Compare or Cover? Feedback Geometry for Budgeted LLM Router Construction

Learned LLM routers are trained on request–model quality scores, but a finite annotation budget reveals only part of the score matrix. This creates a choice: cover more distinct requests, or compare models on shared requests. We formalize this choice as support design at equal annotation cost, varying training-request overlap while holding labels per model, predictors and tuning data fixed. Deferral curves at matched switching rates separate routing quality from switching frequency. An exact coupling identity links the reduction in gain-prediction mean squared error to exactly twice the covariance between separately fitted predictors. For linear smoothers, shared supports turn gain prediction into regression on observed paired gains. Across 10,080 router fits on five archives, full sharing achieves the best mean rank among nine equal-cost designs, including D-optimal, k-center, uncertainty sampling and covariance-aware adaptive allocation. Sharing improves deployed routing quality over independent sampling in all ten archive–predictor pairs and reduces the build-label budget needed to reach zero area under the deferral curve by up to a factor of 1.8. Direct measurements show lower gain-prediction error and selection optimism, with routing gains increasing as model pools grow from 2 to 32 models. The benefit persists across representations and predictor families and replicates on a sixth archive under a frozen protocol. These results establish full sharing as a simple default for budgeted router construction at no additional annotation cost.