Verification with Transfer: Exact Information Frontiers and Their Price in Calls

Hazar Yueksel

cs.LGOct 8, 2026

Verification with Transfer: Exact Information Frontiers and Their Price in Calls

Authors: Hazar Yueksel

Abstract

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 $\mathbb{F}_2$ , optimal accuracy has a closed form, and after a polynomial-time reduction the budget profile is computable in time $2^{O(h^2)}\operatorname{poly}(J,k+h)$ 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.

Figures & tables

Panel of Figure 1 — Figure 1: Verification with transfer. Auxiliary calls reveal information about the world; a verification accepts or rejects a whole response, and an accepted response may be output. We price the auxiliary information $B_{\mathrm{aux}}$ (defined in ( 3 )) and the calls $Q$ needed to succeed with probability $s$ within $n$ verifications.

Figure 2: Information and call frontiers for a Zipf prior

q_{i}\propto i^{-1.2}

M=8

unique answers with

n=1

verification. Black:

\mathcal{R}_{n}(q,s)

. Blue and orange: the exact minimum expected number of binary calls over all interleaved and all source-first protocols with designed sources, computed by dynamic programming over candidate sets. They exceed the information price by at most 0.28 and 0.35 calls; Theorem 4 (ii) guarantees less than

1+\log_{2}5

Figure 3: The frontier as a ruler. (a) Bits a two-layer transformer uses against bits the planner’s sources deliver, pooled over three banks per latent dimension

d

and 64 secrets per point (the zero-error-split fractions in the text use four secrets per bank); the dashed line is the exact posterior. (b) Fraction of delivered bits used at the zero-error split, one point per bank and training run, against the largest XOR degree among the determined target bits (small horizontal jitter). Filled markers are the 15 banks of (a) with their first training seed, open markers the two further seeds of bank 0 at

d\in\{6,7,8\}

, and diamonds the degree-controlled banks at

(k,h,J)=(4,2,12)

, selected and predicted before training.

Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: The optimized bound is exact beyond uniform priors. Unique answers,

q=(0.8,0.1,0.1)

, one target query. Curves are certified finite-alphabet optimization. At zero error, the exact price is

0.2

bits, versus

0.1520

for the unoptimized reference.

Figure 5: The upfront reduction fails for noisy feedback. Uniform binary target, one noisy query, error tolerance

\delta=\epsilon^{2}

. Both curves are exact optimal information costs. At

\epsilon=0.01

, the costs are

0.9900

bits upfront and

0.0793

bits with interleaving.

Source	Information (bits)	Oracle accuracy	Learned accuracy
Balanced partition	4.0	1.000000	1.000000
Reveal–erasure, $\alpha=0.50$	4.0	0.531250	0.531250
Reveal–erasure, $\alpha=0.55$	4.4	0.578125	0.578125
Uninformative	0.0	0.062500	0.062500

Appendix

Table 1: Source quantity versus value. Uniform

M=256

targets,

15

exact queries. Information and oracle accuracy are exact. Learned values span

15

runs (five seeds, three learning rates) at step

400

. A single number means all runs agree at the displayed precision.

Figure 6: Scaling with available sources and nuisance dimension. Each point averages three generated inputs. Target dimension is

k=J/4

. The number of subspace states is at most

\mathcal{N}_{h}=2^{O(h^{2})}

(

5

67

and

2825

for

h=2,4,6

; Appendix M ).

Bank family	$(k,h,J)$	Banks	Agree	QSP (ms)	GHWs (ms)	Done
Low nuisance	$(4,2,12)$	6	6/6	0.37	30.91	6/6
Wider low nuisance	$(8,3,24)$	6	–	1.49	timeout	0/6
Intermediate nuisance	$(4,5,12)$	6	6/6	6.87	50.72	6/6
High nuisance, short bank	$(3,7,12)$	6	6/6	35.46	18.25	6/6
RM(1,3)	$(2,2,8)$	1	1/1	0.37	13.13	1/1
RM(1,4)	$(2,3,16)$	1	1/1	0.78	16.44	1/1

Appendix

Table 2: An independent implementation computes the same profiles. Where both solvers finish, their complete rank profiles agree on every bank. Both compute the complete rank profile from identical inputs on one CPU, with two fresh-worker repetitions and a common 15-second wall cap. Times are medians over per-bank medians, excluding imports/worker startup but including input conversion. QSP completes all 27 banks; timeout entries are censored. RM rows are standard Reed–Muller constructions.

		Two-exchange run		Best released baseline		QSP
Family	Banks	Acc. at $C^{\star}$	Gap (pp)	Acc. at $C^{\star}$	Gap (pp)	Acc. at $C^{\star}$
Independent linear	16	0.734	2.83	0.750	2.28	1.000
Clustered linear	16	0.812	2.36	0.875	1.10	1.000
Coded scramblings	8	1.000	0.01	1.000	0.00	1.000

Appendix

Table 3: Local search on every family, against the best released baseline. Every method receives the same label-fitted model and optimizes its own source/target split. The two-exchange run uses a larger neighborhood than the lookahead and one-exchange heuristics of Appendix P but different random starts, and loses to one-step lookahead at

C^{\star}

on three of the forty banks; the right-hand pair therefore reports the per-bank best over all released source-aware baselines. Gap averages QSP-minus-baseline population accuracy over all integer budgets from zero through source rank plus one. QSP has accuracy one at

C^{\star}

by definition of

C^{\star}

. The coded rows are scramblings of one geometry.

Figure 7: Every external-comparison instance. Each point is the median of two repetitions. Crosses at 15 seconds are censored GHWs runs. Points on both sides of the diagonal show the crossover.

Family	Acc. 95% interval	Gap interval (pp)	QSP (ms)	Two-exchange (ms)
Independent	[0.625,0.844]	[2.13,3.50]	10.56	500.65
Clustered	[0.688,0.938]	[1.34,3.40]	8.70	480.65
Coded	[1.000,1.000]	[0.00,0.03]	18.33	76.81

Appendix

Table 4: Bank-bootstrap percentile intervals and mean runtimes for the expanded local-search comparison. Coded intervals reflect only eight scramblings of one geometry. These timings come from a different host than those of Table 2 .

Noise (%)	Reps	Recovered	Planned	Yield	Conditional	Online cap	Train calls
0	1	24/24	24/24	1.0000	1.0000	9.25	432
0	3	24/24	24/24	1.0000	1.0000	25.25	1296
0	9	24/24	24/24	1.0000	1.0000	73.25	3888
0.1	1	14/24	24/24	0.9914	0.9914	9.25	432
0.1	3	24/24	24/24	1.0000	1.0000	25.25	1296
0.1	9	24/24	24/24	1.0000	1.0000	73.25	3888

Appendix

Table 5: All source-noise conditions. Yield counts abstaining banks as unsuccessful; conditional accuracy excludes them. “Recovered” means the fitted support equals the true support, which is used for evaluation only. Online cap is the mean physical cap over planned banks, charging every repeated source measurement. Every row also costs 18 noiseless target demonstrations per bank.

Figure 8: The span learner is not noise-robust. Bands are bank-bootstrap intervals for yield, with abstention counted as failure. Repetition improves some conditions but multiplies acquisition cost, so the curves are not at equal physical budgets. Target feedback stays exact.

Train bit $p$	Contexts	Recovered	Planned	Same-prior yield	Uniform-test yield	Train calls
0.05	18	0/24	4/24	0.1436	0.0405	432
0.05	22	0/24	8/24	0.2834	0.0731	528
0.05	44	9/24	22/24	0.8761	0.6416	1056
0.2	18	17/24	24/24	0.9213	0.8516	432
0.2	22	22/24	24/24	0.9836	0.9563	528
0.2	44	24/24	24/24	1.0000	1.0000	1056

Appendix

Table 6: All nonuniform-training and shift conditions. Each training context costs one noiseless target demonstration in addition to the source calls shown. Rows share the same 24 banks and test random numbers. The uniform-data theorem does not provide its stated finite-sample recovery guarantee at

p<1/2

Figure 9: Coverage of the latent support matters. Uniform-test yield after fitting under each training prior; bands resample banks. More training contexts overcome moderate bias but not the strongest bias tested. Each training context costs 24 source calls and one noiseless target demonstration.

acquisition	curriculum	runs	calls	coset mass	bits used	up/down	wrecked	rollouts (ref.)
SFT	none	9	0	1.000	-0.03	0/0	0	65.6 (64.0)
	qsp	9	5	0.054	-2.06	0/9	3	26893 (2.7)
	greedy	9	5	0.049	-0.41	2/4	2	$3.1\times 10^{11}$ (2.7)
	random	9	5	0.076	-0.08	3/5	0	81.6 (8.0)
GRPO	none	9	0	1.000	-0.03	0/0	0	65.6 (64.0)
	qsp	9	437	0.000	-3.75	0/9	7	$2.9\times 10^{18}$ (2.7)

Appendix

Table 7: In-weight acquisition at

k=6

(three banks, three secrets each). Calls are verifier calls spent on sources (mean over runs); coset mass is the median over runs; bits used is

k+\log_{2}

of the mean over runs of

P(A^{\star})

after acquisition; up/down counts runs whose own bits used moved above

+0.2

or below

-0.2

; a run is wrecked when the coset mass falls below

10^{-2}

; rollouts is the expected number of policy samples until the first verified success,

\mathbb{E}\,[1/P(A^{\star})]

over runs, with the exact posterior in parentheses. GRPO runs whose source becomes confidently wrong receive no reward gradient afterwards (all-fail groups) and spend their remaining call budget without acquiring the label.

width	layers	steps	lr	final training loss	bits used at $b^{\star}$
64	2	6,000	0.001	0.693	-0.01 of 4
128	2	6,000	0.0003	0.693	-0.04 of 4
128	2	6,000	0.001	0.693	0.00 of 4
128	2	6,000	0.003	0.693	0.00 of 4
256	2	6,000	0.001	0.692	-0.03 of 4
256	2	20,000	0.001	0.693	0.00 of 4

Appendix

Table 8: In-context decoder capacity sweep at

k=6

(

d=8

): 8 configurations in 14 runs, 3 of them on all three banks and the rest on bank 0, one pretraining seed per row. The Bayes-optimal training loss for banks 0, 1, 2 is 0.477, 0.476, 0.472 nats against the uninformed value

\ln 2=0.693

, so every configuration that stays at

\ln 2

has left the whole learnable signal on the table.

Figure 10: Diversity does not buy structure for the in-weight learner.

k=6

, one bank; left, target accuracy on known contexts whose target prompt was held out of pretraining; right, coset mass after SFT on the planner’s sources, per run and median, against the prior and the algebraic agent.

Figure 11: Fraction of delivered bits used against latent dimension, at the zero-error split with the planner’s sources. In-context decoder: three banks per point at width

128

(one pretraining run per bank). Diamonds: the scratchpad executors, each on its own canonical bank, which differs from the in-context banks. The smaller

d=5

executor puts less probability on

A^{\star}

than the uniform prior, so its fraction is negative; it is drawn at the bottom edge with its value.

Figure 12: Capacity sweep at

d=8

. In-context decoder on each of the three

d=8

banks, one pretraining seed per configuration: bits used at the zero-error split against width (two layers,

6000

steps) and against training steps (width

256

). Dashed lines mark each bank’s delivered bits (4, 5 and 5). No configuration reaches the frontier, and with one seed per configuration there is no clear trend in width or steps.

$d$	sources	$\|S\|$	delivered	used	on frontier	coset mass	samples (exact)	guesses (exact)
5	none	0	0.00	0.00	1.00	1.00	16.0 (16.0)	8.0 (8.5)
	random basis	2–3	1.74	1.40	0.75	0.86	7.8 (5.3)	4.2 (3.2)
	greedy	2–3	2.74	2.65	0.92	0.96	2.8 (2.7)	1.8 (1.8)
	planner	2–3	2.74	2.74	1.00	1.00	2.7 (2.7)	1.6 (1.8)
8	none	0	0.00	0.00	1.00	1.00	64.0 (64.0)	34.8 (32.5)
	random basis	4–6	3.58	0.44	0.00	0.14	51.8 (8.0)	24.0 (4.5)

Appendix

Table 9: Curricula at the zero-error split (in-context decoder, three banks, four secrets each;

|S|

ranges over banks, whose

b^{\star}

differ). Bits are

k+\log_{2}\mathbb{E}

over contexts of

2^{\rho-k}

(delivered) and of

P(A^{\star})

(used). On frontier is the fraction of contexts whose own bits used are within

0.2

of their own

\rho(S)

; coset mass is the mean probability the learner puts on the targets compatible with the acquired labels. Samples is the expected number of draws, with replacement, from the unchanged policy until a verification accepts,

\mathbb{E}[1/P(A^{\star})]

; guesses is the number of distinct verifications made by a learner that verifies its most probable answers first, as in Theorem 9 , averaged over contexts. The exact posterior is in parentheses:

\mathbb{E}[2^{k-\rho}]

samples and

\mathbb{E}[(2^{k-\rho}+1)/2]

guesses. Means are over twelve contexts and are therefore noisy: without sources every learner has the same expected number of guesses as the exact posterior,

(2^{k}+1)/2

, so the difference in those rows is sampling noise.

set	bank	degrees at $b^{\star}$	per-bit use at $b^{\star}$	used (64)	used (4)	per-bit use, all sources	at $b^{\star}$ , $24{,}000$ steps
low	202609500	$(2,2,\cdot,1)$	$0.13,1.00,0.00,1.00$	2.17	1.97	$0.02,1.00,0.03,1.00$	$1.00,1.00,0.00,1.00$
low	202609503	$(2,2,\cdot,2)$	$0.00,1.00,0.00,1.00$	2.00	2.00	$0.00,1.00,1.00,1.00$	$1.00,1.00,0.00,1.00$
low	202609512	$(\cdot,2,1,2)$	$0.00,0.00,1.00,1.00$	2.00	2.00	$0.00,0.00,1.00,1.00$	$0.00,0.00,1.00,1.00$
high	202609506	$(\cdot,3,4,2)$	$0.00,0.02,0.11,1.00$	1.16	1.27	$0.01,0.02,0.03,1.00$	$0.00,1.00,0.72,1.00$
high	202609518	$(\cdot,4,2,1)$	$0.00,0.00,1.00,1.00$	2.00	2.00	$0.00,0.00,1.00,1.00$	$0.00,0.01,1.00,1.00$
high	202609534	$(\cdot,4,2,2)$	$0.00,0.00,1.00,1.00$	2.00	2.00	$0.02,0.02,1.00,1.00$	$0.00,0.00,1.00,1.00$

Appendix

Table 10: Which bit is lost (post hoc). Controlled banks, first training seed,

6{,}000

pretraining steps unless stated, rerun with the same seeds (reproducing the recorded runs exactly) and scored on all

2^{d}=64

secrets. Per-bit use of

a_{i}

1+\mathbb{E}\log_{2}P(a_{i}\mid a_{<i},\text{labels})

; the four terms add up to

k+\mathbb{E}\log_{2}P(A^{\star})

, which is at most the bits used

k+\log_{2}\mathbb{E}P(A^{\star})

by Jensen’s inequality. Bits used over all 64 secrets are compared with the four-secret estimate of Table 11 . The column for all sources scores the largest budget on the planner’s profile, where every target bit is determined; the last column repeats

b^{\star}

after

24{,}000

steps.

set	$d$	bank	degrees $(a_{0},\dots,a_{k-1})$	delivered	used
released	5	0	$(\cdot,1,\cdot,2)$	2.00	2.01
released	5	1	$(\cdot,2,2,2)$	3.00	2.99
released	5	2	$(\cdot,2,2,2)$	3.00	3.00
released	6	0	$(\cdot,1,4,3)$	3.00	1.46, 1.98, 1.90
released	6	1	$(\cdot,\cdot,2,2)$	2.00	1.99
released	6	2	$(\cdot,2,2,2)$	3.00	3.00

Appendix

Table 11: XOR degrees and bits used at the zero-error split. Degree of target bit

a_{i}

: the fewest rows among the planner’s acquired source rows and the earlier target rows whose XOR is

T_{i}

(

\cdot

= not determined). Bits are

k+\log_{2}\mathbb{E}

2^{\rho-k}

(delivered) and of

P(A^{\star})

(used) over four secrets with the planner’s sources. Released banks list one training seed, except bank 0 at

d\in\{6,7,8\}

, which lists three; controlled banks list both training seeds.

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