfe5db51764696c665f11cbf5539be46db591a"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}