WHAT DOES ROBUST SELECTION REVEAL ABOUT THE ALTERNATIVES?

Choosing a good candidate need not require estimating every candidate well. We ask how much uncertainty can remain after a choice is already justified. We study unit directions whose quality is the squared positive part of their cosine similarity with an unknown vector. We observe that vector through exact linear measurements. A choice is robustly $\epsilon$-optimal if it stays within $\epsilon$ of the best candidate for every nonzero vector consistent with the measured history. Once the selected direction has been measured, we derive the exact worst-case uncertainty in an alternative's quality, conditional on the selected quality $\theta$. For $0<\epsilon \le 1/2$, this diameter is $\theta +\epsilon$ below $\theta =\epsilon /(1+2\epsilon )$ and $2\sqrt{\epsilon (1+\epsilon )\theta (1-\theta )}$ above it. Matching histories attain both branches. For an already valid selected confidence interval, we derive a location-sensitive bound on simultaneous intervals obtained from the same observations and coverage event. A complementary construction shows that, for $0<\epsilon \le 1/10$, randomized adaptive continuation can require \[ \Omega\!\left(\min\left{m,\varepsilon\eta^2\right}\right) \] additional queries in the worst case, even with the exact norm and selected quality supplied. Here $m$ is the unmeasured dimension, $\eta >0$ is the target interval width, the failure probability is fixed below $1/2$, and $\eta /\sqrt{\epsilon }$ is sufficiently small. These results quantify what robust selection reveals about unselected alternatives and what it can leave unmeasured.