Quotient-First Decoding: Sharp Truncation Gaps Without Support Loss

Can a truncation decoder change its output law when only the representation of probability mass changes? We analyze exact mass-preserving refinements relative to a supplied outcome quotient. We characterize exactly the outcome laws reachable by nucleus sampling without changing its support. At each fixed parent law, the fine-uniform-refinement limit simultaneously maximizes every convex f-divergence from the reference decoder. The sharp one-step total-variation (TV) supremum is $\min \{1/2,1-\rho \}$; its horizon-$H$ counterpart is $1-\max \{\rho ,1/2{\}}^H$. We identify the mechanism under threshold-prefix compatibility: refinement thins the boundary outcome, preserves the conditional law above it, and can strictly sharpen the distribution. A binary likelihood-ratio factorization explains how this distortion propagates along autoregressive paths. Quotient-first decoding removes it by aggregating before truncation; classical conditional-preserving projection characterizes its unique faithful lift. In state-preserving experiments on three real language models and 120 held-out ARC-Challenge, GSM8K, and MMLU prompt families, Qwen3-8B reaches path TV $D_{64}=0.185$ (95% confidence interval (CI) [0.176, 0.195]). Uniform refinement reaches 0.256 ([0.245, 0.267]) with identical path support. An exact, locally canonical byte quotient gives $D_{32}=0.038$; quotient-first replay agrees with the reference kernel to $4.44\times 10^{-16}$. These results explain representation dependence and establish an exact, conditional-faithful remedy for the supplied outcome quotient.