Verification with Transfer: Exact Information Frontiers and Their Price in Calls
Abstract
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.
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BibTeX
@misc{yueksel2026verification,
title = {{Verification with Transfer: Exact Information Frontiers and Their Price in Calls}},
author = {Hazar Yueksel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12211},
eprint = {2610.12211},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12211},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Verification with Transfer: Exact Information Frontiers and Their Price in Calls
AU - Hazar Yueksel
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12211
UR - https://arxiv.org/abs/2610.12211
AB - A verifier that accepts or rejects whole answers reveals little: under a flat prior over k-bit answers, zero error needs 2^k−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+log_25 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 𝔽_2, optimal accuracy has a closed form, and after a polynomial-time reduction the budget profile is computable in time 2^O(h^2)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.
ER -