Fresh paired probe designs and delayed cues

The radius is a mathematical population statement only when the rows are genuinely independent of the fixed checkpoints or conditionally fresh as specified in the main text. Function names and pseudorandom seeds do not prove that hypothesis.

For paired differences, use the saved per-example correctness vectors and the appropriate interval-length radius. For extra input corruption, a tightened ceiling can use Theorem thm:measurementbayes. The stored uncorrupted task ceiling remains only a data-processing upper reference when additional corruption is introduced.