How Much of the Routing Gap Is Real? Decomposing the Router-to-Oracle Gap into Reproducible Specialist Advantage and Single-Draw Label Noise

Abstract

On real open-model pools, 12--36% of the reported router-to-oracle gap is single-draw label noise that no single-commit router can capture, while the majority is genuine, recoverable specialist advantage; this work proves why (a recoverability asymmetry) and releases a protocol to measure it. Routing among large language models (LLMs) trades cost for quality, motivated by the gap between learned routers and a per-instance oracle. But under stochastic decoding that oracle is a single Bernoulli draw, not a reproducible property. We recast the question structurally: the expected oracle decomposes as $O^{\exp}=O^{\mathrm{repro}}+Δ$, into reproducible single-commit headroom $O^{\mathrm{repro}}$ and a non-negative single-commit selection floor $Δ$. Our main result is a recoverability asymmetry: this floor is closed by no single-commit router (deterministic or randomized), yet is provably recovered by test-time sampling: best-of-$K$ on the committed model, at the oracle's own budget, dominates the independent-pool single-draw oracle. This cap needs no cross-model independence, pinning "not recoverable" to single-commit selection, not to information. The floor's magnitude is a prospective, conservative localization, not an audit: LLMRouterBench (33 models, 391,645 instances) builds its oracle as a per-query union of single $T=0.2$ draws, so its 20-point gap is by construction a union of stochastic draws; since $O^{\mathrm{repro}}$ is non-identifiable at $k=1$, we re-estimate by fresh $k\ge20$ resampling under one-sided, dependence-corrected bounds. Across three controlled open-model re-generations (arithmetic, competition math, and non-math science), single-draw noise is a substantial minority of the gap, larger on unsaturated benchmarks and approaching half on the hardest queries. We release a multi-sample oracle protocol that routing benchmarks can adopt.