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\title{\bf The Validator's Dividend: Measuring the Cost--Quality Frontier of\\
Validation-Gated Resampling versus k-Sample Consensus for Agentic\\
Tool-Call Output on Self-Hosted H200}
\author{ThakiCloud AI Research (Hyojung Han)}
\date{}

\begin{document}
\maketitle

\begin{abstract}
Unattended agent loops on quantized checkpoints must trust every tool call. We price one
percentage point of tool-call accuracy in sample cost when a free deterministic validator
(schema plus AST-level check) is on the path. We decompose each draw into correct (C),
detectable-wrong (D), and silent-wrong (S) outcomes (detectable fraction $\delta$) and compare
four sampling policies against a model-tier lever. The gate captures a dividend
$\Delta = q\,p_d/(1-p_d)$, bounded by $\delta p$, at marginal cost $c/q$ per quality point, and
cannot cross the silent floor $1 - p_s/(1-p_d)$ under any retry budget. We give the
gate-versus-majority crossover $p_d^{*}(q) = 1 - 1/(q(3-2q))$, show the cheap-judge arm is
dominated below a conditional-accuracy bound $\beta^{*}$, and place model upgrades via the tier
crossover $(r-1)\Delta Q_{\mathrm{resample}} > 2(q' - q)$. A pre-registered five-arm BFCL
protocol on an NVFP4 27B teacher on H200 is specified for follow-up. This is an analytical
study; no new measurements are reported.
\end{abstract}




\section{Introduction}

Consider an unattended agent loop on a quantized, self-hosted H200 checkpoint. Each planning
instance requires a \emph{tool-call plan}: an ordered list of tool calls with named arguments,
serialized as JSON or an AST. The only deployment correctness signal is a \emph{free
deterministic validator} $V$ --- a schema/grammar plus AST-level structural check, as in the
Berkeley Function-Calling Leaderboard (BFCL) harness~\citep{bfcl2024leaderboard} --- at
negligible cost. Our question is narrowly quantitative: \emph{what does one percentage point of
tool-call accuracy cost, in self-hosted sample cost, when a free deterministic validator is
already on the path?} Four sampling levers compete for that budget, and a fifth is exogenous:
(i)~validate-then-retry (redraw on $V$-rejection, abstain if all fail); (ii)~whole-plan majority
over $k$; (iii)~per-argument consensus; (iv)~cheap-judge selection of the $k$; (v)~model-tier
upgrade ($r>1$ per sample).

This is an analytical study (not an empirical run) and the execution-side upgrade of our
ensemble-voting routing work~\citep{thaki2026ensamblerouter}: no new measurements are reported
(protocol pre-registered in Section~8); the cost-quality question moves from \emph{routing} to
structured tool-call output, where a free validator and a self-hosted quantized model sit.

Our contributions are threefold.
\begin{itemize}
\item \textbf{System.} We price each point of tool-call accuracy --- validate-then-retry, $k$-sample consensus, model-tier upgrade --- with their crossovers, yielding a deployable policy.
\item \textbf{Society.} A free deterministic validator with the right policy lets a small org reach large-model tool-call reliability on cheap self-hosted inference.
\item \textbf{Science.} We lift self-consistency from free-form answers to structured actions and fix the quality-per-dollar frontier over sample count and the resampling-vs-tier-upgrade crossover.
\end{itemize}

Section~2 surveys related work; Section~3 fixes the setup; Sections~4--6 give the gate dividend, $k$-sample consensus, and the cost-quality frontier with its tier crossover; Section~7 the judge arm; Section~8 the policy and protocol; Sections~9--10 close.\section{Background and Related Work}

\subsection{Self-consistency and multi-sample aggregation}

Self-consistency samples $k$ chain-of-thought trajectories and returns the majority answer,
trading inference compute for accuracy~\citep{arxiv220311171}. Its limits are probed: wider
sampling shows diminishing returns~\citep{arxiv260805643}, hybrid CoT+PoT ensembling matches it
from two samples~\citep{arxiv260417433}, and another line grounds consensus in
execution~\citep{arxiv260508680}. \textbf{Gap.} All of this is measured on free-form answers;
structured agentic actions --- tool-call plans with named arguments --- under a free
deterministic validator remain uncharted.

\subsection{Tool-calling evaluation and reliability}

BFCL provides a deterministic AST/graded harness for tool-call
correctness~\citep{bfcl2024leaderboard}; recent work contrasts programmatic against native JSON
tool calling across 14 LLMs on BFCL v4~\citep{arxiv260806370}. Agents are hard to reproduce:
multi-step pipelines repeat tools, order, and arguments
inconsistently~\citep{arxiv260528840}, and results depend on undocumented seed and
history~\citep{arxiv260600135}. Grammar-constrained decoding and runtime control with
rollback~\citep{arxiv260327905} raise first-attempt success. \textbf{Gap.} Evaluation and
reliability measurement exist; the sampling-policy economics \emph{on top of} such an
evaluator --- which resampling rule buys how much accuracy per dollar --- do not.

\subsection{Low-bit serving for agentic workloads}

Low-bit quantization dominates self-hosted serving, with phase-aware techniques targeting
agentic workloads. Our prior work localizes the quality cost on self-hosted H200: the tool-call
cliff (low-bit accuracy decay)~\citep{thaki2026toolcallcliff}, the reasoning verbosity tax
(explicit-versus-compact frontiers)~\citep{thaki2026verbositytax}, and the prefix-reuse economics
of manifest and schema ordering~\citep{thaki2026manifestorder}; together they show low-bit
agentic serving carries a quality tax this paper buys back at inference time.

\subsection{Cost-quality routing, cascades, and judges}

A mature line routes queries across model tiers under quality constraints via closed-loop,
difficulty-aware routing and conformal cascading~\citep{arxiv260423577}. Closest to us,
UniScale finds model routing (coarse) and test-time scaling (fine) decoupled dimensions of one
quality-cost trade-off, optimized jointly online~\citep{arxiv260530898}; carbon-aware routing
adds energy~\citep{arxiv260913559}. For LLM-as-a-judge best-of-$n$ candidates, a judge with
moderate correlation ($\rho{=}0.47$) captures only 21.0\% of the gain over random on a
5{,}000-prompt best-of-4 benchmark~\citep{arxiv260312520}; on a controlled deterministic testbed
no judge-score policy beat the deterministic baseline~\citep{arxiv260713347}. Small models need
strong verifiers to self-correct~\citep{arxiv240417140}: without a deterministic validator, a
cheap judge is the only fallback, the regime our earlier routing work
addressed~\citep{thaki2026ensamblerouter}.

\subsection{Our prior work}

This paper extends a small family of self-hosted H200 studies. \citet{thaki2026ensamblerouter}
built a confidence-weighted, heterogeneous-tier ensemble for skill \emph{routing} with a
break-even inequality and a 50\% per-voter accuracy floor; we move the question from
routing-side voters to execution-side sampling of tool-call plans. \citet{thaki2026toolcallcliff}
measured the tool-call cliff (its tax $\tau_q$, NVFP4 at 89.6\% on 800 cases); we convert that
tax into a resampling dividend: validate-then-retry recovers the detectable fraction, up to
$q/(1-p_d)\le 1$. \citet{thaki2026verbositytax} derived per-turn cost-quality frontiers for
explicit versus compact chain-of-thought; we make the sample count $k$ a first-class budget.
\citet{thaki2026manifestorder} gave closed-form prefix-cache economics for manifest and schema
ordering; we show the sampling levers compose the same way.\section{Problem Setup and Notation}
\label{sec:setup}

\paragraph{Plans, validator, and grader.}
A task is one tool-call planning instance in a BFCL category (simple, multiple, parallel)~\citep{bfcl2024leaderboard}. One draw is one \emph{plan}: an ordered list of tool calls with named arguments (JSON or AST). Pipeline: draw at the self-hosted checkpoint; run the free deterministic validator $V$ (schema/grammar plus AST-level checks); accept the first passing plan, redrawing up to a budget. Offline, the deterministic AST grader $G$~\citep{bfcl2024leaderboard} scores full correctness, including argument values; $G$ is an evaluation instrument, not a deployment signal.

\begin{definition}[Outcome classes]
For one draw on a fixed task, let $C$, $D$, $S$ be the disjoint outcome classes --- \emph{correct} ($G$ passes), \emph{detectable wrong} ($V$ fails, $G$ fails), \emph{silent wrong} ($V$ passes, $G$ fails) --- with probabilities $q$, $p_d$, $p_s$, so $q + p_d + p_s = 1$; the total error mass is $p = p_d + p_s$, and the detectable fraction $\delta = p_d/p$ ($p>0$). The limiting case $\delta = 0$ is the \emph{validator blind spot}: all error mass is silent, undetectable by any gate based on $V$.
\end{definition}

\paragraph{Cost model.}
One sample costs $c$ (sample-cost units) at fixed prompt and task category, the same per-sample accounting as~\citep{thaki2026manifestorder}; the validator costs $v \approx 0$, negligible against $c$. A tier upgrade $M_2$ has accuracy $q'$ at cost $rc$ per sample ($r>1$); in the quantized regime it is the same model at a higher-precision tier, $r$ set by throughput, $q' - q$ by the accuracy tax. A cheap judge costs $j$ per selection ($j \approx 0.2c$ for a small local judge).

\paragraph{Assumptions.}
Three assumptions, used throughout:

\noindent
\fbox{\begin{minipage}{0.94\linewidth}
\begin{assumption}[A1 --- i.i.d.\ draws]
At temperature $T>0$, plans on a fixed task are i.i.d.\ draws from the model. Measured non-reproducibility of multi-step tool-calling pipelines --- same tools, same order, same arguments across repeated identical invocations, far less consistently than assumed~\citep{arxiv260528840} --- anchors this idealization; its violation is flagged in Section~9.
\end{assumption}
\begin{assumption}[A2 --- canonical plan]
Each case has a single canonical correct plan, and whole-plan majority is exact-match; with multiple valid plans, majority quality can only rise, so the binomial models below are conservative lower bounds on achievable quality.
\end{assumption}
\begin{assumption}[A3 --- deterministic gate]
The verdict of $V$ is class-deterministic: it fails on every $D$ draw and passes on every $C$, $S$ draw --- no false passes, no false rejections. An imperfect $V$ is a perturbation (Remark~\ref{rem:imperfectV}); token-level runtime control that enforces structure during decoding~\citep{arxiv260327905} reduces the false-pass rate.
\end{assumption}
\end{minipage}}
\section{The Validator's Dividend}
\label{sec:dividend}

\paragraph{Baseline (Arm 0).}
One draw, no gate: $Q_0 = q$, $C_0 = c$. Figure~\ref{fig:policy} depicts the policy family studied in this and the following sections.

\begin{theorem}[Gate dividend]\label{thm:gate}
Let $n_{\max} \ge 1$ be the resampling budget. Validate-then-retry (VTR) draws a plan; if $V$ fails it redraws, accepting the first passing plan, and if all $n_{\max}$ draws fail it abstains, counted as an error. Under A1--A3:
\begin{enumerate}[label=(\alph*)]
\item \emph{Acceptance.} The per-draw acceptance probability is $a = q + p_s = 1 - p_d$.
\item \emph{Gate dividend.} The accuracy of the returned plan is
\[
Q_{\mathrm{VTR}}(n_{\max}) \;=\; \frac{q}{a}\,\bigl(1 - (1-a)^{n_{\max}}\bigr),
\]
and the uncapped limit ($n_{\max}\to\infty$) is
\[
Q^{*} \;=\; \frac{q}{1-p_d} \;=\; q + \Delta, \qquad \Delta \;=\; \frac{q\,p_d}{1-p_d}.
\]
The whole dividend is realized on the first accepted draw: by A1 the accepted draw has conditional quality $q/a$, so $Q_{\mathrm{VTR}}(n_{\max}) = Q^{*}\,\Pr(\text{no abstention})$.
\item \emph{Silent floor.} $Q^{*} = 1 - p_s/(1-p_d)$: the residual error at the floor is entirely silent (the argument-value class), since the only wrong class $V$ can accept is $S$. The finite-budget error contains the detectable abstention tail $(1-a)^{n_{\max}}$, so $Q_{\mathrm{VTR}}(n_{\max}) < Q^{*}$ for every finite $n_{\max}$.
\item \emph{Dividend bound.} $\Delta \le \delta\, p$. In particular, $\delta \to 0$ (validator blind spot) forces $\Delta \to 0$.
\item \emph{Expected draws.} $\mathbb{E}[n] = \dfrac{1-(1-a)^{n_{\max}}}{a}$; for $n_{\max} = 2$, $\mathbb{E}[n] = 1 + p_d$ exactly.
\end{enumerate}
\end{theorem}

\begin{proof}
Part (a) follows from A3: $V$ passes exactly $C \cup S$, of total mass $a = q + p_s = 1 - p_d$.
For (b), by A1 the accepted draw is, on every attempt, drawn from the class distribution conditioned on $V$-pass,
\[
\Pr\bigl(\text{accepted draw} \in C \;\big|\; V \text{ passes}\bigr)
\;=\; \frac{q}{q + p_s} \;=\; \frac{q}{a};
\]
at least one of the $n_{\max}$ draws is accepted with probability $1-(1-a)^{n_{\max}}$, so $Q_{\mathrm{VTR}}(n_{\max}) = \frac{q}{a}\bigl(1-(1-a)^{n_{\max}}\bigr)$, and $n_{\max}\to\infty$ gives $Q^{*} = q/(1-p_d) = q + \Delta$.
For (c), $1 - Q^{*} = p_s/(1-p_d)$, and the only wrong class ever accepted is $S$, so the floor's residual is entirely silent; for finite $n_{\max}$,
\[
1 - Q_{\mathrm{VTR}}(n_{\max})
\;=\; \frac{p_s}{a}\,\bigl(1-(1-a)^{n_{\max}}\bigr) + (1-a)^{n_{\max}},
\]
the second term being the abstention tail (all draws in $D$ by A3, hence detectable), strictly positive for $a<1$ and vanishing as $n_{\max}\to\infty$, so $Q_{\mathrm{VTR}}(n_{\max}) < Q^{*}$ for every finite budget.
For (d), $q \le 1-p_d$ gives $\Delta = \frac{q\,p_d}{1-p_d} \le p_d = \delta\, p$, and $\delta \to 0$ forces $\Delta \to 0$.
For (e), $\Pr(n \ge i) = (1-a)^{i-1}$ for $1 \le i \le n_{\max}$, so $\mathbb{E}[n] = \sum_{i=1}^{n_{\max}} (1-a)^{i-1} = \frac{1-(1-a)^{n_{\max}}}{a}$, and for $n_{\max} = 2$ this is $(1-(1-a)^2)/a = 2 - a = 1 + p_d$ exactly.
\end{proof}

\begin{lemma}[Marginal cost per quality point]\label{lem:mcd}
Define the \emph{marginal cost per quality point} (MCD) of an arm at the baseline $(Q_0, C_0)$ as $\mathrm{MCD} = (C - C_0)/(Q - Q_0)$, in sample-cost units per full quality unit. For validate-then-retry,
\[
\mathrm{MCD}_{\mathrm{VTR}}
\;=\; \frac{(\mathbb{E}[n]-1)\,c}{\Delta}
\;=\; \frac{\dfrac{p_d}{1-p_d}\,c}{\dfrac{q\,p_d}{1-p_d}}
\;=\; \frac{c}{q},
\]
where the middle ratio uses the uncapped quantities $\mathbb{E}[n]-1 = p_d/(1-p_d)$ and the $\Delta$ of Theorem~\ref{thm:gate}.
\end{lemma}

\begin{proof}
For $n_{\max}=2$, Theorem~\ref{thm:gate}(e) gives $\mathbb{E}[n]-1 = p_d$ and Theorem~\ref{thm:gate}(b) gives $Q_{\mathrm{VTR}}(2) - q = q\,p_d$, so $\mathrm{MCD} = p_d\,c/(q\,p_d) = c/q$; in the uncapped limit, $\mathbb{E}[n]-1 = 1/a - 1 = p_d/(1-p_d)$, giving the displayed chain. In general,
\[
Q_{\mathrm{VTR}}(n_{\max}) - Q_0
\;=\; \frac{q}{a}\bigl(1-(1-a)^{n_{\max}}\bigr) - q
\;=\; q\,\frac{1-(1-a)^{n_{\max}} - a}{a}
\;=\; q\,(\mathbb{E}[n]-1),
\]
so $\mathrm{MCD}_{\mathrm{VTR}} = c/q$ for every $n_{\max}$.
\end{proof}

\begin{remark}[Average quality per dollar is unchanged; the dividend is marginal]\label{rem:mcdread}
The gate renormalizes quality and expected cost by the same factor $1/(1-p_d)$ (uncapped: $Q^{*} = q/(1-p_d)$ at $\mathbb{E}[n] = 1/(1-p_d)$), so average quality per sample-cost stays $q/c$; in that coarse sense it is not a quality-per-dollar improvement, but its value is twofold. First, the \emph{marginal} price $\mathrm{MCD}_{\mathrm{VTR}} = c/q$ is independent of $p_d$ and $n_{\max}$ --- the quantity competing with the consensus and tier levers of Sections~5--6. Second, latency and energy: with probability $a = 1-p_d$ the loop finishes in one sample, so the extra expected sample cost is only $p_d$ (for $n_{\max}=2$; uncapped, $p_d/(1-p_d)$) --- GPU-hours and watts at fixed quality, the same axis as carbon-aware routing~\citep{arxiv260913559} and joint routing/test-time scaling~\citep{arxiv260530898}.
\end{remark}

\begin{remark}[Imperfect validator]\label{rem:imperfectV}
Suppose $V$ falsely passes a fraction $\epsilon$ of $D$ draws, so $p_d' = p_d(1-\epsilon)$, $p_s' = p_s + \epsilon\,p_d$, and $q$ is unchanged. Every quantity above then holds with $(p_d, p_s)$ replaced by $(p_d', p_s')$: the dividend becomes $\Delta' = q\,p_d'/(1-p_d')$, the silent floor $1 - p_s'/(1-p_d')$, the bound $\Delta' \le p_d' = \delta'\,p'$ of Theorem~\ref{thm:gate}(d) applies to the adjusted quantities, and the MCD stays $c/q$. Token-level runtime control with drift detection and rollback~\citep{arxiv260327905} is the deployment mechanism driving $\epsilon$ down.
\end{remark}

\paragraph{Numerical anchor.}
(Analytical; not measured.) Take $q = 0.9$, $p_d = 0.07$, $p_s = 0.03$, so $p = 0.10$ and $\delta = 0.7$: $\Delta = q\,p_d/(1-p_d) = 0.9 \times 0.07/0.93 \approx 6.77$ percentage points (pp), $Q^{*} \approx 96.8\%$, $\mathbb{E}[n] = 1 + p_d = 1.07$ draws for $n_{\max} = 2$, and $\mathrm{MCD}_{\mathrm{VTR}} = c/q \approx 1.11$ sample-costs per full quality unit ($0.0111\,c$ per pp). The bound of Theorem~\ref{thm:gate}(d) reads $\Delta = 6.77\,\mathrm{pp} \le \delta\,p = 7.0\,\mathrm{pp}$.
\begin{figure}[t]
\centering
\begin{tikzpicture}[scale=0.75, 
  box/.style={draw, rounded corners=1.5pt, minimum height=0.8cm, align=center, font=\small},
  acc/.style={draw, fill=blue!6, rounded corners=1.5pt, minimum height=0.8cm, align=center, font=\small},
  diam/.style={draw, diamond, aspect=0.55, inner sep=1pt, align=center, font=\small},
  arr/.style={-{Stealth[length=2.2mm, width=1.6mm]}, thick},
  dasharr/.style={dashed, -{Stealth[length=2.2mm, width=1.6mm]}, gray}
]
\node[box] (draw) at (0,0) {draw plan\\(cost $c$)};
\node[diam] (V) at (4.2,0) {validator $V$ (free)};
\node[box] (redo) at (4.2,2.4) {redraw\\(up to $n_{\max}$)};
\node[acc, minimum width=3.2cm] (pool) at (8.2,0) {accept / forward to\\consensus pool};
\node[acc, minimum width=3.2cm] (fuse) at (8.2,-2.6) {fusion of $k$ gated draws\\(MAJ-$k$ / ARG-$k$)};
\node[acc] (out) at (8.2,-4.8) {accept plan};
\node[box] (tier) at (0,-2.6) {tier upgrade $M_2$\\(cost $rc$, accuracy $q'$)};
\node[box, dashed, gray] (judge) at (12.7,-2.6) {cheap judge\\(dominated, Thm.~\ref{thm:judge})};
\draw[arr] (draw.east) -- (V.west);
\draw[arr] (V.north) -- node[right, pos=0.5]{fail} (redo.south);
\draw[arr] (redo.west) -- node[above]{next draw} (draw.north);
\draw[arr] (V.east) -- node[above]{pass} (pool.west);
\draw[arr] (pool.south) -- node[right]{pool of $k$} (fuse.north);
\draw[arr] (fuse.south) -- (out.north);
\draw[arr] (tier.north) -- node[left]{swap checkpoint} (draw.south);
\draw[dasharr] (judge.west) -- node[above, font=\footnotesize]{would select} (fuse.east);
\draw[gray, line width=0.6pt] ([xshift=-1.35cm, yshift=-0.5cm]judge.center) -- ([xshift=1.35cm, yshift=0.5cm]judge.center);
\end{tikzpicture}
\caption{(Conceptual diagram of the policy family; no measurement involved.)}
\label{fig:policy}
\end{figure}

\section{$k$-Sample Consensus for Structured Output}\label{sec:consensus}

The second execution-side lever spends extra samples on consensus: whole-plan exact-match majority (Theorem~\ref{thm:maj}), the structured-action analogue of self-consistency~\citep{arxiv220311171}; gate--consensus composition (Proposition~\ref{prop:compose}); the quality-side crossover (Theorem~\ref{thm:crossover}); and the per-argument refinement whole-plan majority loses (Proposition~\ref{prop:argk}).

\begin{theorem}[Whole-plan majority]\label{thm:maj}
Fix an odd $k \ge 3$ and draw $k$ plans i.i.d.\ (A1); let $Q_{\mathrm{MAJ}}(k)$ be the probability that the exact-match majority of the $k$ plans --- the plan with more than $k/2$ votes, if one exists --- passes $G$; on a tie the arm abstains, counted as an error. Under A2,
\begin{equation}\label{eq:maj}
Q_{\mathrm{MAJ}}(k) \;=\; \sum_{j=(k+1)/2}^{k} \binom{k}{j}\, q^{j}(1-q)^{k-j},
\end{equation}
and the cost is exactly $kc$ (all draws are made). In particular, $Q_{\mathrm{MAJ}}(3) = q^{2}(3-2q)$. For $q \in (1/2,1)$ the marginal dividend $d_{k} := Q_{\mathrm{MAJ}}(k+2) - Q_{\mathrm{MAJ}}(k)$ is strictly positive and strictly decreasing in $k$ over odd $k$; for $q \le 1/2$ it is non-positive (the vote-dilution regime of Theorem~\ref{thm:crossover}).
\end{theorem}

\begin{proof}
Under A2 each draw passes $G$ independently with probability $q$, so the majority plan passes iff at least $(k{+}1)/2$ draws are correct, giving~\eqref{eq:maj}; writing $k = 2m{-}1$, the dividend is the tipping band's binomial mass around the majority threshold, $d_{2m-1} = \tfrac{1}{2}\binom{2m}{m}\bigl(q(1-q)\bigr)^{m}(2q-1)$, strictly positive for $q > 1/2$, non-positive for $q \le 1/2$, and strictly decreasing in odd $k$ since the $m$-ratio $\frac{2(2m+1)}{m+1}\,q(1-q) < 1$ for $q \in (1/2,1)$, as $q(1-q) < 1/4$.
\end{proof}

\begin{remark}[Relaxing A2]\label{rem:a2}
When a case admits several semantically correct plans, exact-match counting splits the correct mass across textually distinct variants: $Q_{\mathrm{MAJ}}(k)$ is a conservative lower bound --- multiple valid plans can only raise, never lower, majority quality.
\end{remark}

\begin{proposition}[Composition of the gate and consensus]\label{prop:compose}
Run MAJ-$k$ over $k$ \emph{gated} draws (each first through the validate-then-retry gate). Each gated draw has quality $q' = q/(1-p_d)$ at expected cost $c/(1-p_d)$ (Theorem~\ref{thm:gate}), so the ensemble quality is the majority binomial~\eqref{eq:maj} evaluated at $q'$, at expected cost $kc/(1-p_d)$: the levers compose multiplicatively, the gate renormalizing $(q,c) \to (q',c')$ so every consensus MCD at $(q',c')$ applies unchanged.
\end{proposition}

\paragraph{Analytical example (derived from the closed-form model; not measured).}
At $q = 0.9$, $p_d = 0.07$, gate$+$MAJ-3 reaches $99.7\%$ at $3.23$ expected samples --- three gated draws at $1/(1-p_d)$ expected samples each, uncapped budget (Proposition~\ref{prop:compose}) --- versus MAJ-3 alone at $97.2\%$ for $3$ samples and the tier arm ($r = 2$, accuracy $0.95$) at $95.0\%$ for $2$ samples; the composition is multiplicative, as the manifest-ordering lever composes with the model-tier and quantization levers~\citep{thaki2026manifestorder}.

\begin{theorem}[Gate--majority crossover, quality side]\label{thm:crossover}
Fix $q$ and let $p_d$ range over $[0, 1-q]$ with $p_s = 1 - q - p_d$. Then
\[
 Q^{*} \;>\; Q_{\mathrm{MAJ}}(3) \quad\Longleftrightarrow\quad p_d \;>\; p_d^{*}(q), \qquad p_d^{*}(q) \;:=\; 1 - \frac{1}{q(3-2q)},
\]
and for $q \in (0.5, 1)$ the threshold is interior: $0 < p_d^{*}(q) < 1 - q$. For $q \le 0.5$, $p_d^{*}(q) \le 0$ and $Q_{\mathrm{MAJ}}(3) \le q$: exact-match majority at a per-draw accuracy of 50\% or less cannot beat the single draw, matching the routing side's 50\% floor~\citep{thaki2026ensamblerouter}.
\end{theorem}

\begin{proof}
Substituting $Q^{*} = q/(1-p_d)$ and $Q_{\mathrm{MAJ}}(3) = q^{2}(3-2q)$, dividing by $q > 0$, and inverting the strictly increasing map $p_d \mapsto (1-p_d)^{-1}$ gives $p_d > p_d^{*}(q) = 1 - 1/(q(3-2q))$. With $h(q) := q(3-2q)$, $h(q) > 1$ on $(0.5,1)$ ($h(0.5) = h(1) = 1$, maximum $9/8$ at $q = 3/4$) together with $q^{2}(3-2q) < 1$ on $(0,1)$ gives $0 < p_d^{*}(q) < 1 - q$; for $q \le 0.5$, $h(q) \le 1$ gives $p_d^{*}(q) \le 0$ and is likewise the condition $Q_{\mathrm{MAJ}}(3) \le q$.
\end{proof}

\paragraph{Analytical example (derived from the closed-form model; not measured).}
At $q = 0.9$, $Q_{\mathrm{MAJ}}(3) = 97.2\%$ and $Q_{\mathrm{MAJ}}(5) = 99.1\%$; the first consensus step $d_{1\to 3} = 7.2\,\mathrm{pp}$ costs $2c$ ($\mathit{MCD} = 27.8\,c$) and the second, $d_{3\to 5} = 1.9\,\mathrm{pp}$, the same $2c$ ($\mathit{MCD} = 103\,c$) --- the steepening returns echo the self-consistency-efficiency line~\citep{arxiv260417433, arxiv260805643}. The crossover threshold is $p_d^{*}(q) = 7.41\,\mathrm{pp}$, and just below it, at $p_d = 7\,\mathrm{pp}$ ($p_s = 3\,\mathrm{pp}$), the majority keeps a $0.4\,\mathrm{pp}$ edge ($97.2\%$ versus $Q^{*} = 96.77\%$) while the gate keeps the MCD edge ($\mathit{MCD}_{\mathrm{VTR}} = c/q = 1.11\,c$ versus $27.8\,c$, Lemma~\ref{lem:mcd}). Gate$+$MAJ-3 reaches $99.7\%$ at $3.23$ expected samples (Proposition~\ref{prop:compose}).

\begin{proposition}[Per-argument consensus (ARG-$k$)]\label{prop:argk}
A plan is a call skeleton plus $m$ named arguments; assume per-argument error $\alpha \in (0,1)$, independent across arguments and draws (A1), skeleton correctness $s \ge 1/2$, silent errors passing $V$. ARG-$k$ ($k$ odd) fuses by skeleton $k$-majority and per-argument strict majority: each fused argument is correct with probability $\sum_{j=(k+1)/2}^{k} \binom{k}{j}\, (1-\alpha)^{j} \alpha^{k-j}$, the fused skeleton with probability $\sum_{j=(k+1)/2}^{k} \binom{k}{j}\, s^{j}(1-s)^{k-j}$, their product by independence for the fully-correct plan. In the single-silent-argument regime the $k$ plans are pairwise distinct so MAJ-$k$ abstains, while each argument is correct in $k-1 \ge (k+1)/2$ draws, so ARG-$k$ returns the true plan at cost $kc$.
\end{proposition}

\section{Cost-Quality Frontier and the Model-Tier Crossover}\label{sec:frontier}

The levers of Sections~4 and~5 compare on a common price axis, the \emph{marginal cost per quality (MCD)} at $(q,c)$: for incremental cost $\Delta C$ and incremental quality $\Delta Q>0$, the MCD is $\Delta C/\Delta Q$, the sample cost per full quality unit. The three levers have MCDs
\begin{equation}\label{eq:mcds}
\begin{aligned}
\mathit{MCD}_{\mathrm{VTR}} &= \frac{c}{q}
\quad\bigl(\text{Lemma~\ref{lem:mcd}}\bigr),\\
\mathit{MCD}_{\mathrm{resample}}(k{\to}k{+}2) &= \frac{2c}{Q_{\mathrm{MAJ}}(k{+}2)-Q_{\mathrm{MAJ}}(k)} = \frac{2c}{d_k}
\quad\bigl(\text{Theorem~\ref{thm:maj}}\bigr),\\
\mathit{MCD}_{\mathrm{tier}} &= \frac{(r-1)c}{q'-q}\,.
\end{aligned}
\end{equation}
The cost--quality frontier of Sections~4--5 is the set of $(\text{cost},\text{quality})$ points reachable by these levers; \autoref{fig:frontier} traces it.

\begin{theorem}[Model-tier versus resampling crossover]\label{thm:tier}
Let $M$ be the operating model at $(q,c)$ and $M'$ a tier upgrade with accuracy $q'>q$ at sample cost $rc$, $r>1$. Under A1--A2, for any odd $k\ge 1$ the resampling increment $k{\to}k{+}2$ is the cheaper lever if and only if
\begin{equation}\label{eq:tier-cross}
(r-1)\bigl(Q_{\mathrm{MAJ}}(k{+}2)-Q_{\mathrm{MAJ}}(k)\bigr) \;>\; 2\,(q'-q)\,.
\end{equation}
\end{theorem}

\emph{Proof.} Both MCDs in~\eqref{eq:mcds} are positive, so the resampling increment is cheaper exactly when $2c/d_k < (r-1)c/(q'-q)$, and clearing the positive factors $c\,d_k\,(q'-q)$ yields~\eqref{eq:tier-cross} (equality being the knife-edge indifference). $\blacksquare$

As an analytical example (derived from the closed-form model; not measured), fix $q=0.9$, $r=2$, $q'-q=5\,\mathrm{pp}$: then $\mathit{MCD}_{\mathrm{tier}}=c/0.05=20\,c$, $\mathit{MCD}_{\mathrm{resample}}(1{\to}3)=2c/d_1=27.8\,c$ ($d_1=Q_{\mathrm{MAJ}}(3)-Q_{\mathrm{MAJ}}(1)=7.2\,\mathrm{pp}$, Theorem~\ref{thm:maj}), and $\mathit{MCD}_{\mathrm{VTR}}=c/q=1.11\,c$, the ranking at high $q$ being gate $\ll$ tier $<$ majority. At $q=0.9$ the $1{\to}3$ increment satisfies $(r-1)d_1=7.2\,\mathrm{pp}<2(q'-q)=10\,\mathrm{pp}$, and since $d_k$ decreases in $k$ (Theorem~\ref{thm:maj}), no odd budget $k$ makes resampling cheaper than the $2\times$-cost $5\,\mathrm{pp}$ tier. Consensus is competitive only when the next tier is expensive (large $r$) or the gap is small ($q'-q$ small), as for adjacent quantization steps.

\begin{corollary}[Cheapness of the gate]
For $q\in(1/2,1)$ and $r\ge 2$, the gate lever is strictly the cheapest of the three levers at every odd $k\ge 1$: $\mathit{MCD}_{\mathrm{VTR}}<\mathit{MCD}_{\mathrm{resample}}(k{\to}k{+}2)$ and $\mathit{MCD}_{\mathrm{VTR}}<\mathit{MCD}_{\mathrm{tier}}$.
\end{corollary}

\emph{Proof.} Majority is a noise filter: for $q\ge 1/2$, $Q_{\mathrm{MAJ}}(k)\ge q$ (equality at $k=1$), so $d_k\le 1-Q_{\mathrm{MAJ}}(k)\le 1-q<2q$ and $\mathit{MCD}_{\mathrm{VTR}}=c/q<2c/d_k=\mathit{MCD}_{\mathrm{resample}}$; and $q'-q\le 1-q<q\le(r-1)q$, so $\mathit{MCD}_{\mathrm{VTR}}=c/q<(r-1)c/(q'-q)=\mathit{MCD}_{\mathrm{tier}}$. $\blacksquare$

In the motivating setting, the realistic upgrade of a quantized self-hosted checkpoint is the same model at higher precision --- NVFP4 $\to$ BF16 on a single H200. There $q'-q$ is the accuracy tax $\tau_q$ of \citet{thaki2026toolcallcliff} and $r$ the throughput ratio of the two precisions, $r\approx 2$ in Section~8's pre-registered protocol. The gate's dividend $\Delta=q\,p_d/(1-p_d)$ (Section~4) is the detectable portion of that tax, up to the renormalization: reading the tax as the checkpoint's total error mass, $p=\tau_q$, gives $\Delta=\frac{q}{1-p_d}\,\delta\,\tau_q$ --- the detectable fraction $\delta\,\tau_q$ scaled by the renormalization $q/(1-p_d)\le 1$. The gate thus recovers the detectable portion of the quantization tax, scaled by $q/(1-p_d)\le 1$ --- by Section~4's silent floor, no more --- at marginal price $c/q$; this is a cross-paper, falsifiable prediction, carried to P1 of Section~8, which bounds the measured gate lift by $\Delta$, hence by the detectable mass $\delta\,p$, concentrated where that mass is largest.

These levers sit inside the per-turn cost frame of the self-hosted agent loop. Each extra draw re-sends the fixed agent prefix at the cached hit-fraction price of \citet{thaki2026manifestorder} and carries the per-turn reasoning-token bill of \citet{thaki2026verbositytax}; $k$ thus enters the per-turn cost model as a first-class budget, and the levers compose with the prefix-ordering, reasoning-budget, and quantization levers multiplicatively~\citep{thaki2026manifestorder,thaki2026verbositytax}. In self-hosted serving they also buy latency and energy: with probability $a=1-p_d$ the gated loop finishes in a single sample (Section~4), the axis of carbon-aware function-calling routing~\citep{arxiv260913559}, and, with the model-tier lever, the joint routing/test-time-scaling objective~\citep{arxiv260530898}.
\begin{figure}[!t]
\centering
\begin{tikzpicture}
\begin{axis}[
  width=0.52\textwidth,
  xmin=0, xmax=5.4,
  ymin=85, ymax=100,
  xtick={0,1,2,3,4,5},
  ytick={85,88,91,94,97,100},
  xlabel={expected cost (sample units)},
  ylabel={quality (\%)},
  tick label style={font=\footnotesize},
  label style={font=\small},
]
\addplot[gray, dashed, line width=0.8pt, domain=0:5.4] {90*x};
\node[font=\footnotesize, gray, anchor=west] at ([xshift=2pt]axis cs:4.4,93) {baseline $q/c$};
\node[draw=blue!70!black, fill=blue!70!black, circle, inner sep=1.2pt] at (axis cs:1,90) {};
\node[draw=blue!70!black, fill=blue!70!black, circle, inner sep=1.2pt] at (axis cs:1.07,96.3) {};
\node[draw=blue!70!black, fill=blue!70!black, circle, inner sep=1.2pt] at (axis cs:2,95.0) {};
\node[draw=blue!70!black, fill=blue!70!black, circle, inner sep=1.2pt] at (axis cs:3,97.2) {};
\node[draw=blue!70!black, fill=blue!70!black, circle, inner sep=1.2pt] at (axis cs:3.23,99.7) {};
\node[draw=blue!70!black, fill=blue!70!black, circle, inner sep=1.2pt] at (axis cs:5,99.1) {};
\node[font=\footnotesize, anchor=north] at (axis cs:1,90) {A0 ($k{=}1$)};
\node[font=\footnotesize, anchor=west] at ([xshift=6pt]axis cs:1.07,96.3) {VTR-2};
\node[font=\footnotesize, anchor=north] at (axis cs:2,95.0) {tier ($r{=}2$, $q'{=}0.95$)};
\node[font=\footnotesize, anchor=north] at (axis cs:3,97.2) {MAJ-3};
\node[font=\footnotesize, anchor=south] at (axis cs:3.23,99.7) {gate+MAJ-3};
\node[font=\footnotesize, anchor=south] at (axis cs:5,99.1) {MAJ-5};
\end{axis}
\end{tikzpicture}
\caption{Quality versus expected cost for the policy family at $q=0.9$, $p_d=0.07$, $p_s=0.03$, $r=2$, $q'=0.95$. The dashed line is the baseline quality-per-dollar $q/c$; the VTR-2 point lies on it (Lemma~\ref{lem:mcd}). (Analytical model (not measured); computed from the closed-form expressions of Sections~4--6 at $q=0.9$, $p_d=0.07$, $p_s=0.03$, $r=2$, $q'=0.95$.)}
\label{fig:frontier}
\end{figure}
\section{The Cheap-Judge Arm}\label{sec:judge}

The fourth arm replaces the deterministic gate with a learned scorer: a cheap judge (small local model or heuristic, cost $j \approx 0.2c$ per selection, an analytical parameter) picks one of the $k$ i.i.d.\ draws at temperature $T>0$. One conditional parameter $\beta$ models it: if the pool holds a correct plan (probability $1-(1-q)^k$), the judge picks it with probability $\beta$; otherwise every choice is wrong. Hence
\begin{equation}
Q_J(k) \;=\; \beta\bigl(1-(1-q)^k\bigr), \qquad C_J(k) \;=\; k\,c + j \;=\; c\!\left(k + \tfrac{j}{c}\right).
\end{equation}

\begin{theorem}[Judge dominance bound]\label{thm:judge}
JUDGE-$k$ is strictly dominated by VTR ($n_{\max}=2$) in quality-per-dollar whenever
\begin{equation}
\beta \;\le\; \beta^* \;:=\; \frac{q/(1-p_d)}{\,1-(1-q)^k\,}.
\end{equation}
\end{theorem}

\begin{proof}[Proof sketch]
If $\beta^* < 1$, since $\beta^*(1-(1-q)^k) = q/(1-p_d) = Q^*$ (Theorem~\ref{thm:gate}), for $\beta \le \beta^*$ the judge's quality $\beta(1-(1-q)^k) \le Q^*$ while its cost $c(k + j/c)$ exceeds VTR's expected cost $(1+p_d)c$ for every $k\ge2$ (and $p_d < 1$ implies $(1+p_d)c \le 2c$), so its quality-per-dollar is strictly below the gate's $c/q$. If $\beta^* \ge 1$, then the pool-attainable quality $1-(1-q)^k$ already exceeds the gate ceiling $Q^* = q/(1-p_d)$; any judge has quality at most $1-(1-q)^k$ (it cannot pick a correct plan that is not in the pool), and its cost $kc + j$ with $k\ge2$ still exceeds the gate's $(1+p_d)c < 2c$, so even perfect pool selection pays a strictly higher price per quality point than the gate; the dominance in quality-per-dollar holds a fortiori for $\beta \le \beta^*$. The bound therefore holds for all $p_d \in (0, 1-q)$.
\end{proof}

For $q=0.9$, $p_d=0.07$, $k=3$, $\beta^*\approx96.9\%$---beyond the conditional accuracy of any calibrated cheap judge (analytical example; closed-form; not measured). A judge at moderate global correlation ($\rho=0.47$) captures only $21.0\%$ of the improvement that perfect selection achieves over random choice~\citep{arxiv260312520}, and on a controlled testbed no judge-score policy improved the deterministic baseline~\citep{arxiv260713347}.

The judge shares the main model's silent-error class: its selections fail in common mode with the sampled plans, bounded by the co-failure ceiling~\citep{thaki2026ensamblerouter}. The arm is relevant only where \emph{no} deterministic validator exists---the routing-decision regime of~\citep{thaki2026ensamblerouter}, not the tool-call-output regime of this paper.
\section{Deployment Policy and Pre-Registered Protocol}\label{sec:protocol}

\subsection{Deployment Policy}\label{subsec:deploy}

\begin{enumerate}[leftmargin=2.4em, itemsep=2pt, topsep=2pt]
\item \textbf{Always gate; $n_{\max}=2$.} The cheapest lever: marginal cost $c/q$ per quality unit, dividend $\Delta = q p_d/(1-p_d)$, one sample with probability $a=1-p_d$ (Theorem~\ref{thm:gate}, Lemma~\ref{lem:mcd}).
\item \textbf{Test the silent floor.} If the target exceeds the ceiling $q/(1-p_d)$, add MAJ-3 (ARG-3 under the single-silent-argument regime of Proposition~\ref{prop:argk}); the extra samples are the cheapest available consensus (Theorem~\ref{thm:maj}).
\item \textbf{Upgrade the tier only on crossover.} Move to $M_2$ only when residual silent mass after consensus exceeds the target, or the consensus MCD exceeds the tier MCD (Theorem~\ref{thm:tier}), i.e.\ when $(r-1)\Delta Q_{\mathrm{resample}} > 2(q'-q)$ no longer holds.
\item \textbf{No judge arm while a deterministic validator exists.} Selection by a learned scorer is dominated (Theorem~\ref{thm:judge}) and reserved for the no-validator regime.
\end{enumerate}

\subsection{Pre-Registered Measurement Protocol}\label{subsec:protocol}

This is an analytical study (not an empirical run). No new measurements are reported.

\begin{itemize}[leftmargin=2.4em, itemsep=2pt, topsep=2pt]
\item \textbf{Setting.} BFCL (simple, multiple, parallel), deterministic AST grading, no judge in the grading path~\citep{bfcl2024leaderboard}; self-hosted NVFP4 27B on one H200 ($k{=}1$ baseline rows exist: 89.6\%, 800 cases~\citep{thaki2026toolcallcliff}); $k\in\{1,2,3,5\}$, temperature per harness; tier control: BF16 ($r\approx2$ from throughput).
\item \textbf{Arms.} A0: $k=1$; A1: VTR ($n_{\max}=2$); A2: MAJ-3; A3: ARG-3; A4: JUDGE-5 (8B local judge, $j\approx0.2c$).
\end{itemize}

\emph{Predictions.} All derive from the closed-form model of Sections~4--6; none is a measurement.
\begin{enumerate}[label=P\arabic*., leftmargin=2.6em, itemsep=2pt, topsep=2pt, parsep=0pt]
\item \textbf{Dividend bound.} A1 lift $\le \hat\delta\,\hat p$, where $\hat p$ is the $k{=}1$ grader-fail rate (total error mass), $\hat\delta$ the share of those failures detected by $V$ (validation-fail rate $=\hat\delta\,\hat p$); the lift peaks where the detectable mass is largest (parallel category).
\item \textbf{Silent floor.} Post-A1 residuals are dominated by the silent (argument-value) class in all categories~\citep{thaki2026toolcallcliff}; A1 lift smallest in simple (low $p_d$), $\approx$ identity at $k{=}1$.
\item \textbf{Crossover.} With $\hat q\approx0.896$: (i) MAJ-3 exceeds VTR-2 in quality by $\le2$~pp whenever $\hat p_d\ge6$~pp (gap $1.1$~pp at $\hat p_d=7$~pp; the gap grows as $\hat p_d$ shrinks), while VTR-2 beats MAJ-3 in MCD by $\ge2\times$; (ii) since $d_1\approx7.4$~pp at $\hat q$ and $d_k$ decreases in $k$, the resampling increment beats the $r\approx2$ tier only if $q_{\mathrm{BF16}}-q_{\mathrm{NVFP4}}<\hat d_1/2\approx3.7$~pp; at the tax $\approx5$~pp the model predicts the tier dominates every odd-$k$ consensus budget (no crossover exists).
\item \textbf{Judge dominated.} JUDGE-5 achieves lower quality at no lower cost than VTR-2 ($\beta<\beta^*$).
\end{enumerate}

\emph{Refutation criteria.} These ex-ante criteria falsify the model.
\begin{enumerate}[label=R\arabic*., leftmargin=2.6em, itemsep=2pt, topsep=2pt, parsep=0pt]
\item \textbf{Independence.} A1 lift $>\hat\delta\,\hat p+3$~pp $\Rightarrow$ correlated detectable errors (co-failure).
\item \textbf{Co-failure.} MAJ-3 $<$ VTR-2 while $\hat p_d<7$~pp $\Rightarrow$ silent-class correlation across draws (ceiling binds; routing caveat~\citep{thaki2026ensamblerouter}, now execution-side).
\item \textbf{Quantization tax.} If $\hat d_1/2 > q_{\mathrm{BF16}}-q_{\mathrm{NVFP4}}$ (the tier's MCD falls below MAJ-3's MCD) $\Rightarrow$ the tax is smaller than the $5$~pp used in the model; if $q_{\mathrm{BF16}}-q_{\mathrm{NVFP4}}>\hat d_1/2$ the tier-dominance prediction (P3) is confirmed.
\item \textbf{Judge information.} The judge beats the gate on the detectable class $\Rightarrow$ schema-level information the validator misses; re-derive $\beta^*$ feature-conditioned.
\end{enumerate}

\emph{Robustness.} Tool-calling results are sensitive to the random seed, the system prompt, and history carry-forward~\citep{arxiv260600135}; the protocol fixes and reports all of these, treating evaluation-implementation sensitivity as a known threat~\citep{arxiv260600135}.
\section{Limitations}\label{sec:limits}

\textbf{Analytical, not empirical.} This paper reports no measurements; every number is a closed-form derivation at stated parameters, and the analytical examples are labeled as such.

\textbf{Independence (A1).} The i.i.d.\ draw model is an approximation: multi-step tool-calling exhibits measured non-reproducibility across identical invocations~\citep{arxiv260528840}, and residual inconsistency persists even on simple tasks~\citep{arxiv250618781}. Correlated (co-failure) errors reduce the realized dividend, bounded by the co-failure ceiling~\citep{thaki2026ensamblerouter}; R2 is the pre-registered check.

\textbf{Perfect validator (A3).} Real validators have nonzero false-pass rates on the silent class; the $\varepsilon$ perturbation of Section~4 applies, and token-level runtime control reduces the false-pass rate~\citep{arxiv260327905}.

\textbf{Single canonical plan (A2).} Exact-match majority is conservative when several valid plans exist; allowing multiple correct plans can only raise majority quality, so the binomial bound is a lower bound.

\textbf{BFCL-specific structure.} Category structure and grading semantics are those of BFCL~\citep{bfcl2024leaderboard}; other suites may shift $p_d$ and $p_s$ (the regime varies across models and conditions~\citep{arxiv260806370, arxiv260528840}).

\textbf{Cost lattice.} The ratios $r$, $j$, and throughput multipliers are approximate and must be re-derived from the serving stack before deployment, as in~\citep{thaki2026ensamblerouter}.

\textbf{Judge calibration.} $\beta$ is a model parameter, not a measurement; no judge-calibration data is assumed.


The rendered figures for this work (Figures~\ref{fig:fig-policy-family},~\ref{fig:fig-cost-quality-frontier},~\ref{fig:fig-mcd-comparison}) are collected here.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.92\linewidth]{figures/fig_policy_family.png}
\caption{The execution-side sampling levers and the model-tier lever compose around a free deterministic validator on a self-hosted quantized checkpoint.  (Conceptual example)}
\label{fig:fig-policy-family}
\end{figure}

\begin{figure}[htbp]
\centering
\input{figures/fig_cost_quality_frontier.tex}
\caption{Relative quality-versus-cost positions of the six sampling arms under the analytical model at q=0.9, p\_d=0.07, p\_s=0.03.  (Analytical model (not measured))}
\label{fig:fig-cost-quality-frontier}
\end{figure}

\begin{figure}[htbp]
\centering
\input{figures/fig_mcd_comparison.tex}
\caption{The validate-then-retry gate is the cheapest lever, far below the model-tier upgrade and the first consensus increment at q=0.9.  (Analytical model (not measured))}
\label{fig:fig-mcd-comparison}
\end{figure}

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

On a self-hosted quantized model, the free deterministic validator is the cheapest reliability lever: the gate dividend at marginal cost $c/q$, capped by the silent floor no retry budget crosses. Consensus buys the remaining silent class; tier upgrades sit on the same quality-per-dollar axis; a judge is dominated whenever a validator exists. Gate, then consensus, then tier, never a judge: deployable for the unattended agent loop; every pre-registered criterion falsifiable.
\bibliography{refs}
\end{document}