{"paper":{"title":"Hierarchical Copula-Gumbel-Top-\\texorpdfstring{$K$}{K} Routing: Two-Sided Dependence Control for Frozen Mixture-of-Experts at Fixed Per-Token Routing Laws","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Richard Yi Da Xu","submitted_at":"2026-07-25T12:43:47Z","abstract_excerpt":"A stochastic Gumbel-Top-$K$ router defines, for every token of a mixture-of-experts (MoE) model, a \\emph{routing law}: a distribution over ordered expert lists and mixture weights. We ask which \\emph{joint} distributions over the routing choices of different tokens are reachable while every individual token's complete routing law is held exactly fixed. We give a two-sided construction, \\emph{Hierarchical Copula-Gumbel-Top-$K$} (\\CGA{}). Within a group of related tokens, an exchangeable Gaussian copula positively correlates the Gumbel perturbations at each expert coordinate, which can increase "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.28670","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/2607.28670/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"}