Heteroskedastic Signals in Budgeted LLM Verification: Structural Heterogeneity Limits Optimization Gains

Selective-compute LLM systems decide which outputs merit verification, additional reasoning, tool execution, or human audit under a limited budget. It is natural to expect that stronger online optimization over a shared uncertainty or reward signal should improve these decisions. We take a critical look at this assumption and ask: when does optimizing harder fail because the signal is not decision-comparable across inputs? In budgeted LLM verification, we find that uncertainty quality is heteroskedastic across cost strata: some regions exhibit near-random discriminability while concentrating many errors. Under an explicit local model, we characterize the resulting distortion of global allocation and show that its upper bound scales with cross-stratum signal-quality dispersion. To separate weak signals from optimizer instability and structural mismatch, we introduce a controlled intervention hierarchy: Threshold, MP-Adapt, MP-Strat, and cost-stratified thresholding (CST). We then turn the diagnosis into Heterogeneity-Gated Allocation (HGA), which uses a warm-up comparability test to choose between global and cost-stratified allocation. Across MBPP and MATH using Qwen3-8B, LLaMA3-8B, and GPT-4o-mini, global online adaptation yields inconsistent gains over static thresholding; CST improves hit rate by up to 17 percentage points in strongly heterogeneous settings, while HGA preserves most gains and avoids blind stratification when the partition is not useful. These findings suggest a resource-allocation principle for LLM systems: before optimizing harder over a shared proxy, test whether the proxy is decision-comparable across observable operating regimes, and gate structural specialization on that test.