A verifier that accepts or rejects whole answers reveals little: under a flat prior over
k-bit answers, zero error needs
2k−1 verifications. The usual remedy is to solve related source tasks, either all first, as a curriculum does, or interleaved with verification. We price this remedy in information and in calls. With an exact verifier, the least causal information that any interleaving of source calls and
n verifications needs to succeed with probability
s is a list rate-distortion function, attained by one observation before any verification. It lower-bounds the expected number of binary source calls, which designed sources meet within
1+log25 calls for unique answers and within a logarithmic term in general, where no additive constant suffices. With an exact verifier and fixed sources, moving every call before the first verification preserves all hard caps on calls, although interleaving can save unboundedly many expected calls; under a noisy verifier, source-first protocols can lose unbounded factors in information and in error. For linear banks over
F2, optimal accuracy has a closed form, and after a polynomial-time reduction the budget profile is computable in time
2O(h2)poly(J,k+h) for
J sources and nuisance dimension
h. In these banks, for zero error under a hard cap, the calls beyond the rounded-up information price are exactly those spent on nuisance. Every numbered result apart from two clauses about the planner is machine-checked in Lean 4, assuming two published results. Used as a ruler, the frontier shows a small transformer using all delivered bits at latent dimension
5 and none at
11 within fixed training budgets; in a test with predictions recorded before training, low XOR degree of the target bits did not suffice for their use.