\documentclass[11pt]{article}

\usepackage{../../template/ifitsmanu-paper}

%% --- Paper metadata ---
\title{Calibration Drift Under Verifier Composition.\\
       A Joint Scoring-Rule Mechanism for Pipeline-Level\\
       Cost-Correct Minimization.}
\author{Manu Bhardwaj}
\date{May 2026}

\renewcommand{\shortauthor}{Bhardwaj}
\renewcommand{\shorttitle}{Calibration Drift Under Verifier Composition}

%% --- Compact math macros used in this paper ---
\newcommand{\Apipe}{\alpha^{\mathrm{pipe}}}
\newcommand{\Aminpipe}{\alpha^{\mathrm{pipe}}_{\min}}
\newcommand{\Cost}{\mathrm{CostCorrect}}
\newcommand{\CPM}{\mathrm{CPM}_{1{:}1}}
\newcommand{\Reg}{\mathrm{Reg}}
\newcommand{\CH}{C_{\mathrm{H}}}

\begin{document}

\maketitle

\begin{abstract}
Production large language model verification is composed. A process
reward model gates trajectories, an outcome verifier accepts the final
answer, and an LLM judge gates the reject-or-revise loop. The deployer
pays Cost-correct on the composed pipeline, not on any single verifier.
The procurement mechanism of \citet{bhardwaj2026procurement} elicits one
verifier at a time. We show that per-verifier strictly proper
elicitation does not compose. Pipeline-level miscalibration under any
monotone Boolean composition rule equals the within-instance
verifier-disagreement covariance exactly. Per-verifier strictly proper
elicitation is dominant-strategy IC for the marginal reports it asks
for, but the resulting selection rule does not implement pipeline
cost-correct minimization. Candidate pairs with matched marginals and
mismatched joint distributions are paid identically and selected at
chance, while their pipeline accept rates differ by the disagreement
covariance. Under strategic commitment to a joint distribution within
the calibration-monotone class, verifiers are paid identically across
cost-correct-favorable and cost-correct-unfavorable joints, so the
deployer's selection on the composed pipeline is dominated by exogenous
noise. A joint scoring-rule mechanism over the cross-product report
space restores dominant-strategy incentive compatibility, ex post
individual rationality, and budget feasibility on the joint elicitation.
The deployer's expected gap to first-best Cost-correct on the composed
pipeline is at most $\CH \cdot \sqrt{(\log K_1 + \log K_2) / N}$ over
$K_1 \cdot K_2$ candidate pairs, by Hoeffding plus a union bound. A
matching lower bound holds on a calibration-monotone-pair family by Le
Cam's two-point method. The mechanism is therefore minimax optimal up to
log factors. Simulation on MATH, GSM8K, and HumanEval with
$K_1, K_2 \in \{4, 8, 16\}$ and probe budget
$N \in \{16, \ldots, 4096\}$ shows the joint mechanism reaching Paper~\#1's
5\%-of-first-best operational target at $N = 512$ under unknown joint
correlation, roughly double Paper~\#1's $N = 256$, and at $N = 256$ when
correlation is supplied as a side channel. The per-verifier baseline
does not reach the target at any $N$ tested when conditional
disagreement covariance exceeds $0.1$. The compliance corollary is
sharp. Per-component procurement records are not sufficient evidence
under the European Union AI Act high-risk obligations entering force on
August 2, 2026 \citep{euaiact2024}. The audit trail must include the
joint-report ledger.
\end{abstract}

% =========================================================================
\section{Introduction}
\label{sec:intro}

The verification-economics framing of \citet{bhardwaj2026fieldnote2}
treats the verifier accept rate $\alpha$ as the binding lever in
cost-per-correct-answer for large language model deployments. The
companion analysis on the $\alpha$-asymmetry
\citep{bhardwaj2026fieldnote3} shows that the partial of Cost-correct
with respect to $\alpha$ dominates the partials with respect to
per-token price, the reasoning multiplier $R$, and the rollout ratio
$\bar\rho$ in the rStar-Math regime \citep{guan2025rstar}. The
procurement mechanism of \citet{bhardwaj2026procurement} gives a
dominant-strategy incentive-compatible scoring-rule mechanism that
selects a single verifier with provable regret $\sqrt{\log K / N}$
versus the oracle-best in a candidate population of size $K$ on $N$
adversarially constructed probes.

A typical production verification stack is not a single verifier. The
deployer runs a process reward model that scores intermediate
trajectories \citep{lightman2024lets, uesato2022solving}, an outcome
verifier that accepts the final answer \citep{cobbe2021training}, and
one or more LLM judges that gate a reject-or-revise loop
\citep{zheng2023judging}. Each component can be procured under the
one-verifier mechanism. The composed pipeline is what the deployer pays
Cost-correct on. The economic question this paper answers is whether
per-verifier procurement composes. The answer is no, in a precise
sense, and the fix is a joint scoring-rule mechanism on the
cross-product report space.

Four contributions.

\paragraph{Theorem 1 (composition identity).} For any two binary
verifiers with conditional accept rates $\alpha_1(x)$ and $\alpha_2(x)$
and within-instance disagreement covariance
$C(x) = \mathrm{Cov}(V_1(x), V_2(x) \mid x)$, the AND-rule pipeline
accept rate satisfies
$\mathbb{E}[V_1 \wedge V_2 \mid x] = \alpha_1(x) \alpha_2(x) + C(x)$
identically. The same identity, with sign flips and additive constants,
holds for OR and for arbitrary monotone Boolean composition by
inclusion-exclusion.

\paragraph{Theorem 2 (non-implementation of pipeline cost-correct).}
Per-verifier strictly proper elicitation is dominant-strategy IC at each
slot in isolation but does not implement pipeline cost-correct
minimization. Under any non-degenerate joint distribution over verifier
reports, applying the one-verifier scoring-rule mechanism of
\citet{bhardwaj2026procurement} independently to each slot and composing
the selected verifiers under a monotone Boolean rule yields a selection
rule that, under truthful marginal reporting, does not separate
candidate pairs with matched marginal accept rates and mismatched joint
distributions. The pairs are paid identically and selected at chance,
while their pipeline accept rates differ by exactly the within-instance
disagreement covariance. The non-implementation is ex ante undetectable
from marginal reports.

\paragraph{Theorems 3 and 4 (joint mechanism with matching regret
bounds).} A joint scoring-rule mechanism that pays each candidate
verifier-pair the value of a strictly proper scoring rule
\citep{gneiting2007strictly, frongillo2021general} applied to the joint
report distribution on the cross-product space $\{0, 1\}^2$ is
dominant-strategy IC, ex post IR, and budget feasible under a per-probe
payment cap. The deployer who selects the verifier-pair with highest
empirical joint score incurs expected regret of at most
$\CH \cdot \sqrt{(\log K_1 + \log K_2) / N}$ versus the oracle-best
pair, by Hoeffding \citep{hoeffding1963probability} plus a union bound.
A matching lower bound holds on a calibration-monotone-pair family by Le
Cam's two-point method
\citep{lecam1973convergence, tsybakov2009introduction}. The mechanism is
minimax optimal up to log factors.

\paragraph{Simulation result.} Synthesized verifier pairs on MATH
\citep{hendrycks2021measuring}, GSM8K \citep{cobbe2021training}, and
HumanEval \citep{chen2021evaluating}, with controlled disagreement
covariance $C \in \{-0.2, -0.1, 0, +0.1, +0.2\}$ and
$K_1, K_2 \in \{4, 8, 16\}$. The joint mechanism reaches a
5\%-of-first-best regret target at $N = 512$ under unknown $C$ and at
$N = 256$ under known $C$ supplied as a side channel. The per-verifier
baseline does not reach the target at any $N$ tested when
$|C| \geq 0.1$.

The contribution that goes beyond \citet{bhardwaj2026procurement} is the
move from single-verifier procurement to pipeline procurement. The
companion paper characterizes the verifier the deployer ends up with
under unobservable quality. This paper characterizes the pipeline the
deployer ends up with under unobservable joint quality. The shift
requires the disagreement-covariance correction, the joint scoring
rule, and a strengthened calibration-monotone-pair assumption.

The contribution beyond classical peer prediction
\citep{miller2005peer, witkowski2012peer, kong2019informationtheoretic,
       frongillo2021general}
is the procurement framing. Peer prediction elicits truthful reports
from agents whose joint distribution generates the signal. This paper
elicits truthful reports from two procured verifiers whose joint
distribution is the operational artifact the deployer pays Cost-correct
on, in a setting with adversarial probes and known ground truth. The
grounded-probe assumption inherited from \citet{bhardwaj2026procurement}
rules in strict propriety in dominant strategies, not Nash, and rules
out the common-prior assumptions that the peer-prediction tradition
spent fifteen years removing.

The contribution beyond the recent process-reward-modeling literature
\citep{lightman2024lets, uesato2022solving, cobbe2021training} is the
composition analysis. That literature establishes that production
stacks do compose process and outcome verifiers, but treats verifiers
as in-house artifacts. This paper analyzes the composed pipeline under a
procurement mechanism and shows that the procurement game is
structurally different from the in-house composition game.

The result has an external forcing function. The European Union AI Act
high-risk obligations apply from August 2, 2026 \citep{euaiact2024}.
High-risk deployers must produce accept-rate evidence at a documented
threshold under Article 15. The companion paper's per-component
mechanism produces this evidence for a single procured verifier. The
composition identity of Theorem 1 implies that per-component evidence
drifts from the pipeline-level accept rate by exactly $C(x)$. An auditor
who accepts per-component records accepts an accept-rate misstatement of
up to $|C(x)|$. The joint-mechanism audit trail closes that gap.

The rest of the paper is organized as follows. \Cref{sec:model} sets up
the model. \Cref{sec:composition} proves the composition identity.
\Cref{sec:nonimpl} proves the non-implementation result for per-verifier
elicitation. \Cref{sec:mechanism} constructs the joint scoring-rule
mechanism. \Cref{sec:regret} proves matching regret bounds.
\Cref{sec:probes} develops probe-correlated label noise as the new
binding cost. \Cref{sec:simulation} reports the simulation.
\Cref{sec:euaiact} returns to the EU AI Act forcing function.
\Cref{sec:limitations} records limitations and future work.

% =========================================================================
\section{Model}
\label{sec:model}

We extend the single-verifier setup of \citet{bhardwaj2026procurement}
to a two-slot setting. Three-and-up composition follows by induction for
AND and OR; the general monotone case is handled in
\Cref{app:monotone}.

\paragraph{Players.} A single deployer faces $K_1$ candidate verifier
providers for slot 1, indexed $k_1 \in \{1, \ldots, K_1\}$, and $K_2$
candidate verifier providers for slot 2, indexed
$k_2 \in \{1, \ldots, K_2\}$. The deployer commits to a procurement
mechanism before observing any private information. Each verifier
provider knows its own type and observes the mechanism.

\paragraph{Task distribution.} The deployer faces a known task
distribution $D$ over prompts $x$ and a known target quality threshold
$\theta$. A response $y$ is correct at threshold $\theta$ if a fixed
programmatic check $c(x, y, \theta) \in \{0, 1\}$ returns $1$.

\paragraph{Verifier type.} Each verifier $k_i$ in slot
$i \in \{1, 2\}$ has a private decision function
$V_{k_i} : \mathcal{X} \times \mathcal{Y} \to \{0, 1\}$, drawn from a
known family $\mathcal{F}_i$. The function $V_{k_i}$ specifies whether
verifier $k_i$ accepts a candidate response as correct at threshold
$\theta$. Verifier types are private. The families
$\mathcal{F}_1, \mathcal{F}_2$ and the per-prompt cost-of-quality
functions $\{\kappa_{k_i}\}$ are common knowledge.

\paragraph{Joint distribution.} Verifier reports from the two slots are
not assumed independent. We write
$\alpha_{k_i}(x) = \Pr[V_{k_i}(x, y) = 1 \mid x]$ for the marginal
accept rate of verifier $k_i$ on prompt $x$ and
$C_{k_1, k_2}(x) = \mathrm{Cov}(V_{k_1}(x, y), V_{k_2}(x, y) \mid x)$
for the within-instance disagreement covariance.

\paragraph{Composition rule.} A fixed monotone Boolean function
$f : \{0, 1\}^2 \to \{0, 1\}$ aggregates the per-slot reports. The
default rule is AND: $f(r_1, r_2) = r_1 \wedge r_2$. The OR rule and
the generic monotone case are treated in appendices.

\paragraph{Pipeline accept rate.} Under composition rule $f$ and
verifier pair $(k_1, k_2)$,
\begin{equation}
  \Apipe_{k_1, k_2}(x)
  = \mathbb{E}\!\left[f(V_{k_1}(x, y), V_{k_2}(x, y)) \mid x\right].
  \label{eq:pipeaccept}
\end{equation}
For the AND rule,
$\Apipe_{k_1, k_2}(x) = \alpha_{k_1}(x) \alpha_{k_2}(x) + C_{k_1, k_2}(x)$
by \Cref{thm:composition} below.

\paragraph{Cost-correct on the pipeline.} Per-task cost under pair
$(k_1, k_2)$ is, extending \citet{bhardwaj2026fieldnote2},
\begin{equation}
  \Cost(k_1, k_2)
  = \frac{\CPM \cdot R \cdot (1 + \bar\rho)}
         {\mathbb{E}_x[\Apipe_{k_1, k_2}(x)]},
  \label{eq:costpipe}
\end{equation}
with $\CPM$, $R$, and $\bar\rho$ held fixed across pair choice. The
deployer minimizes $\Cost$, which is equivalent to maximizing the
expected pipeline accept rate.

\paragraph{Probe set.} The deployer has a budget of $N$ probes drawn
from a probe distribution $P$ over $\mathcal{X} \times \mathcal{Y}$
with known ground-truth labels $\ell_i \in \{0, 1\}$. Probes may be
adversarial with respect to $\mathcal{F}_1 \times \mathcal{F}_2$. We
treat the probe-construction cost as exogenous in
\Cref{sec:nonimpl,sec:mechanism,sec:regret} and endogenize it in
\Cref{sec:probes}.

\paragraph{Mechanism.} A direct mechanism is a pair $(s, t)$ where
$s : \{0, 1\}^{K_1 \cdot K_2 \cdot 2 \cdot N} \to \{1, \ldots, K_1\}
\times \{1, \ldots, K_2\}$
is a selection rule mapping joint reports to a chosen verifier-pair,
and
$t : \{0, 1\}^{K_1 \cdot K_2 \cdot 2 \cdot N} \to \mathbb{R}^{K_1 \cdot K_2}$
is a payment rule. We restrict to mechanisms that depend only on
reported joint decisions on probes.

\paragraph{Solution concept.} We seek mechanisms that satisfy
dominant-strategy incentive compatibility (DSIC), ex post individual
rationality (IR), and budget feasibility under a per-probe payment cap
$\bar t$. We measure performance by expected regret to first-best on
the composed pipeline,
\begin{equation}
  \Reg(s, t)
  = \mathbb{E}\!\left[\,\Cost(s) - \min_{(k_1, k_2)} \Cost(k_1, k_2)\,\right],
  \label{eq:regpipe}
\end{equation}
and by worst-case regret over $\mathcal{F}_1 \times \mathcal{F}_2$.

\paragraph{Calibration-monotone-pair family.} A family
$\mathcal{F}_1 \times \mathcal{F}_2$ is \emph{calibration-monotone-pair}
if there exists a partial order $\succeq$ on pairs such that
$(k_1, k_2) \succeq (k_1', k_2')$ implies
$\Apipe_{k_1, k_2}(x) \geq \Apipe_{k_1', k_2'}(x)$ for all $x$ in the
support of $D$. The condition is a strict strengthening of the
calibration-monotone assumption of \citet{bhardwaj2026procurement}. It
is more restrictive than per-slot calibration monotonicity because it
constrains the joint ordering, not just the marginal orderings.

% =========================================================================
\section{The composition identity}
\label{sec:composition}

We give the composition identity for the AND rule first and indicate
the generalization to OR and arbitrary monotone Boolean rules in
\Cref{cor:monotone}.

\begin{theorem}[composition identity for AND]
\label{thm:composition}
Let $V_1, V_2 : \mathcal{X} \times \mathcal{Y} \to \{0, 1\}$ be binary
verifiers with marginal accept rates
$\alpha_1(x) = \Pr[V_1 = 1 \mid x]$ and
$\alpha_2(x) = \Pr[V_2 = 1 \mid x]$, and within-instance disagreement
covariance $C(x) = \mathrm{Cov}(V_1, V_2 \mid x)$. Then
\begin{equation*}
  \mathbb{E}[V_1 \wedge V_2 \mid x]
  = \alpha_1(x) \alpha_2(x) + C(x).
\end{equation*}
\end{theorem}

\begin{proof}
For binary random variables, $V_1 \wedge V_2 = V_1 \cdot V_2$
pointwise. Take conditional expectation given $x$,
\begin{equation*}
  \mathbb{E}[V_1 V_2 \mid x]
  = \mathbb{E}[V_1 \mid x] \mathbb{E}[V_2 \mid x] + \mathrm{Cov}(V_1, V_2 \mid x)
  = \alpha_1(x) \alpha_2(x) + C(x).
\end{equation*}
The first equality is the definition of covariance for binary random
variables. The second substitutes $\alpha_i(x) = \mathbb{E}[V_i \mid x]$
and $C(x) = \mathrm{Cov}(V_1, V_2 \mid x)$.
\end{proof}

\begin{corollary}[composition identity for OR]
\label{cor:or}
Under the same hypotheses,
\begin{equation*}
  \mathbb{E}[V_1 \vee V_2 \mid x]
  = \alpha_1(x) + \alpha_2(x) - \alpha_1(x) \alpha_2(x) - C(x).
\end{equation*}
\end{corollary}

\begin{proof}
$V_1 \vee V_2 = V_1 + V_2 - V_1 V_2$ pointwise for binary $V_i$. Apply
linearity and \Cref{thm:composition}.
\end{proof}

\begin{corollary}[composition identity for arbitrary monotone Boolean rules]
\label{cor:monotone}
Let $f : \{0, 1\}^m \to \{0, 1\}$ be a monotone Boolean function on $m$
binary verifiers. Then $\mathbb{E}[f(V_1, \ldots, V_m) \mid x]$ is a
polynomial in the marginal accept rates $\{\alpha_i(x)\}$ and the
higher-order joint moments
$\{\mathbb{E}[V_{i_1} \cdots V_{i_k} \mid x] : k \geq 2\}$, where the
polynomial coefficients are given by Mobius inversion over the lattice
of monotone Boolean functions \citep[ch.~2]{grimmett2014probability}.
\end{corollary}

\begin{proof}
Standard from the indicator-function representation of monotone Boolean
functions; see \Cref{app:monotone} for the explicit two-and-three-verifier
formulas.
\end{proof}

\paragraph{Discussion.} \Cref{thm:composition} is elementary. Its
content is not the algebra; the algebra is the bilinear identity for
binary random variables. The content is that the additive correction
term is \emph{exactly} the within-instance covariance, not a bounded
error term or a worst-case slack. The pipeline accept rate is determined
by the per-verifier accept rates only when the per-verifier reports are
conditionally independent on each prompt. Production verifier stacks
are not conditionally independent. A process reward model and an
outcome verifier may share trajectory features and have positive
disagreement covariance in the rank-1-aligned regime documented by
\citet{ye2026implicit}; the construction protocols of
\citet{lightman2024lets, cobbe2021training} do not separate the two
verifiers' training-trajectory distributions.

The implication for procurement is that any calibration argument
applied to $V_1$ and $V_2$ in isolation is silent on the pipeline. The
reverse is also true. Per-component reports can be miscalibrated in the
marginal Brier sense while the pipeline is well-calibrated, if the
marginal miscalibrations cancel through $C(x)$. Neither direction is
the safe one to assume in production.

We use \Cref{thm:composition} in \Cref{sec:nonimpl} to construct the
matched-marginal candidate pair that the per-verifier mechanism fails to
separate, and in \Cref{sec:mechanism} to define the joint scoring rule
that implements pipeline cost-correct minimization in dominant
strategies.

% =========================================================================
\section{Per-verifier elicitation does not implement pipeline cost-correct}
\label{sec:nonimpl}

\paragraph{Setup.} The deployer runs the one-verifier mechanism of
\citet{bhardwaj2026procurement} independently for slot 1 and slot 2.
Each candidate verifier in each slot reports a probability of acceptance
on each of the $N$ probes. Per-slot payment is a strictly proper
scoring rule applied to the reports against ground-truth labels. The
deployer selects the verifier in each slot with highest empirical
per-slot score and composes the selected pair under the AND rule. We
call this the \emph{per-verifier mechanism}.

The per-verifier mechanism is DSIC at each slot in isolation, by
\citet[Theorem~2]{bhardwaj2026procurement}, because strict propriety
makes truthful marginal reporting dominant on each slot's payment rule.
We show that DSIC at the per-slot level is not sufficient for
implementation of pipeline cost-correct minimization.

\begin{theorem}[non-implementability of pipeline cost-correct under per-verifier elicitation]
\label{thm:nonimpl}
There exists a two-verifier instance with non-degenerate joint
distribution over verifier reports in which the per-verifier mechanism,
under its unique truthful equilibrium, selects a verifier pair that is
strictly suboptimal under pipeline Cost-correct. The per-verifier
selection rule on truthful marginal reports does not identify the
pipeline cost-correct-optimal pair.
\end{theorem}

\begin{proof}[Construction]
Take a uniform task distribution over two prompts $x_1, x_2$, each with
ground-truth label $\ell = 1$. Fix one slot-2 verifier $V_2$ with
marginal accept rate $\alpha_2 = 0.6$ on every prompt. Consider two
slot-1 candidates $V_1, V_1'$, both with marginal accept rate
$\alpha_1 = 0.6$ on every prompt, distinguished only by their joint
distribution with $V_2$.

\begin{center}
\begin{tabular}{@{}lccccc@{}}
\toprule
Joint state & $(V{=}1, V_2{=}1)$ & $(V{=}1, V_2{=}0)$
            & $(V{=}0, V_2{=}1)$ & $(V{=}0, V_2{=}0)$
            & $C(x)$ \\
\midrule
$V_1$       & $0.40$ & $0.20$ & $0.20$ & $0.20$ & $+0.04$ \\
$V_1'$      & $0.36$ & $0.24$ & $0.24$ & $0.16$ & $\hspace*{0.7em}0.00$ \\
\bottomrule
\end{tabular}
\end{center}

Both candidates have marginal $\alpha = 0.6$. Under truthful reporting,
both achieve identical expected Brier score on the marginal labels,
since the score depends only on the marginal $\alpha$ and the label
distribution. By the indistinguishability of the per-slot empirical
Brier on the truthful equilibrium, the per-verifier mechanism selects
between $V_1$ and $V_1'$ uniformly at random.

By \Cref{thm:composition}, the AND-pipeline accept rate is
$\alpha_1 \alpha_2 + C$. The pair $(V_1, V_2)$ achieves
$0.6 \cdot 0.6 + 0.04 = 0.40$. The pair $(V_1', V_2)$ achieves
$0.6 \cdot 0.6 + 0 = 0.36$. The cost-correct-optimal pair is strictly
$(V_1, V_2)$ by an $\alpha$-gap of $0.04$, which translates to a
Cost-correct gap of $0.04 / 0.36 \approx 11\%$. The per-verifier
mechanism selects this pair with probability $1/2$, leaving an expected
gap of $5.5\%$ on the table.

The gap is not closed by collecting more probes. The marginal
indistinguishability is exact at the population level, not a
finite-sample artifact. Larger $N$ tightens the empirical Brier
concentration but does not separate $V_1$ from $V_1'$ on the marginal
score.
\end{proof}

\paragraph{Why this is the right negative result.} The non-implementation
requires conditional correlation. When $C(x) = 0$ for all $x$, the
joint accept rate is determined by the marginal accept rates, so
marginal selection implements pipeline selection. The construction is
non-trivial only when $C(x) \neq 0$, which is the realistic regime
where PRMs and outcome verifiers project onto correlated trajectory
features \citep{ye2026implicit}. The negative result bites in
production.

\begin{corollary}[no per-verifier rescue]
\label{cor:nopvfix}
No per-verifier scoring rule, including any strictly proper rule in the
class of \citet{gneiting2007strictly}, implements pipeline Cost-correct
minimization on a non-degenerate joint distribution.
\end{corollary}

\begin{proof}
The per-slot payment under any per-verifier rule depends only on the
slot's marginal reports against the ground-truth labels. The marginal
reports identify only the marginal accept rate. The pipeline accept
rate is the marginal accept rate plus the disagreement covariance by
\Cref{thm:composition}. The covariance is not identified by any
per-slot rule. Therefore no per-slot rule can break the selection tie
between candidates with matched marginals but mismatched covariance.
The standard payoff-equivalence argument \citep{myerson1981optimal}
formalizes the identifiability claim. The full argument is in
\Cref{app:thm-nonimpl}.
\end{proof}

\paragraph{Strategic refinement.} A stronger negative result holds when
the verifier is permitted to \emph{commit} to a joint distribution
before the mechanism runs. A strategic verifier with private knowledge
of the deployer's slot-2 verifier $V_2$ can choose the joint
distribution within its calibration-monotone class. Under per-verifier
elicitation, the verifier is paid only on marginals, so it is
indifferent across joint distributions consistent with its marginal. A
verifier that commits to the cost-correct-optimal joint distribution
receives no reward over one that commits to a worse joint distribution.
The deployer's selection is then dominated by exogenous noise. Under
the joint mechanism of \Cref{sec:mechanism}, the verifier is paid on
joint reports and strictly prefers the cost-correct-optimal joint
distribution. The strategic-refinement result is stated as Theorem~2.b
and proved in \Cref{app:thm-nonimpl}.

\paragraph{Remark on the parallel literature.} The non-implementation
result is structurally analogous to the well-known non-identification
of joint distributions from marginal observations in classical
statistics \citep{cover2006elements} and to the failure of pairwise
scoring in the setting of \citet{frongillo2021general}. The novelty is
the procurement framing. Peer prediction studies agents who report on a
common signal; our setting studies verifiers procured for cost-correct
minimization on a fixed task distribution. The implementation of
pipeline cost-correct minimization under conditional verifier
correlation has not appeared in the literature in this form.

The next section gives the joint scoring-rule mechanism that implements
pipeline cost-correct minimization in dominant strategies on the joint
report space.

% =========================================================================
\section{The joint scoring-rule mechanism}
\label{sec:mechanism}

\paragraph{Construction.} Fix a strictly proper scoring rule
$S : \Delta(\{0, 1\}^2) \times \{0, 1\}^2 \to \mathbb{R}$ on the joint
distribution over the cross-product report space, for instance the joint
Brier score
\begin{equation}
  S(\hat q, (r_1, r_2))
  = -\sum_{(a, b) \in \{0, 1\}^2}
    \left(\hat q(a, b) - \mathbf{1}[(r_1, r_2) = (a, b)]\right)^2,
  \label{eq:jointbrier}
\end{equation}
which is strictly proper by the multidimensional extension of
\citet[Theorem~2]{gneiting2007strictly}. Each candidate pair
$(V_{k_1}, V_{k_2})$ reports a joint distribution
$\hat q_{(k_1, k_2), n} \in \Delta(\{0, 1\}^2)$ on each probe $n$,
optionally constrained to the four point masses for accept-or-reject
verifiers. The mechanism pays the pair
\begin{equation}
  t_{(k_1, k_2)}(\hat q, r)
  = a + b \cdot \frac{1}{N} \sum_{n=1}^N
    S\!\left(\hat q_{(k_1, k_2), n}, (V_{k_1}(x_n, y_n), V_{k_2}(x_n, y_n))\right),
  \label{eq:jointpayment}
\end{equation}
for constants $a \geq 0$ and $b > 0$ chosen to enforce ex post IR and
the per-probe payment cap. The selection rule is empirical $\arg\max$
over pairs.

The construction scores the \emph{joint observed outcome}
$(V_{k_1}, V_{k_2})$ against the reported joint distribution. The
ground-truth label $\ell$ enters through the conditional structure of
$\hat q$: under truthful reporting, $\hat q$ matches the conditional
joint distribution of $(V_{k_1}, V_{k_2})$ given $\ell$ on the probe
distribution. Atomic commitment of the joint report (both components
submitted simultaneously, with no observability between components at
report time) is part of the mechanism. Sealed-bid joint submission with
a commit-reveal hash makes atomic commitment enforceable in deployment.

\begin{theorem}[joint mechanism]
\label{thm:joint}
Under the joint scoring-rule mechanism with $a$ chosen so that
$a + b \cdot \min_S \geq 0$, where $\min_S$ is the infimum of $S$ on
its domain, the mechanism is dominant-strategy incentive-compatible,
ex post individually rational, and budget feasible under per-probe
payment cap $\bar t = a / N + b \cdot \max_S / N$.
\end{theorem}

\begin{proof}
Strict propriety of $S$ on $\Delta(\{0, 1\}^2)$ implies that for any
belief $q$ a verifier pair holds about the joint distribution of
$(V_{k_1}, V_{k_2})$ given $(x, y, \ell)$, the unique maximizer of
$\mathbb{E}_{(V_{k_1}, V_{k_2})} S(\hat q, (V_{k_1}, V_{k_2}))$ over
$\hat q$ is $\hat q = q$. The multidimensional version of strict
propriety is established in
\citet[Theorem~3]{frongillo2021general} via convex analysis of the
Bregman-divergence representation. Atomic commitment of the joint
report (\Cref{sec:mechanism} construction) rules out post-observation
conditioning, so the dominant strategy is truthful joint reporting on
the cross-product space, which is the report space that identifies the
pipeline accept rate by \Cref{thm:composition}. Individual rationality
follows from the choice of $a$. Budget feasibility follows from the
per-probe payment cap.
\end{proof}

\paragraph{Identifiability condition.} The joint scoring-rule mechanism
requires the joint distribution over $(V_{k_1}, V_{k_2})$ to be
identifiable from probe reports. Formally, the empirical joint-report
correlation matrix on the probe set must be full rank. The condition is
generically satisfied under adversarial probe construction
(\Cref{sec:probes}) but can fail on natural probe distributions;
\Cref{sec:simulation} reports one such failure on HumanEval.

\begin{proposition}[identifiability sufficient condition]
\label{prop:identif}
If the probe distribution $P$ contains at least two probe types
$(x_a, y_a, \ell_a), (x_b, y_b, \ell_b)$ whose conditional joint
distributions over $(V_{k_1}, V_{k_2})$ differ as distributions on
$\{0, 1\}^2$, equivalently if the empirical joint-report correlation
matrix on the probe set has rank at least two, then the joint scoring
rule is identifying in the sense that the unique strategy maximizing
expected payment is truthful joint reporting.
\end{proposition}

The condition is straightforward to check at deployment time: enumerate
the joint reports on the probe set and check that the empirical
correlation matrix has rank $\geq 2$. \Cref{sec:simulation} implements
the check as a pre-flight gate and documents the failure mode when it
does not hold.

\paragraph{Connection to peer prediction.} The joint elicitation extends
multi-task peer prediction \citep{dasgupta2013crowdsourced} to the
grounded-probe setting. The grounded-probe assumption eliminates the
common-prior dependence that peer prediction requires in the
no-ground-truth setting and yields strict propriety in dominant
strategies rather than only in Bayesian equilibrium. The mechanism is
structurally close to the information-theoretic framework of
\citet{kong2019informationtheoretic}, with the joint-report space
playing the role of the complementarity carrier.

\paragraph{Connection to scored AI oversight.} \citet{loven2026honest}
proves DSIC for a parametric pseudospherical scoring family in scored
AI oversight via the Prekopa principle. The Lov\'en result is for the
single-agent setting. The joint mechanism inherits the strict-propriety
guarantee per slot and extends it to the cross-product report space;
the Prekopa-principle proof technique adapts to the joint setting under
the identifiability condition.

% =========================================================================
\section{Regret bounds for the joint mechanism}
\label{sec:regret}

We now bound the deployer's expected gap from first-best Cost-correct
on the composed pipeline under the joint mechanism of
\Cref{sec:mechanism}. Throughout this section, verifier pairs report
truthfully, by \Cref{thm:joint}.

\begin{theorem}[upper bound]
\label{thm:upper}
Let
$\Apipe_{k_1, k_2}(Q) := \mathbb{E}_{(x, y) \sim Q}[f(V_{k_1}(x, y),
V_{k_2}(x, y))]$
denote the population pipeline accept rate of pair $(k_1, k_2)$ under
distribution $Q$. Let
$(k_1^*, k_2^*) = \arg\max_{(k_1, k_2)} \Apipe_{k_1, k_2}(D)$ be the
oracle-best pair on the deployer's task distribution. Suppose probes
are drawn iid from a probe distribution $P$ with
$\Apipe_{k_1, k_2}(P) = \Apipe_{k_1, k_2}(D)$ for all pairs. Then the
expected gap of the empirical $\arg\max$ rule is
\begin{equation}
  \mathbb{E}\!\left[\Apipe_{k_1^*, k_2^*}(D) - \Apipe_{\hat k_1, \hat k_2}(D)\right]
  \leq \CH \cdot \sqrt{\frac{\log K_1 + \log K_2}{N}}
  \label{eq:upperbound}
\end{equation}
for a universal Hoeffding constant $\CH$, distinct from the
disagreement covariance $C(x)$ of \Cref{thm:composition}.
\end{theorem}

\begin{proof}
The empirical pipeline accept rate
$\hat \Apipe_{k_1, k_2}
 = \frac{1}{N} \sum_n f(V_{k_1}(x_n, y_n), V_{k_2}(x_n, y_n))$
is a bounded iid average in $[0, 1]$ for each pair. By Hoeffding's
inequality \citep{hoeffding1963probability},
$\Pr[|\hat \Apipe_{k_1, k_2} - \Apipe_{k_1, k_2}(P)| > \epsilon]
 \leq 2 e^{-2 N \epsilon^2}$.
By a union bound over $K_1 \cdot K_2$ pairs,
\begin{equation*}
  \Pr\!\left[\max_{(k_1, k_2)}
  \left|\hat \Apipe_{k_1, k_2} - \Apipe_{k_1, k_2}(P)\right| > \epsilon\right]
  \leq 2 K_1 K_2 e^{-2 N \epsilon^2}.
\end{equation*}
Setting the right-hand side to $\delta$ gives
$\epsilon = \sqrt{(\log K_1 + \log K_2 + \log(2/\delta)) / (2 N)}$. On
the complement event, the standard $\arg\max$ regret argument yields
$\Apipe_{k_1^*, k_2^*} - \Apipe_{\hat k_1, \hat k_2} \leq 2\epsilon$.
Integrating the tail and using the unbiasedness assumption closes the
bound on the deployer's distribution with $\CH = O(1)$. The full
computation is in \Cref{app:thm-upper}.
\end{proof}

The translation to Cost-correct units follows the Paper~\#1 argument.
On the event that pipeline accept rates are bounded below by
$\Aminpipe > 0$, the gap in $1 / \Apipe$ is bounded by
$|\Apipe_{k_1^*, k_2^*} - \Apipe_{\hat k_1, \hat k_2}|
 / (\Aminpipe)^2$,
which scales as $\sqrt{(\log K_1 + \log K_2) / N}$ up to a Lipschitz
constant determined by $\Aminpipe$.

\begin{theorem}[lower bound]
\label{thm:lower}
Suppose $\mathcal{F}_1 \times \mathcal{F}_2$ is calibration-monotone-pair
and contains at least two distinct pairs
$(k_a^{(1)}, k_a^{(2)}) \succ (k_b^{(1)}, k_b^{(2)})$ with
$\sup_{x, y} |\Apipe_{k_a^{(1)}, k_a^{(2)}}(x, y) -
\Apipe_{k_b^{(1)}, k_b^{(2)}}(x, y)| > 0$. Then for any mechanism
$(s, t)$ and any $K_1, K_2 \geq 2$, there exists a profile of types in
$(\mathcal{F}_1 \times \mathcal{F}_2)^{K_1 K_2}$ such that
\begin{equation}
  \mathbb{E}\!\left[\Apipe_{k_1^*, k_2^*}(D) - \Apipe_{s}(D)\right]
  \geq c \cdot \sqrt{\frac{\log K_1 + \log K_2}{N}}
  \label{eq:lowerbound}
\end{equation}
for a constant $c > 0$ depending on $\mathcal{F}_1 \times \mathcal{F}_2$
but not on $K_1$, $K_2$, or $N$.
\end{theorem}

\begin{proof}[Proof sketch]
Apply Le Cam's two-point method
\citep{lecam1973convergence, tsybakov2009introduction}. Construct a
packing of $\Theta(K_1 \cdot K_2)$ profiles of pair types that are
pairwise indistinguishable on probe sets of size $N$ at total variation
distance $O(\sqrt{N} \cdot \Delta_{\mathrm{pipe}})$, where
$\Delta_{\mathrm{pipe}} = \sup |\Apipe_{k_a} - \Apipe_{k_b}|$. The
reduction from selection regret to estimation error follows from the
calibration-monotone-pair assumption: an order in $\succeq$ implies an
order in expected pipeline accept rate. Standard Le Cam arguments yield
expected error of order $\sqrt{(\log K_1 + \log K_2) / N}$ on the
implied estimation problem. Full argument in \Cref{app:thm-lower}.
\end{proof}

\Cref{thm:upper,thm:lower} together imply that the joint scoring-rule
mechanism is minimax optimal up to log factors over
calibration-monotone-pair families.

\paragraph{Comparison to Paper~\#1.} The $K$ counts enter additively in
the log, reflecting the union bound over the cross product
$K_1 \times K_2$. The $N$ dependence is unchanged at $\sqrt{1/N}$. The
hidden constant is larger than the Paper~\#1 constant by a factor that
scales with $\max_x |C(x)|$. In sample-complexity terms, achieving a
target gap $\epsilon$ in pipeline accept rate requires
$N \geq \CH^2 (\log K_1 + \log K_2) / \epsilon^2$, where $\CH$ is the
universal Hoeffding constant of \Cref{thm:upper}. At $K_1 = K_2 = 16$
and $\epsilon = 0.05$, the joint mechanism budget is approximately
$N \approx 2200$, against Paper~\#1's $N \approx 1100$ at $K = 16$. The
factor-of-two probe budget relative to Paper~\#1 is the price of joint
elicitation under unknown conditional correlation.
\Cref{fig:regret-jointvspv} pins these rates against the simulation.

\begin{figure}[t]
  \centering
  \begin{tikzpicture}
    \begin{axis}[
      width=0.82\linewidth, height=6cm,
      xlabel={Probe budget $N$},
      ylabel={Pipeline Cost-correct gap to oracle},
      xmin=16, xmax=4096,
      ymin=0, ymax=0.20,
      xmode=log, log basis x={2},
      xtick={16,64,256,1024,4096},
      xticklabels={16,64,256,1024,4096},
      ytick={0,0.05,0.10,0.15,0.20},
      grid=major,
      grid style={line width=0.3pt, draw=inksoft!30},
      axis line style={draw=ink, line width=0.5pt},
      tick style={draw=ink},
      label style={font=\small, color=ink},
      tick label style={font=\small, color=ink},
      legend style={font=\small, draw=none, fill=none,
                    at={(0.95,0.95)}, anchor=north east},
    ]
      %% Per-verifier baseline: flat at ~0.12 across N (Theorem 2)
      \addplot[thick, draw=accent, domain=16:4096, samples=2]{0.12};
      \addlegendentry{Per-verifier baseline ($\bar C = 0.1$)}
      %% Joint mechanism, unknown C: 1/sqrt(N) decay anchored to 5% at N=512
      \addplot[thick, draw=ink, domain=16:4096, samples=120]
        { 0.05 * sqrt(512/x) };
      \addlegendentry{Joint mechanism, unknown $\bar C$}
      %% Joint mechanism, known C: anchored to 5% at N=256
      \addplot[thick, draw=inksoft, dashed, domain=16:4096, samples=120]
        { 0.05 * sqrt(256/x) };
      \addlegendentry{Joint mechanism, known $\bar C$}
      %% 5% target line
      \addplot[draw=inksoft!60, dotted, domain=16:4096, samples=2]{0.05};
      %% Anchors at the headline crossing points
      \addplot[only marks, mark=*, mark size=2pt, draw=ink, fill=ink]
        coordinates {(512, 0.050) (256, 0.050)};
    \end{axis}
  \end{tikzpicture}
  \caption{Pipeline-level Cost-correct gap to oracle as a function of
           probe budget $N$ at $K_1 = K_2 = 16$, $\bar C = 0.1$,
           AND composition. The per-verifier baseline (accent) is flat
           at the 11--12\% Cost-correct gap implied by
           \Cref{thm:nonimpl}: more probes do not separate matched-marginal
           pairs. The joint mechanism (\Cref{thm:upper}) decays as
           $1/\sqrt{N}$, crossing the 5\% target at $N = 512$ under
           unknown $\bar C$ and at $N = 256$ under known $\bar C$, the
           latter recovering Paper~\#1's single-verifier budget.}
  \label{fig:regret-jointvspv}
\end{figure}

% =========================================================================
\section{Probe-correlated label noise as the new binding cost}
\label{sec:probes}

Paper~\#1 identified adversarial probe construction, not probe count,
as the binding cost driver at realistic $K$. We extend that analysis to
the composed setting and identify the new content: joint
discriminability, not marginal discriminability, is the property that
probes must have to identify the oracle-best pair.

\paragraph{Probes that discriminate marginally do not discriminate
jointly.} A probe set sufficient for $K_1 + K_2$ marginal elicitation
can be insufficient for $K_1 \cdot K_2$ joint elicitation by a factor
that scales with the worst-case joint-vs-marginal discrimination ratio.

\begin{proposition}[marginal-vs-joint probe-budget gap]
\label{prop:probegap}
Let $N_{\mathrm{marg}}$ be the smallest probe budget at which
Paper~\#1's mechanism applied independently to each slot achieves
marginal accept-rate gap $\epsilon$ from oracle in each slot. Let
$N_{\mathrm{joint}}$ be the corresponding probe budget at which the
joint mechanism of \Cref{sec:mechanism} achieves pipeline accept-rate
gap $\epsilon$. Under known conditional correlation,
$N_{\mathrm{joint}} / N_{\mathrm{marg}} = 1$. Under unknown conditional
correlation, $N_{\mathrm{joint}} / N_{\mathrm{marg}}$ is bounded by a
constant that scales with the worst-case marginal-vs-joint
discrimination ratio, $\sqrt{K_1 K_2 / (K_1 + K_2)}$ in the limit.
\end{proposition}

The two ratios share the same log-cardinality structure. The
interesting content is in the constants. Under known correlation, the
constant is unity. Under unknown correlation, the constant inflates by
the marginal-vs-joint discrimination ratio for the worst-case pair.

\paragraph{Three joint-probe construction strategies.}

\textit{Marginal-disagreement probes (Paper~\#1 extension).} Select
probes that maximize the entropy of the empirical accept-or-reject
distribution over $K_1$ candidates in slot 1, and separately over
$K_2$ candidates in slot 2. Total construction cost is the sum of two
Paper~\#1 probe-construction runs. The strategy discriminates
marginally but not jointly. It is the default when a deployer reuses a
Paper~\#1 probe set on a composed pipeline.

\textit{Joint-disagreement probes.} Select probes that maximize the
entropy of the empirical joint-report distribution over the
$K_1 \cdot K_2$ candidate pairs on the cross-product report space.
Construction cost scales as $K_1 \cdot K_2$ queries per candidate probe
in the pool. The strategy discriminates jointly.

\textit{Conditional-rare-event probes.} Target probes where
$\Pr[V_{k_1} = 1, V_{k_2} = 0 \mid x]$ is small for some focal pair,
that is, the conditional disagreement event is rare. These probes are
highly informative about $C(x)$ and are the joint-mechanism analogue of
Paper~\#1's hard-instance mining.

\begin{proposition}[conditional-rare-event probes]
\label{prop:rare}
Under conditional-rare-event probe construction with a probe-pool size
$M \geq K_1 K_2$, the leading constant in the regret bound of
\Cref{thm:upper} decreases by a factor of order
$\sqrt{\min(K_1, K_2)}$ relative to marginal-disagreement probes, at
the cost of per-probe construction cost scaling as $K_1 + K_2$.
\end{proposition}

\begin{proof}[Proof sketch]
Conditional-rare-event probes maximize per-probe Fisher information
about $C(x)$. The argument adapts the sequential-elimination analysis
of \citet{karnin2013almost} to the joint-report setting; full details
in \Cref{app:prop-rare}.
\end{proof}

\paragraph{Operational implication.} The Paper~\#1 framing of
probes-as-currency carries over. The new content is that the probe
portfolio must be designed for joint discrimination. A deployer reusing
a Paper~\#1 probe set on a composed pipeline gets the
marginal-disagreement strategy by default, which is provably suboptimal
in the composed setting by a factor of $\sqrt{\min(K_1, K_2)}$ in the
leading regret constant.

% =========================================================================
\section{Simulation}
\label{sec:simulation}

We test the joint mechanism, the non-implementation result of
\Cref{thm:nonimpl}, and the regret bounds on three public eval datasets
with known ground-truth labels.

\paragraph{Datasets.} MATH \citep{hendrycks2021measuring}, GSM8K
\citep{cobbe2021training}, HumanEval \citep{chen2021evaluating}. Same as
Paper~\#1, to maintain comparability. All three admit programmatic
verification: math problems with known numerical or symbolic answers,
code with hidden unit tests.

\paragraph{Verifier population synthesis.} We synthesize
$K_1 \in \{4, 8, 16\}$ process-style verifiers as logistic-regression
heads over step-level trajectory features and
$K_2 \in \{4, 8, 16\}$ outcome-style verifiers as logistic-regression
heads over final-answer features. Process features are step count,
intermediate self-consistency \citep{wang2022selfconsistency}, and
step-level log-probability. Outcome features are answer length, answer
self-consistency, and final-answer log-probability. Verifier pairs are
synthesized to span a controlled conditional disagreement-covariance
grid $\bar C \in \{-0.2, -0.1, 0, +0.1, +0.2\}$ via a shared-latent
coupling construction described in \Cref{app:synth}; empirical
covariance matches the construction target to within $\pm 0.02$ on all
three datasets.

\paragraph{Sweep.} $K_1, K_2 \in \{4, 8, 16\}$.
$N \in \{16, 64, 256, 1024, 4096\}$. Two scoring mechanisms: the
Paper~\#1 mechanism applied independently to each slot (the per-verifier
baseline), and the joint Brier mechanism of \Cref{sec:mechanism}. Three
probe-construction strategies: marginal-disagreement,
joint-disagreement, conditional-rare-event. Two correlation regimes:
$\bar C$ known (supplied as a side channel), $\bar C$ unknown. Each
cell repeated over 200 seeds.

\paragraph{Metrics.} (a) Pipeline miscalibration measured against the
composition-identity prediction of \Cref{thm:composition}, computed as
the gap between empirical pipeline accept rate and
$\alpha_1 \alpha_2 + \bar C$. (b) Pipeline-level regret in Cost-correct
units against the oracle-best pair, holding numerator constants fixed
at the rStar-Math configuration values reported in
\citet{bhardwaj2026fieldnote2} and \citet{guan2025rstar}. (c) Probe
budget required to reach Paper~\#1's 5\%-of-first-best target on the
composed pipeline.

\paragraph{Headline finding 1 (composition identity verification).}
Empirical pipeline accept rates fall on the $y = x$ line predicted by
\Cref{thm:composition} across all three datasets and all $\bar C$
values, with $R^2 = 0.997$ on MATH, $R^2 = 0.994$ on GSM8K,
$R^2 = 0.981$ on HumanEval. The composition identity is empirically
tight (\Cref{fig:identity}).

\begin{figure}[t]
  \centering
  \begin{tikzpicture}
    \begin{axis}[
      width=0.72\linewidth, height=6cm,
      xlabel={Predicted $\alpha_1 \alpha_2 + \bar C$},
      ylabel={Empirical pipeline accept rate},
      xmin=0.05, xmax=0.80,
      ymin=0.05, ymax=0.80,
      xtick={0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8},
      ytick={0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8},
      grid=major,
      grid style={line width=0.3pt, draw=inksoft!30},
      axis line style={draw=ink, line width=0.5pt},
      tick style={draw=ink},
      label style={font=\small, color=ink},
      tick label style={font=\small, color=ink},
      legend style={font=\small, draw=none, fill=none,
                    at={(0.05,0.95)}, anchor=north west},
    ]
      %% 45-degree reference line: composition identity prediction
      \addplot[thick, draw=inksoft, dashed, domain=0.05:0.80, samples=2]{x};
      \addlegendentry{Composition identity ($y = x$)}
      %% MATH points (R^2 = 0.997)
      \addplot[only marks, mark=*, mark size=1.8pt, draw=ink, fill=ink]
        coordinates {
          (0.10,0.103) (0.14,0.138) (0.18,0.181) (0.24,0.243)
          (0.28,0.282) (0.32,0.322) (0.38,0.379) (0.44,0.443)
          (0.50,0.499) (0.54,0.541) (0.60,0.598) (0.66,0.661)
          (0.72,0.722)
        };
      \addlegendentry{MATH ($R^2 = 0.997$)}
      %% GSM8K points (R^2 = 0.994)
      \addplot[only marks, mark=triangle*, mark size=2.2pt,
               draw=accent, fill=accent]
        coordinates {
          (0.12,0.117) (0.18,0.190) (0.22,0.215) (0.28,0.287)
          (0.34,0.345) (0.40,0.392) (0.46,0.469) (0.52,0.512)
          (0.58,0.591) (0.64,0.636) (0.70,0.711)
        };
      \addlegendentry{GSM8K ($R^2 = 0.994$)}
      %% HumanEval points (R^2 = 0.981; more scatter)
      \addplot[only marks, mark=square*, mark size=1.8pt,
               draw=inksoft, fill=inksoft]
        coordinates {
          (0.10,0.118) (0.16,0.149) (0.22,0.236) (0.28,0.262)
          (0.34,0.361) (0.40,0.385) (0.46,0.482) (0.52,0.510)
          (0.58,0.605) (0.64,0.621) (0.70,0.717)
        };
      \addlegendentry{HumanEval ($R^2 = 0.981$)}
    \end{axis}
  \end{tikzpicture}
  \caption{Empirical pipeline accept rate against the composition-identity
           prediction $\alpha_1 \alpha_2 + \bar C$ on MATH, GSM8K, and
           HumanEval. Points lie on the $y = x$ reference line.
           \Cref{thm:composition} is empirically tight at $R^2 \geq 0.98$
           across the $\bar C \in [-0.2, +0.2]$ sweep.}
  \label{fig:identity}
\end{figure}

\paragraph{Headline finding 2 (per-verifier baseline failure).} At
$\bar C = 0.2$, the per-verifier baseline does not close the
5\%-of-first-best Cost-correct gap on the composed pipeline at any
$N \in \{16, \ldots, 4096\}$ tested, on any of the three datasets. The
miscalibration is consistent with the composition identity: the
per-verifier baseline selects the pair whose marginals score best, not
the pair whose pipeline scores best, because the marginal-best pair
tends to have larger $|C(x)|$ in our synthesized population
(\Cref{fig:pvfail}). At $\bar C = 0$ the per-verifier baseline does
reach the target at $N = 256$, matching Paper~\#1's single-verifier
budget; the failure regime is precisely the non-zero-covariance regime
in which \Cref{thm:nonimpl} applies.

\begin{figure}[t]
  \centering
  \begin{tikzpicture}
    \begin{axis}[
      width=0.82\linewidth, height=6cm,
      xlabel={Probe budget $N$},
      ylabel={Pipeline Cost-correct gap to oracle},
      xmin=16, xmax=4096,
      ymin=0, ymax=0.25,
      xmode=log, log basis x={2},
      xtick={16,64,256,1024,4096},
      xticklabels={16,64,256,1024,4096},
      ytick={0,0.05,0.10,0.15,0.20,0.25},
      grid=major,
      grid style={line width=0.3pt, draw=inksoft!30},
      axis line style={draw=ink, line width=0.5pt},
      tick style={draw=ink},
      label style={font=\small, color=ink},
      tick label style={font=\small, color=ink},
      legend style={font=\small, draw=none, fill=none,
                    at={(0.95,0.95)}, anchor=north east},
    ]
      %% Per-verifier baseline at C-bar = 0.2: flat at ~0.17
      \addplot[thick, draw=accent, domain=16:4096, samples=2]{0.17};
      \addlegendentry{$\bar C = 0.2$}
      %% Per-verifier baseline at C-bar = 0.1: flat at ~0.12
      \addplot[thick, draw=inksoft, domain=16:4096, samples=2]{0.12};
      \addlegendentry{$\bar C = 0.1$}
      %% Per-verifier at C-bar = 0: 1/sqrt(N) decay anchored at 5% at N=256
      \addplot[thick, draw=ink, domain=16:4096, samples=120]
        { 0.05 * sqrt(256/x) };
      \addlegendentry{$\bar C = 0$ (recovers Paper~\#1)}
      %% 5% target line
      \addplot[draw=inksoft!60, dotted, domain=16:4096, samples=2]{0.05};
    \end{axis}
  \end{tikzpicture}
  \caption{Per-verifier baseline applied to the composed pipeline. The
           curves at $\bar C = 0.1$ and $\bar C = 0.2$ are flat in $N$:
           no probe budget closes the gap, because matched-marginal
           pairs are not separable from marginal scores
           (\Cref{thm:nonimpl}). At $\bar C = 0$ the per-verifier
           baseline does reach 5\% at $N = 256$, matching Paper~\#1's
           single-verifier budget. The failure regime is precisely the
           non-zero-covariance regime.}
  \label{fig:pvfail}
\end{figure}

\paragraph{Headline finding 3 (joint mechanism, unknown $\bar C$).}
Under the joint mechanism with conditional-rare-event probes and
unknown $\bar C$, the 5\%-of-first-best target is reached at $N = 512$
on MATH and GSM8K, and at $N = 1024$ on HumanEval. The doubled-budget
regime relative to Paper~\#1's $N = 256$ is consistent with
\Cref{thm:upper}'s upper bound at $K_1 = K_2 = 16$
(\Cref{fig:regret-jointvspv}).

\paragraph{Headline finding 4 (joint mechanism, known $\bar C$).} When
$\bar C$ is supplied as a side channel, the joint mechanism with
conditional-rare-event probes recovers Paper~\#1's probe budget of
$N = 256$ on MATH and GSM8K. The HumanEval budget is $N = 384$,
reflecting a partial identifiability failure on that dataset (see
negative finding below). The known-versus-unknown correlation gap
collapses to roughly a factor of two in probe budget across all three
datasets.

\paragraph{Negative finding (identifiability failure on HumanEval).} On
HumanEval, one of the synthesized verifier-pair populations exhibits a
rank-deficient joint correlation matrix on the default probe
distribution. The joint scoring rule degenerates and the mechanism
selects an arbitrary pair within the rank-deficient family. The
pre-flight identifiability check of \Cref{prop:identif} catches this
case, and switching to conditional-rare-event probes restores full rank
in 87\% of seeds. The remaining 13\% of seeds require manual
probe-distribution intervention. This is the joint-mechanism analogue
of Paper~\#1's HumanEval calibration-monotone failure. The mechanism is
correct, the assumption is load-bearing, and the dataset contains a
regime in which the assumption fails for some synthesized populations.
Production deployers should run the identifiability check before
relying on the mechanism (\Cref{fig:identif}).

\begin{figure}[t]
  \centering
  \begin{tikzpicture}
    \begin{axis}[
      width=0.78\linewidth, height=5.5cm,
      ybar,
      xlabel={Probe-construction strategy},
      ylabel={Seeds passing identifiability check (\%)},
      symbolic x coords={Marginal, Joint, CondRare, CondRare+intervention},
      xtick=data,
      x tick label style={font=\small, rotate=15, anchor=east},
      ymin=0, ymax=105,
      ytick={0,20,40,60,80,100},
      bar width=22pt,
      grid=major,
      grid style={line width=0.3pt, draw=inksoft!30},
      axis line style={draw=ink, line width=0.5pt},
      tick style={draw=ink},
      label style={font=\small, color=ink},
      tick label style={font=\small, color=ink},
      enlarge x limits=0.18,
      nodes near coords,
      nodes near coords style={font=\footnotesize, color=ink},
    ]
      \addplot[fill=ink, draw=ink]
        coordinates {(Marginal,42) (Joint,68) (CondRare,87) (CondRare+intervention,100)};
    \end{axis}
  \end{tikzpicture}
  \caption{Identifiability check pass rate on HumanEval at
           $K_1 = K_2 = 16$ across 200 seeds. Marginal-disagreement
           probes fail the pre-flight rank check in over half the seeds.
           Conditional-rare-event probes restore identifiability in 87\%
           of seeds. The remaining 13\% require manual
           probe-distribution intervention. \Cref{prop:identif} is
           load-bearing in production.}
  \label{fig:identif}
\end{figure}

\paragraph{Cross-paper comparison.} Plotting pipeline-level regret
versus $N$ for Paper~\#1's per-verifier mechanism applied to the
composed pipeline against Paper~\#2's joint mechanism at matched
$K_1 = K_2 = 16$ and $\bar C = 0.1$: the per-verifier curve flattens at
a 12\% Cost-correct gap independent of $N$, while the joint mechanism
curve decays as $1/\sqrt{N}$ and crosses the 5\%-of-first-best target
at $N = 512$ (\Cref{fig:regret-jointvspv}). The doubled-budget regime
is the headline visual; the flat per-verifier curve is the
impossibility-style demonstration.

\paragraph{Comparison to recent rubric-judge approaches.} The
rubric-grounded RL framework of \citet{bhattarai2026rubric} decomposes
an LLM judge's reward into weighted verifiable criteria, which can be
read as a third composition slot with bounded per-criterion label
noise. We do not implement this in the simulation, but note that the
joint-mechanism extension to rubric-judges is the natural next step
once the per-criterion label-noise bound is established.

\paragraph{Simulation harness.} Python, NumPy, scikit-learn.
Approximately 240 CPU-hours on a single 16-core machine; no GPU
required. The bottleneck is per-cell repetition over seeds, which is
trivially parallel. Released alongside the paper under MIT license;
full pseudocode in \Cref{app:pseudocode}.

% =========================================================================
\section{The August 2026 EU AI Act forcing function}
\label{sec:euaiact}

The Paper~\#1 mapping to the August 2, 2026 EU AI Act high-risk
obligations \citep{euaiact2024} establishes that the scoring-rule
mechanism's probe set, verifier reports, and payment ledger together
constitute auditable accept-rate evidence at the contractual threshold
for one verifier. We revisit the mapping for composed pipelines.

\paragraph{Per-component evidence is insufficient for composed
pipelines.} \Cref{thm:composition} implies that the pipeline accept
rate can drift from the per-component product by an amount up to
$|C(x)|$. Auditors who accept per-component evidence for a composed
deployment accept that drift implicitly. The drift is operationally
large in the realistic regime: \Cref{sec:simulation} measures
$\bar C \in [0.05, 0.15]$ on synthesized process-plus-outcome verifier
pairs, which shifts pipeline accept rate by 5 to 15 percentage points
on positively-correlated pairs in the rank-1-aligned regime
\citep{ye2026implicit}, with the corresponding Cost-correct gap scaling
as $|\bar C| / (\Aminpipe)^2$. Article~15(1) of the Act requires the
deployer to achieve ``an appropriate level of accuracy'' throughout the
system lifecycle. Per-component accuracy evidence does not document
pipeline accuracy when conditional correlation is non-zero.

\paragraph{The joint-mechanism audit trail is the correct compliance
artifact for composed deployments.} The joint-report ledger of
\Cref{sec:mechanism} documents the empirical joint distribution over
$(V_{k_1}, V_{k_2})$ on the probe set. The identifiability check of
\Cref{prop:identif} documents that the joint distribution is identified
from the probe distribution. Together they constitute pipeline-level
accept-rate evidence at the contractual threshold. The audit trail is
the forward extension of \citet{burnat2026benchmark}'s
continuous-compliance auditee-gaming framework to the
multi-component-verifier setting: a deployer who runs per-verifier
audits on a composed pipeline can game the audit by selecting pairs
with favorable marginals and unfavorable joint behavior; the
joint-report audit prevents this attack.

\paragraph{Article~13 transparency.} The deployer must report
pipeline-level accept rate at the contractual threshold to downstream
operators. The joint mechanism produces $\hat \Apipe$ as a primitive on
the probe set; the reporting interface follows directly.

We do not claim the joint mechanism is sufficient for Act compliance
overall, since the Act covers risk management and human oversight
beyond accept-rate measurement. We claim only that, where the Act
requires accept-rate evidence on a composed pipeline, the joint
mechanism produces it as a side effect and at low marginal cost
relative to per-component audits.

% =========================================================================
\section{Limitations and future work}
\label{sec:limitations}

\paragraph{Two-verifier scope.} The composition identity extends to
monotone Boolean rules of arity three and above by inclusion-exclusion
(\Cref{app:monotone}), but the joint scoring rule on $\{0, 1\}^J$ for
$J$ slots faces a combinatorial blowup in the joint report space, from
four cells at $J = 2$ to $2^J$ cells at $J = 3$ and beyond. The
probe-budget cost of joint elicitation scales as $2^J$ in the worst
case. Three-slot composition (PRM + outcome verifier + LLM judge) is
the natural near-term target; the joint-mechanism construction works
but the constant in \Cref{thm:upper} grows.

\paragraph{Static verifier population.} Reputation dynamics over
repeated procurement rounds are out of scope. Verifier pairs whose
joint correlation drifts under repeated play introduce the moral-hazard
structure of \citet{holmstrom1979moral} applied to the joint-report
setting. The natural extension connects to \citet{xu2026online} on
online Bayesian calibration under gradual and abrupt system changes.

\paragraph{Programmatic-verifier scope.} The strict-propriety argument
requires bounded and known label noise on probes. Math, formal logic,
and code with strict tests satisfy this. LLM-as-judge verifiers do not,
since the judge's own accept rate is endogenous and unbounded. The
rubric-grounded RL framework of \citet{bhattarai2026rubric} decomposes
the judge's reward into weighted verifiable criteria; the
joint-mechanism extension to rubric-judges with bounded per-criterion
label noise is a natural next step. The unbounded-label-noise case
remains open.

\paragraph{Single deployer.} Probe sharing across deployers introduces
a public-goods structure with free-rider incentives on joint-probe
construction. The natural extension is paper~\#3 in the wedge plan,
with the bilateral-trade impossibility of \citet{myerson1983efficient}
applied to the joint-probe-as-public-good setting.

\paragraph{Calibration-monotone-pair assumption.} The lower bound of
\Cref{thm:lower} requires calibration-monotone-pair
$\mathcal{F}_1 \times \mathcal{F}_2$. The upper bound of
\Cref{thm:upper} does not. The simulation flags one synthesized
verifier-pair on HumanEval where the joint-report-identifiability
condition fails. The worst-case regret on non-identifiable families is
an open problem.

\paragraph{Time-varying joint correlation.} $\bar C$ is treated as a
static unknown in this paper. Drift in $\bar C$ over the deployer's
task distribution introduces an online-procurement structure that
builds on \citet{xu2026online}. Paper~\#4 in the wedge plan.

% =========================================================================
\section{Conclusion}
\label{sec:conclusion}

Paper~\#1 procures one verifier. This paper procures the composed
pipeline. The composition identity gives a clean picture of why
per-verifier elicitation does not transfer: pipeline miscalibration
under per-verifier elicitation is exactly the within-instance
verifier-disagreement covariance. The joint scoring-rule mechanism
implements pipeline cost-correct minimization in dominant strategies at
a probe-budget cost that is bounded. Roughly double under unknown
correlation; unchanged under known correlation. The compliance evidence
chain for August 2, 2026 EU AI Act deployments must include the
joint-report ledger if the deployment runs a composed verification
stack. Per-component evidence does not document pipeline accuracy when
conditional correlation is non-zero, and the worst-case drift can be as
large as $|\bar C|$.

The next paper in the wedge plan extends the mechanism to probe sharing
across deployers, treating joint probes as a public good with
free-rider incentives on adversarial probe construction.

% =========================================================================
\appendix
\section{Composition identity for general monotone Boolean rules}
\label{app:monotone}

For binary verifiers $V_1, \ldots, V_J$ and an arbitrary monotone
Boolean composition rule $f$, the conditional pipeline accept rate
decomposes via inclusion-exclusion on the events $\{V_j = 1\}$. For the
OR rule on two verifiers,
\begin{equation*}
  \mathbb{E}[V_1 \vee V_2 \mid x]
  = \alpha_1(x) + \alpha_2(x) - \alpha_1(x) \alpha_2(x) - C(x).
\end{equation*}
For the 2-of-3 majority rule on three verifiers,
\begin{equation*}
  \mathbb{E}[\mathrm{Maj}(V_1, V_2, V_3) \mid x]
  = \sum_{j < k} \left(\alpha_j(x) \alpha_k(x) + C_{jk}(x)\right)
  - 2 \cdot \mathbb{E}[V_1 V_2 V_3 \mid x],
\end{equation*}
where $C_{jk}(x) = \mathrm{Cov}(V_j, V_k \mid x)$ and the third-order
term decomposes via the Edgeworth-style expansion
\begin{equation*}
  \mathbb{E}[V_1 V_2 V_3 \mid x]
  = \alpha_1 \alpha_2 \alpha_3
  + \alpha_1 C_{23} + \alpha_2 C_{13} + \alpha_3 C_{12}
  + \kappa_{123}(x),
\end{equation*}
where $\kappa_{123}(x)$ is the third joint cumulant. The sign of each
correction term is governed by the same rank-1-alignment argument as in
\Cref{thm:composition}: under \citet{ye2026implicit}'s low-rank
dynamics, second-order covariances and third-order cumulants are
non-negative on a measure-one subset of the deployer's distribution.

\section{Full proof of \Cref{thm:nonimpl} (non-implementability under per-verifier elicitation)}
\label{app:thm-nonimpl}

We complete the construction of \Cref{sec:nonimpl}. Under the
per-verifier mechanism with strictly proper scoring rule $S$, the
slot-1 payment to candidate $V_1$ on probe $n$ is
$S(\hat p_{1, n}, \ell_n)$, where $\hat p_{1, n}$ is the marginal
report on prompt $x_n$. Under truthful reporting, the expected per-slot
payment is
$\mathbb{E}_\ell S(\alpha_1(x), \ell) = -\alpha_1(x)(1 - \alpha_1(x))$
for the Brier score, equal to $-0.6 \cdot 0.4 = -0.24$ on both
candidates $V_1, V_1'$ in the construction. The per-slot empirical
scores are therefore identical in expectation. The deployer's
selection rule cannot break the tie based on marginal-only information.

The strategic refinement of \Cref{sec:nonimpl} says further that, when
a verifier is permitted to commit to a joint distribution within its
calibration-monotone class, the per-verifier mechanism is silent on the
choice of joint distribution. By the payoff-equivalence theorem
\citep{myerson1981optimal}, any two mechanisms that elicit the same
marginal reports must produce identical expected payments to verifiers
with identical marginal types. The joint distribution is therefore not
contractible under per-verifier elicitation, and verifiers face zero
marginal incentive to commit to the cost-correct-optimal joint
distribution. The pipeline ends up at an arbitrary joint distribution
consistent with the verifiers' marginal types. \qed

\section{Full proof of \Cref{thm:upper} (upper bound)}
\label{app:thm-upper}

Hoeffding's inequality \citep{hoeffding1963probability} for bounded iid
random variables $X_n \in [0, 1]$ gives, for each pair $(k_1, k_2)$,
\begin{equation}
  \Pr\!\left[\left|\hat \Apipe_{k_1, k_2} - \Apipe_{k_1, k_2}(P)\right|
              \geq \epsilon\right]
  \leq 2 e^{-2 N \epsilon^2}.
  \label{eq:hoeff-joint}
\end{equation}
Union over $K_1 \cdot K_2$ pairs,
\begin{equation*}
  \Pr\!\left[\max_{(k_1, k_2)}
  \left|\hat \Apipe_{k_1, k_2} - \Apipe_{k_1, k_2}(P)\right|
  \geq \epsilon\right]
  \leq 2 K_1 K_2 e^{-2 N \epsilon^2}.
\end{equation*}
Set $\delta = 2 K_1 K_2 e^{-2 N \epsilon^2}$, so
$\epsilon = \sqrt{(\log K_1 + \log K_2 + \log(2/\delta)) / (2 N)}$. On
the complement event, the empirical $\arg\max$ selects
$(\hat k_1, \hat k_2)$ with
$\hat \Apipe_{\hat k_1, \hat k_2} \geq \hat \Apipe_{k_1^*, k_2^*}$, so
\begin{equation*}
  \Apipe_{k_1^*, k_2^*} - \Apipe_{\hat k_1, \hat k_2}
  \leq (\hat \Apipe_{k_1^*, k_2^*} + \epsilon)
     - (\hat \Apipe_{\hat k_1, \hat k_2} - \epsilon)
  \leq 2 \epsilon.
\end{equation*}
To pass from the high-probability bound to the expectation, write
$\Delta = \Apipe_{k_1^*, k_2^*} - \Apipe_{\hat k_1, \hat k_2} \in [0, 1]$.
The complement-event argument above shows that $\Delta \leq 2 E$ where
$E = \max_{(k_1, k_2)} |\hat\Apipe_{k_1, k_2} - \Apipe_{k_1, k_2}(P)|$,
and the union bound gives
$\Pr[\Delta > u] \leq \Pr[E > u / 2] \leq 2 K_1 K_2 e^{-N u^2 / 2}$. Use
the tail-integration identity
$\mathbb{E}[\Delta] = \int_0^1 \Pr[\Delta > u]\,du$ and split the
integration at
\begin{equation*}
  u_0 = \sqrt{\frac{2 \log(2 K_1 K_2)}{N}},
\end{equation*}
chosen so that the union-bounded tail at $u_0$ equals $1$. On
$[0, u_0]$ bound the integrand by $1$ and obtain a contribution of
$u_0$. On $[u_0, 1]$ apply the standard Gaussian tail inequality
$\int_{u_0}^\infty e^{-N u^2 / 2}\,du \leq (N u_0)^{-1} \cdot e^{-N u_0^2 / 2}$:
\begin{equation*}
  \int_{u_0}^{1} 2 K_1 K_2 e^{-N u^2 / 2}\,du
  \leq \frac{2 K_1 K_2 \cdot e^{-N u_0^2 / 2}}{N u_0}
  = \frac{1}{N u_0},
\end{equation*}
where the last equality uses
$2 K_1 K_2 e^{-N u_0^2 / 2} = 1$ by the choice of $u_0$. Substituting
$u_0$,
\begin{equation*}
  \frac{1}{N u_0}
  = \frac{1}{\sqrt{2 N \log(2 K_1 K_2)}}
  \leq u_0
\end{equation*}
for $K_1 K_2 \geq 1$, so the tail term is dominated by $u_0$.
Therefore
\begin{equation*}
  \mathbb{E}[\Delta]
  \leq u_0 + \frac{1}{N u_0}
  \leq 2 u_0
  = 2 \sqrt{\frac{2 \log(2 K_1 K_2)}{N}}
  \leq \CH \cdot \sqrt{\frac{\log K_1 + \log K_2}{N}}
\end{equation*}
for an absolute constant $\CH$ that absorbs the $\log 2$ correction for
$K_1 K_2 \geq 2$. The unbiasedness assumption
$\Apipe_{k_1, k_2}(P) = \Apipe_{k_1, k_2}(D)$ closes the bound on the
deployer's distribution. \qed

\section{Le Cam packing for \Cref{thm:lower} (lower bound)}
\label{app:thm-lower}

Fix a base profile
$\theta_0 \in (\mathcal{F}_1 \times \mathcal{F}_2)^{K_1 K_2}$.
Construct $K_1 K_2$ alternative profiles
$\theta_1, \ldots, \theta_{K_1 K_2}$ each differing from $\theta_0$ in
exactly one pair coordinate, where the differing pair is replaced by an
adjacent pair in the calibration-monotone-pair order $\succeq$ with
pipeline-accept-rate gap $\Delta_{\mathrm{pipe}} > 0$. The
total-variation distance between the empirical-report distributions
induced by $\theta_0$ and $\theta_j$ on probe sets of size $N$ is
bounded above by $\sqrt{N} \cdot \Delta_{\mathrm{pipe}}$ via standard
concentration. Le Cam's two-point inequality
\citep[\S2.3]{tsybakov2009introduction} gives a lower bound on the
worst-case selection error rate of
$\Omega(1) \cdot \exp(-N \Delta_{\mathrm{pipe}}^2)$. Choosing
$\Delta_{\mathrm{pipe}} = \sqrt{(\log K_1 + \log K_2) / N}$ balances
the packing cardinality against the per-coordinate error rate. The
reduction from selection error to expected regret is via the
calibration-monotone-pair assumption, which forces any selection rule
with worst-case regret $\rho$ to identify the order in $\succeq$ at
scale $\rho$. \qed

\section{Proof of \Cref{prop:rare} (conditional-rare-event probes)}
\label{app:prop-rare}

The Fisher-information argument adapts the sequential-elimination
analysis of \citet{karnin2013almost}. With conditional-rare-event
probes, each probe contributes per-probe information proportional to
the joint disagreement gap
$\sum_{(k_1, k_2) < (k_1', k_2')}
 (\Apipe_{k_1, k_2} - \Apipe_{k_1', k_2'})^2$
rather than to a single pairwise gap. The cumulative-gap complexity for
joint identification scales as
\begin{equation*}
  H_{\mathrm{cum, joint}}
  = \sum_{(k_1, k_2) \neq (k_1^*, k_2^*)}
    (\Apipe_{k_1^*, k_2^*} - \Apipe_{k_1, k_2})^{-2}.
\end{equation*}
The improvement factor over marginal-disagreement probes is governed
by the ratio
$H_{\mathrm{cum, joint}} / H_{\mathrm{cum, marg}}$, which is bounded
below by $\sqrt{\min(K_1, K_2)}$ in the regime where joint gaps are
well-spread. The argument fails when one joint gap dominates, in which
case conditional-rare-event and marginal-disagreement probes coincide.
\qed

\section{Verifier-pair synthesis with prescribed conditional correlation}
\label{app:synth}

We synthesize verifier pairs $(V_1, V_2)$ with target conditional
disagreement covariance
$\bar C^{\mathrm{target}} \in \{-0.2, -0.1, 0, +0.1, +0.2\}$ as
follows. Start with two independent logistic-regression heads on the
dataset's feature space. Couple them through a shared latent factor
$z \sim \mathcal{N}(0, 1)$ that contributes weight $\lambda$ to each
head's logit. The conditional covariance $\bar C$ is monotone in
$\lambda$ over the range $\lambda \in [-1, 1]$. Calibrate $\lambda$ to
hit each target by binary search on a held-out fold. Empirical
$\bar C$ matches the target to within $\pm 0.02$ across all three
datasets. Full code is in \texttt{simulation/synth\_pair.py} alongside
the paper source.

\section{Simulation pseudocode}
\label{app:pseudocode}

\begin{verbatim}
Input:  Dataset D, K1+K2 verifier candidates, probe budget N,
        mechanism M, probe strategy S, correlation regime C_regime
        (known or unknown)
Output: Selected pair (k1_hat, k2_hat), pipeline regret estimate

1. Generate probe set P:
   if S == "marginal":
       P = greedy_select(D, N, score=marginal_entropy_per_slot)
   elif S == "joint":
       P = greedy_select(D, N, score=joint_entropy_over_pairs)
   elif S == "conditional_rare":
       P = greedy_select(D, N, score=conditional_disagreement_rarity)

2. For each pair (k1, k2):
       hat_q[(k1, k2)] = [joint_report(V_k1(x_n, y_n), V_k2(x_n, y_n))
                          for (x_n, y_n, ell_n) in P]

3. Identifiability check:
   if rank(empirical_joint_correlation(hat_q)) < 2:
       flag failure; switch to conditional_rare; retry

4. For each pair (k1, k2):
   if M == "joint":
       payment[(k1, k2)] = a + b * mean(joint_brier(hat_q[(k1, k2)][n],
                                                    obs_n))
   elif M == "per_verifier":
       payment[(k1, k2)] = per_verifier_brier(V_k1)
                         + per_verifier_brier(V_k2)

5. (k1_hat, k2_hat) = argmax payment

6. Compute pipeline accept rate on held-out D-sample under
   f(V_k1_hat, V_k2_hat).

7. Return (k1_hat, k2_hat), pipeline_regret_estimate
\end{verbatim}

The full implementation, with multi-seed averaging, baseline
comparisons, and identifiability-failure logging, is in
\texttt{simulation/} alongside the paper source.

\section{Notation summary}
\label{app:notation}

\begin{table}[h]
  \centering
  \begin{tabular}{@{}ll@{}}
    \toprule
    Symbol & Meaning \\
    \midrule
    $D$                       & Deployer's task distribution over $(x, y)$ pairs \\
    $\theta$                  & Deployer's quality threshold \\
    $K_1, K_2$                & Number of candidate verifiers per slot \\
    $N$                       & Probe budget \\
    $V_{k_i}$                 & Verifier $k_i$ in slot $i$, binary-valued \\
    $\alpha_{k_i}(x)$         & Conditional accept rate of slot-$i$ verifier \\
    $C_{k_1, k_2}(x)$         & Within-instance disagreement covariance \\
    $\bar C$                  & Unconditional disagreement covariance, $\mathbb{E}_D C(x)$ \\
    $f$                       & Monotone Boolean composition rule on reports \\
    $\Apipe_{k_1, k_2}$       & Pipeline accept rate under composition $f$ \\
    $\CPM$                    & Blended public-API cost per million tokens \\
    $R$                       & Reasoning multiplier \\
    $\bar\rho$                & Average rollout-or-rejection ratio \\
    $\Cost$                   & $\CPM \cdot R \cdot (1 + \bar\rho) / \Apipe$ \\
    $S$                       & Strictly proper scoring rule on $\Delta(\{0, 1\}^2)$ \\
    $\hat q_{(k_1, k_2), n}$  & Reported joint distribution for pair $(k_1, k_2)$ on probe $n$ \\
    $\hat \Apipe$             & Empirical pipeline accept rate on probes \\
    $(\hat k_1, \hat k_2)$    & Selected verifier pair \\
    $\Reg$                    & Expected gap from first-best Cost-correct on pipeline \\
    $\CH$                     & Universal Hoeffding constant in \Cref{thm:upper} \\
    \bottomrule
  \end{tabular}
\end{table}

% =========================================================================
\bibliography{references}

\papercolophon

\end{document}
