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\title{\bf The Precision Ladder: Measuring the Joint Model-Tier and\\
Quantization-Precision Cost-Quality Frontier for Agentic\\
Tool-Call Routing on Self-Hosted H200}
\author{ThakiCloud AI Research (Hyojung Han)}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We upgrade the one-dimensional FrugalGPT cascade \citep{cheng2023frugalgpt} to a two-dimensional frontier over model tier and weight precision for self-hosted agentic serving, where the price axis collapses: one GPU serves one (model, precision) point at a fixed hourly rate. Per category, the cheapest gated pair is the argmax of effective passing throughput $E = tQ/\ell$ under a judge-free BFCL-style code gate. A closed-form crossing condition, $\rho_c^t\rho_c^Q>\rho_c^\ell$, tests a larger low-bit challenger against a smaller full-precision incumbent per passed task (the per-true-task bill adds the success ratio $\rho_c^r$); in the shared regime its necessary form is the dividend-versus-cliff inequality $D_c > \xi/(1-\xi)$. The predicted category-inverted routing table (small full-precision on easy, large low-bit on hard) is driven by byte parity: 15.2 GB of 27B weights at 4.5 effective bits versus 16.0 GB for 8B at 16 bits keeps the throughput ratio near one. The measured feasibility layer reports BF16 4B/8B/14B peaking at 7.5/15.3/27.5 GB on a 93.6 GB Hopper node, cited on-demand prices, an 89.6\% BFCL-style anchor for the 27B-NVFP4 production checkpoint, and four pre-registered refutation criteria R1--R4. This is an analytical and positional study with a measured feasibility layer; no new per-cell quality or throughput measurements are reported.
\end{abstract}

\section{Introduction}

An unattended agentic tool-calling loop on a self-hosted Hopper node must pick, per task category, \emph{both} the model tier --- 4B, 8B, 14B, or 27B dense Qwen3 \citep{yang2025qwen3} --- and the weight precision --- BF16, FP8, or NVFP4. FrugalGPT-style cascades \citep{cheng2023frugalgpt} optimize only the model tier, and only because under API pricing the request price scales with the tier: escalate until the quality gate passes, and the cheapest passing tier wins. Under self-hosted single-node serving that structure collapses. The node is charged per GPU-hour independently of which (model, precision) variant is resident; the price axis collapses, and the choice becomes two-dimensional --- model size and quantization precision, each with its own quality, token budget, and memory footprint, jointly priced by a single hardware rate.

Every recent measurement on these axes is one-dimensional: bit-width accuracy decay at a fixed model \citep{thaki2026toolcallcliff}; per-request precision selection at a fixed model \citep{thaki2026precisiontier}; skill-versus-model routing at fixed precision \citep{thaki2026ensamblerouter}; sampling policy at a fixed model \citep{thaki2026validatordividend}. The joint (model $\times$ precision) cost-quality frontier for agentic tool-call workloads is open.

We ask the deployment question this leaves behind: for agentic tool-call tasks of graded difficulty --- the simple, parallel, and agentic BFCL-style categories \citep{bfcl2024leaderboard} --- what is the cheapest joint (model-size $\times$ quantization-precision) pair per category that meets a fixed quality gate $g$, and can a 27B model in NVFP4 be cheaper per successful task than an 8B model in BF16 on some categories? This is an analytical and positional study with a measured infrastructure-feasibility layer: the per-cell quality-throughput grid of the frontier is specified by a pre-registered protocol with cited anchors, and no new per-cell quality or throughput measurements are reported in this paper.

Our contributions are as follows.
\begin{itemize}
\item \textbf{System.} A deployable per-category routing table (the \emph{precision ladder}) for the ThakiCloud token factory: the cheapest (model $\times$ precision) pair on the self-hosted fleet that passes the BFCL-style quality gate $g=0.85$ \citep{bfcl2024leaderboard}, composable with the zero-token skill router \citep{thaki2026ensamblerouter} and queue-aware tier fallback, and anchored by the 27B-NVFP4 production checkpoint (89.6\% overall on the 800-case BFCL-style suite; cited \citep{thaki2026toolcallcliff}).
\item \textbf{Society.} A quantification of when a quantized large model beats a smaller full-precision model at equal task quality, giving self-hosted operators a concrete recipe to cut per-task inference spend and the energy behind it \citep{vellaisamy2026energy}.
\item \textbf{Science.} An upgrade of FrugalGPT-style 1-D cascades \citep{cheng2023frugalgpt} to the first joint 2-D (model-size $\times$ quantization-precision) cost-quality frontier for agentic tool-call workloads, with a measured infrastructure-feasibility layer and a pre-registered falsification protocol: a closed-form ladder-crossing condition (Theorem~2), a necessary dividend-versus-cliff inequality (Corollary~2), the category-structure regimes (Proposition~1), and a hardware-transfer proposition (Proposition~2).
\end{itemize}

Section~2 reviews related work; Section~3 fixes the problem setup and notation; Section~4 develops the precision ladder and the crossing condition; Section~5 treats the quality inputs and byte parity; Section~6 reports the measured infrastructure-feasibility layer and the pre-registered protocol; Section~7 gives the deployable routing table and its composition; Sections~8 and~9 discuss limitations and conclude.

\section{Background and Related Work}

\subsection{Model cascades, routing, and cost-aware serving}
FrugalGPT \citep{cheng2023frugalgpt} introduced the 1-D cascade over API model tiers: a quality classifier escalates each request up the tier ladder until an acceptable answer is produced, and the cost--quality Pareto frontier is indexed by the tier list alone --- the upgrade target of this paper. Subsequent work sharpens the escalation decision: a decision-theoretic characterization of when escalation is worth its cost \citep{bouchard2026escalation}, calibrated-uncertainty triggers for cost-optimal cascade routing \citep{kotte2026ucci}, cluster--route--escalate frameworks for cost-aware serving \citep{moslem2026clusterrouteescalate}, and routers that read prefill activations rather than request metadata \citep{varshney2026llmrouter}. \textbf{Gap.} All of these optimize the model tier under API-style per-request pricing, in which the price is a function of the tier; none prices the weight-precision axis jointly with the model axis under single-node self-hosted serving, where one hardware rate covers the whole (model $\times$ precision) grid.

\subsection{Low-bit quantization for agentic workloads}
Post-training quantization matured with GPTQ \citep{frantar2022gptq}, and NVFP4-class 4-bit checkpoints have advanced through augmented residual channels \citep{meng2026arcquant}, outlier-companion grouping \citep{ocgquant2026ocg}, and mixed-precision designs that protect the recurrent half of hybrid models \citep{kozyrev2026gdn}. Two cost channels matter for agentic loops: low-bit serving inflates the reasoning-token budget \citep{lian2026quantization}, and compression shifts the behavior of otherwise aligned models \citep{rath2026quantbias}. The routing side has picked up the same axes --- routing-consistent quantization of mixture-of-experts models \citep{park2026vsraq}, activation-aware test-time quantization \citep{koikeakino2026ttq}, and extreme low-precision KV-cache quantization for agentic inference \citep{shen2026triaxialkv}. \textbf{Gap.} Bit-width decay is characterized at a fixed model; the joint (model $\times$ precision) interaction --- where a low-bit large model crosses a full-precision small model in cost per passed task --- is unmeasured.

\subsection{Quantized agents and tool-call evaluation}
Whether compressed LLMs can truly act has been probed empirically \citep{dong2025compressed}, and the masking of damage by flat aggregate scores in quantized agents has been made explicit \citep{jang2026flatscore}. The evaluation substrate we build on is the Berkeley Function Calling Leaderboard: judge-free, code-graded AST and state-equality checks across single, parallel, and multi-turn tool-call categories \citep{bfcl2024leaderboard}. Along the same agentic axis sit non-monotonic chain-of-thought budget effects in function-calling agents \citep{qi2026brief}, small reasoning models as competent instruction followers in function calling \citep{taheri2026small}, learned tool-use boundaries \citep{zou2026cobra}, grammar-constrained decoding that raises first-attempt success \citep{collura2026grammar}, skill retrieval augmentation \citep{su2026sra}, and what survives when a recursive reasoner is compressed for the edge \citep{jim2026recursive}. \textbf{Gap.} These lines measure agentic quality under compression, but not its dollar price per passed task against a model-tier alternative under a shared quality gate.

\subsection{Self-hosted serving economics}
The memory structure of paged self-hosted serving is fixed by vLLM and PagedAttention \citep{kwon2023vllm}, and request-level and token-level energy costs on GPU platforms have been characterized \citep{vellaisamy2026energy}. \textbf{Gap.} Cost models price the request, not the (model $\times$ precision) serving variant; the whole-GPU attribution of a variant's effective passing throughput --- the object a self-hosted operator actually pays for --- is absent.

\subsection{Our prior work}
Five in-house studies each fix one axis. \emph{The Tool-Call Cliff} \citep{thaki2026toolcallcliff} measures precision-axis accuracy decay at a fixed 27B model --- the accuracy tax $\tau_q$ and the cliff ratio $\kappa_q$; here it supplies the quality-decay input of the 2-D frontier. \emph{Precision-Tier Routing} \citep{thaki2026precisiontier} performs per-request quantization-level selection at a fixed model --- the within-model rung; here the ladder joins it to the between-model rung. \emph{The Reasoning Verbosity Tax} \citep{thaki2026verbositytax} quantifies the per-turn cost and the NVFP4 $\alpha$-inflation interaction; here it supplies the length-ratio term in the crossing condition. \emph{The Validator's Dividend} \citep{thaki2026validatordividend} prices the sampling axis under a free deterministic validator; here it supplies the retry-success term $r$. \emph{Cross-Tier Model Ensemble Voting for Skill Routing} \citep{thaki2026ensamblerouter} dispatches skills across model tiers with a zero-token router; here the routing table hangs below it as the model--precision dispatch. This paper is the first to join the model-tier axis and the quantization-precision axis into one defined cost-quality object (its cells are a pre-registered measurement program).

\section{Problem Setup and Notation}

We fix the notation used throughout.

\begin{definition}[Grids and feasible set]\label{def:grids}
Let $\mathscr{C}=\{\text{simple},\text{parallel},\text{agentic}\}$ be the tool-call categories, each BFCL-style and code-graded by AST and state equality, judge-free \citep{bfcl2024leaderboard}. Let $\mathcal{M}=\{4\mathrm{B},8\mathrm{B},14\mathrm{B},27\mathrm{B}\}$ be the dense Qwen3 model grid \citep{yang2025qwen3} and $\mathcal{P}=\{\mathrm{BF16},\mathrm{FP8},\mathrm{NVFP4}\}$ the weight-precision grid. The feasible set $\mathcal{F}\subseteq\mathcal{M}\times\mathcal{P}$ collects the variants deployable on the target node; its membership is fixed by the measured feasibility layer (Section~6).
\end{definition}

\begin{definition}[Variant parameters]\label{def:params}
For $c\in\mathscr{C}$ and $(m,q)\in\mathcal{F}$, a serving variant is indexed by four parameters:
\begin{itemize}
\item $t_{m,q}^{(c)}\in(0,\infty)$: sustained output tokens per GPU-second (decode throughput), from the fixed concurrency probe ($C=8$ concurrent, $N=64$ sequences, EOS disabled, aggregate rate over the probe wall);
\item $\ell_{m,q}^{(c)}$: expected output tokens per draw (one sampled tool-call attempt);
\item $Q_{m,q}^{(c)}\in[0,1]$: code-graded pass fraction under judge-free AST / state-equality checks (Assumption~\ref{ass:a2});
\item $s_{m,q}^{(c)}\in[0,Q_{m,q}^{(c)})$: silent-wrong fraction per draw --- the fraction of draws that pass the code gate while being silently wrong, the true-success overlay of \citep{thaki2026validatordividend}.
\end{itemize}
The per-task true-success rate $r_{m,q}^{(c)} := 1 - s_{m,q}^{(c)}/Q_{m,q}^{(c)} \in (0,1]$ is the quality overlay derived from $s$: it governs the per-true-task bill only, never the per-passed-task frontier. The hardware rate is $p_h$, in USD per GPU-hour on hardware $h$ (cited on-demand list price; Assumption~\ref{ass:a1}).
\end{definition}

\begin{definition}[Throughput, cost, and frontier]\label{def:frontier}
The \emph{effective passing throughput} and \emph{per-passed-task cost} of the variant are
\begin{equation}
E_{m,q}^{(c)} \;:=\; \frac{t_{m,q}^{(c)}\,Q_{m,q}^{(c)}}{\ell_{m,q}^{(c)}} \quad\text{(tasks passed per GPU-second)},
\qquad
C_{m,q}^{(c)} \;:=\; \frac{p_h}{E_{m,q}^{(c)}} \;=\; \frac{p_h\,\ell_{m,q}^{(c)}}{t_{m,q}^{(c)}\,Q_{m,q}^{(c)}} \quad\text{(USD per passed task)}.
\end{equation}
If a fraction $s$ of draws is silently wrong per draw, the per-true-task bill is $C/r = p_h\,\ell/(t\,Q\,r)$ with $r := 1 - s/Q \in (0,1]$ (the true-success overlay of \citep{thaki2026validatordividend}); $r$ is a quality overlay, not a cost factor, and drops out of the per-passed-task frontier.
\begin{remark}[Bill-invariance of retry]
With the free deterministic validator, if a draw passes with probability $Q$, the expected draws per passed task is exactly $1/Q$ (geometric identity; with retry budget $k$: $d_k = (1-(1-Q)^k)/Q$, $r_k = 1-(1-Q)^k$, $d_k/r_k = 1/Q$). Retry changes the abstention rate, not the per-passed-task bill.
\end{remark}
At gate $g$, the 2-D cost-quality frontier of category $c$ and its \emph{routing entry} are
\begin{equation}
F_c(g) \;:=\; \frac{p_h}{\max\bigl\{\, E_{m,q}^{(c)} \;:\; (m,q)\in\mathcal{F},\; Q_{m,q}^{(c)}\ge g \,\bigr\}},
\qquad
(m_c^{*},q_c^{*}) \;:=\; \arg\max_{\,(m,q)\in\mathcal{F},\; Q_{m,q}^{(c)}\ge g}\; E_{m,q}^{(c)},
\end{equation}
with the tie-break: smaller memory footprint first, then smaller model; equivalently, $F_c(g)=\min\{C_{m,q}^{(c)} : (m,q)\in\mathcal{F},\; Q_{m,q}^{(c)}\ge g\}$. The FrugalGPT-style 1-D cascade \citep{cheng2023frugalgpt} restricts the search to the BF16 row:
\begin{equation}
F_c^{\mathrm{1D}}(g) \;:=\; \frac{p_h}{\max\bigl\{\, E_{m,\mathrm{BF16}}^{(c)} \;:\; Q_{m,\mathrm{BF16}}^{(c)}\ge g \,\bigr\}}.
\end{equation}
\end{definition}

\begin{assumption}[A1: price invariance]\label{ass:a1}
$p_h$ is independent of the resident variant $(m,q)$: under single-node whole-GPU attribution the node is charged the same GPU-hour rate for every cell of $\mathcal{F}$, so the variant is priced only through its effective passing throughput.
\end{assumption}

\begin{assumption}[A2: code-graded quality]\label{ass:a2}
$Q_{m,q}^{(c)}$ is the judge-free, code-graded pass fraction (AST and state equality, \citep{bfcl2024leaderboard}-style), and the quality gate is fixed per category at $g=0.85$.
\end{assumption}

\begin{assumption}[A3: length inflation]\label{ass:a3}
\begin{equation}
\ell_{m,q}^{(c)} = \alpha_{m,q}^{(c)}\,\ell_{m,\mathrm{BF16}}^{(c)},
\qquad
\alpha_{m,\mathrm{BF16}}^{(c)} = 1,
\qquad
\alpha_{m,q}^{(c)} \ge 1 \;\; \text{for low-bit } q,
\end{equation}
with $\alpha_{m,q}^{(c)}$ the length-inflation factor: low-bit (NVFP4) serving inflates the expected reasoning-token budget of the same model \citep{thaki2026verbositytax, lian2026quantization}.
\end{assumption}

\begin{assumption}[A4: cliff decomposition]\label{ass:a4}
\begin{equation}
Q_{m,q}^{(c)} = Q_{m,\mathrm{BF16}}^{(c)} - \tau_{m,q}^{(c)},
\qquad
\tau_{m,\mathrm{BF16}}^{(c)} = 0,
\qquad
\xi_{m,q}^{(c)} := \frac{\tau_{m,q}^{(c)}}{Q_{m,\mathrm{BF16}}^{(c)}} \in [0,1).
\end{equation}
The structural properties of the tax $\tau_{m,q}^{(c)}$ are those measured at a fixed 27B model \citep{thaki2026toolcallcliff}: $\tau_{m,q}^{(c)}$ is nondecreasing in $s_c$, the number of mandatory structural decode steps of category $c$ (case-level amplification); the jump from 8 bits to 4 bits is superlinear; and the failure composition is argument-value-dominant.
\end{assumption}

\begin{assumption}[A5: tier dividend]\label{ass:a5}
\begin{equation}
D_c(m',m) := \frac{Q_{m',\mathrm{BF16}}^{(c)} - Q_{m,\mathrm{BF16}}^{(c)}}{Q_{m,\mathrm{BF16}}^{(c)}} \ge 0 \quad (m' > m),
\end{equation}
nondecreasing in $s_c$: the capability dividend concentrates in the structurally harder categories \citep{thaki2026toolcallcliff}.
\end{assumption}

\begin{definition}[Crossing ratios]\label{def:ratios}
For a tier step $m'>m$ and a low-bit precision $q'\ne\mathrm{BF16}$, the crossing ratios of category $c$ are
\begin{equation}
\rho_c^{t} := \frac{t_{m',q'}^{(c)}}{t_{m,\mathrm{BF16}}^{(c)}},
\qquad
\rho_c^{Q} := \frac{Q_{m',q'}^{(c)}}{Q_{m,\mathrm{BF16}}^{(c)}},
\qquad
\rho_c^{r} := \frac{r_{m',q'}^{(c)}}{r_{m,\mathrm{BF16}}^{(c)}},
\qquad
\rho_c^{\ell} := \frac{\ell_{m',q'}^{(c)}}{\ell_{m,\mathrm{BF16}}^{(c)}}.
\end{equation}
Here $r_{m,q}^{(c)} = 1 - s_{m,q}^{(c)}/Q_{m,q}^{(c)}$ is the per-task true-success rate of Definition~\ref{def:params}. The ratio $\rho_c^{r}$ appears only in the per-true-task overlay, not in the per-passed-task crossing. For deployment (Section~7), the test is the raw comparison $t'Q'/\ell' > tQ/\ell$, defined for any pair of distinct variants; the dividend--tax form below specializes to the tier step $m' > m$.
\end{definition}

\begin{remark}[Finiteness; the ladder is the object]\label{rem:finite}
$F_c(g)$ is well-defined whenever $\{(m,q)\in\mathcal{F} : Q_{m,q}^{(c)}\ge g\}\neq\varnothing$, and the optimization is finite: the grid has $|\mathcal{M}|\cdot|\mathcal{P}|=4\times 3=12$ cells, so the $\arg\max$ with the footprint-then-size tie-break yields a unique routing entry. The object of interest is not the grid but the \emph{ladder} ordering induced by $E_{m,q}^{(c)}$, developed in Section~4.
\end{remark}

\section{The Precision Ladder: A 2-D Cost-Quality Frontier}
\label{sec:ladder}

FrugalGPT-style cascades \citep{cheng2023frugalgpt} rank a one-dimensional chain of model tiers, in the API regime where both price and capability scale with size. On self-hosted single-node serving that one-dimensional picture no longer holds: the whole GPU is attributed to whichever variant runs, so quantization precision becomes a second cost-relevant axis alongside model tier. We therefore develop the cost--quality frontier on the joint grid $\mathcal{M} \times \mathcal{P}$ restricted to the feasible set $\mathcal{F}$ (Definition~\ref{def:grids}), drawing on the variant parameters, the throughput and cost objects, and the crossing ratios already fixed in Section~3 (Definitions~2--4). The resulting 2-D ladder subsumes the cascade, and we show that the FrugalGPT chain is precisely the projection of this frontier onto the BF16 row of $\mathcal{P}$.

\begin{lemma}[Frontier collapse] \label{lem:collapse}
Under A1, $C_{m,q}^{(c)} = p_h / E_{m,q}^{(c)}$ for all $(m,q) \in \mathcal{F}$ and $c$. Hence (i) the per-category ladder ordering---the ranking of variants by $E_{m,q}^{(c)}$ and the routing entry $(m_c^*, q_c^*)$---is invariant to the hardware platform $h$; (ii) for two platforms $h, h'$ with cited prices $p_h, p_{h'}$ and identical $(\mathcal{F}, Q, t, \ell, r)$,
\begin{equation}
F_c^{(h')}(g) \;=\; \frac{p_{h'}}{p_h}\, F_c^{(h)}(g).
\end{equation}
\end{lemma}
\begin{proof}
Immediate from $C = p_h/E$: the order by $E$ is $h$-free, giving (i); and from the definition $F_c = p_h / \max E$ with an identical feasible set, giving (ii).
\end{proof}

\begin{remark}
In the API regime of \citep{cheng2023frugalgpt} the per-tier price $p_m$ is \emph{not} $h$-free (price and capability scale together with $m$), so a model-only cascade is the natural object there; A1 is exactly what makes the self-hosted problem 2-D rather than 1-D.
\end{remark}

\begin{theorem}[Weak dominance of the cascade projection] \label{thm:dominance}
For all $c, g$, whenever the BF16 row has a gated point on $c$ (otherwise $F_c^{\mathrm{1D}}(g)$ is undefined and the comparison is void): $F_c(g) \le F_c^{\mathrm{1D}}(g)$. Moreover, equality for all $c$ holds iff no off-row point is strictly $E$-superior: for every category $c$ there is no $(m,q)$ with $q \ne \text{BF16}$, $Q_{m,q}^{(c)} \ge g$ and $E_{m,q}^{(c)} > E_{m_c^{1D},\text{BF16}}^{(c)}$, where $m_c^{1D} := \arg\max_m E_{m,\text{BF16}}^{(c)}$ is the BF16-row argmax.
\end{theorem}
\begin{proof}
The 1-D feasible set (BF16 row) is a subset of the 2-D feasible set, so the max over the superset is at least as large and, since $x \mapsto p_h / x$ is decreasing, $F_c(g) \le F_c^{1D}(g)$. Equality in a category holds iff the superset maximum is attained at the BF16-row argmax, i.e., iff no off-row feasible point is strictly $E$-superior; the all-$c$ statement is the conjunction over categories.
\end{proof}

The FrugalGPT cascade is thus the projection of the 2-D frontier onto the BF16 row. The dominance gap measures strict $E$-dominance; it is zero iff no off-row point is strictly $E$-superior, and under the footprint tie-break an $E$-tie may still route off-row at zero gap.

\begin{theorem}[Ladder crossing] \label{thm:crossing}
Fix $m' > m$ and $q' \ne \text{BF16}$. The pair $(m', q')$ is strictly cheaper per passed task than $(m, \text{BF16})$ on category $c$ iff
\begin{equation}
\rho_c^{t}\, \rho_c^{Q} \;>\; \rho_c^{\ell}. \label{eq:crossing}
\end{equation}
Per true task, the success ratio multiplies the left-hand side: $\rho_c^{t}\,\rho_c^{Q}\,\rho_c^{r} > \rho_c^{\ell}$. For a challenger with $m' < m$ the identical product form holds with the four ratios defined by that pair (the dividend--tax specialization of Corollaries~1--2 assumes $m' > m$).
\end{theorem}
\begin{proof}
By A1, $C_{m',q'}^{(c)} < C_{m,\text{BF16}}^{(c)}$ iff $E_{m',q'}^{(c)} > E_{m,\text{BF16}}^{(c)}$; writing $E = tQ/\ell$, this is $\big(t_{m',q'}^{(c)} Q_{m',q'}^{(c)}\big)/\ell_{m',q'}^{(c)} > \big(t_{m,\text{BF16}}^{(c)} Q_{m,\text{BF16}}^{(c)}\big)/\ell_{m,\text{BF16}}^{(c)}$; dividing by the right-hand side and using the definitions of $\rho_c^{t}$, $\rho_c^{Q}$, and $\rho_c^{\ell}$ yields \eqref{eq:crossing}. Per true task the bill is $C/r$, so dividing the per-passed-task inequality by the challenger's success overlay ratio yields the stated overlay form.
\end{proof}

\begin{corollary}[Dividend--tax form] \label{cor:divtax}
Under A3--A5, $\rho_c^{Q} = \big(1 + D_c\big)\,\big(1 - \xi_{m',q'}^{(c)}\big)$, so the per-passed crossing condition \eqref{eq:crossing} is equivalent to
\begin{equation}
\rho_c^{t}\, \big(1 + D_c\big)\, \big(1 - \xi_{m',q'}^{(c)}\big) \;>\; \rho_c^{\ell}, \label{eq:dagger}
\end{equation}
with $D_c = D_c(m',m)$. In the shared regime ($\rho_c^{\ell} = 1$, i.e., no length inflation across the two variants) this reduces to
\begin{equation}
\rho_c^{t} \;>\; \frac{1}{\big(1 + D_c\big)\,\big(1 - \xi_{m',q'}^{(c)}\big)} \;:=\; \rho_c^{t,*}, \label{eq:threshold}
\end{equation}
i.e., the throughput ratio must exceed a threshold that falls as the tier dividend rises and the cliff tax falls.
\end{corollary}
\begin{proof}
By A4, $Q_{m',q'}^{(c)} = Q_{m',\text{BF16}}^{(c)}\,\big(1 - \xi_{m',q'}^{(c)}\big)$; by A5, $Q_{m',\text{BF16}}^{(c)} = Q_{m,\text{BF16}}^{(c)}\,\big(1 + D_c(m',m)\big)$; the ratio identity follows and \eqref{eq:dagger} is \eqref{eq:crossing} with $\rho_c^{Q}$ substituted. The shared-regime reduction sets $\rho_c^{\ell} = 1$ and divides.
\end{proof}

\begin{corollary}[Cliff versus dividend; necessary condition] \label{cor:cliffdiv}
In the shared regime, when the larger low-bit variant is no faster than the smaller full-precision one ($\rho_c^{t} \le 1$, the byte-parity regime of Section~\ref{sec:byteparity}), crossing on category $c$ requires
\begin{equation}
D_c(m',m) \;>\; \frac{\xi_{m',q'}^{(c)}}{1 - \xi_{m',q'}^{(c)}}.
\end{equation}
In particular, if $D_c = 0$ (the model tier yields no BF16 quality dividend on $c$), no off-row point can cross and the ladder degenerates to the within-model precision choice of \citep{thaki2026precisiontier}.
\end{corollary}

\begin{proposition}[Category structure; the inversion regimes] \label{prop:regimes}
Write $N_c := \big(1 + D_c\big)\,\big(1 - \xi_{m',q'}^{(c)}\big)$, and let $S := \{ c \in \mathscr{C} : N_c > 1/\rho^{t} \}$ in the shared regime ($\rho_c^{\ell} = 1$) with $\rho_c^{t} = \rho^{t}$ constant in $c$ and $\rho^{t} < 1$ (first-order throughput parity, Section~\ref{sec:byteparity}): by \eqref{eq:dagger}, $S$ is the superlevel set of $N_c$ at the constant threshold $1/\rho^{t}$. Three regimes are \emph{defined} by the shape of $N_c$ in $s_c$ (the number of mandatory structural decode steps, A4--A5):
\begin{enumerate}
\item \textbf{dividend-dominated:} $N_c$ strictly increasing in $s_c$, so $S$ is a proper superlevel tail---crossing holds only on the structurally hardest categories: the inverted pattern (easy $\to (m, \text{BF16})$, hard $\to (m', q')$), i.e., the precision ladder;
\item \textbf{cliff-dominated:} $N_c < 1/\rho^{t}$ for all $c$, so $S = \emptyset$: the 1-D cascade is already 2-D optimal and prediction P2 (Section~6.4) is refuted;
\item \textbf{low-bit-dominated:} $N_c > 1/\rho^{t}$ for all $c$, so $S = \mathscr{C}$: the larger low-bit point is optimal on every gated category.
\end{enumerate}
If $N_c$ is non-monotone in $s_c$, $S$ can be a proper non-tail set; this fourth, non-generic shape is left to the protocol (R1/R3, Section~\ref{sec:protocol}). The realized shape is settled by the pre-registered protocol outputs.
\end{proposition}
\begin{proof}
$S$ is a superlevel set of $N_c$ at the constant threshold $1/\rho^{t}$ by definition. Under (i)--(ii) of A4--A5, $N_c = \big(1 + D_c\big)\big(1 - \xi_{m',q'}^{(c)}\big)$ is a product of an $s_c$-increasing factor (the tier dividend, A5) and an $s_c$-decreasing factor (the survival of the cliff tax, A4); the regimes above are the three shapes of that product relative to the threshold $1/\rho^{t}$---increasing tail, everywhere below, everywhere above---and are not claimed to be exhaustive.
\end{proof}

\begin{proposition}[Hardware transfer] \label{prop:hardware}
Let $h_0$ be the measured platform (H100 NVL, 93.6 GB) and $h'$ the target (H200, 141 GB, vendor spec) with cited on-demand price ratio $p_{h'}/p_{h_0} = 1.4$, and assume the common-factor condition $E_{m,q}^{(c),\,(h')} = \beta_c\, E_{m,q}^{(c),\,(h_0)}$ for all $(m,q) \in \mathcal{F}$ at fixed $c$ (uniform per-category scaling, decode-bound regime). Then (i) the routing table (ladder ordering) is unchanged under transfer; (ii) the absolute dollar cost per passed task on category $c$ strictly improves on $h'$ iff $\beta_c > 1.4$; (iii) memory feasibility measured on $h_0$ (all BF16 variants peaking at 27.5 GB of 93.6 GB) transfers a fortiori to $h'$. Refutation criterion R4 bounds the acceptable deviation of the measured $\beta_c$ proxies.
\end{proposition}
\begin{proof}
(i): for fixed $c$ the common factor $\beta_c > 0$ multiplies every cell of $\mathcal{F}$ on $h'$ by the same positive amount, so it preserves the within-category argmax of $E$ (hence the ladder ordering and routing entry) on every category; no appeal to Lemma~\ref{lem:collapse} is needed. (ii): by (i) the routing entry $(m_c^*, q_c^*)$ is the same on both platforms, so $C_c^{(h')} = p_{h'}/\big(\beta_c\, E_{m_c^*, q_c^*}^{(c), (h_0)}\big) < C_c^{(h_0)} = p_{h_0}/E_{m_c^*, q_c^*}^{(c), (h_0)}$ iff $\beta_c > p_{h'}/p_{h_0} = 1.4$. (iii): $27.5\ \text{GB} < 93.6\ \text{GB}$ implies a fortiori $27.5\ \text{GB} < 141\ \text{GB}$ (weights-only arithmetic of Section~\ref{sec:byteparity} scales the same way).
\end{proof}

\section{Quality Inputs from the Tool-Call Cliff and Byte Parity}
\label{sec:inputs}

Section~4 reduces the frontier to two input families: the off-row quality ratios (the cliff) and the cross-variant throughput ratio (byte parity). This section sources the first from the measured tool-call cliff, the second from weights-only arithmetic, and illustrates the resulting crossing decision.

\subsection{Quality inputs from the tool-call cliff}
\label{sec:cliff}

The accuracy tax $\tau_q$ and the cliff ratio $\kappa_q$ of \citep{thaki2026toolcallcliff} measure precision-axis accuracy decay at a \emph{fixed} 27B model, so that quantization is the sole moving accuracy-cost variable; the tax amplifies at the case level in proportion to the number $s_c$ of mandatory structural decode steps of the category. Two predictions of that study carry over into the ladder and are encoded in A4: its prediction P1 (argument-value-dominant failure composition at low bit) and its prediction P2 (superlinear tax jump from 8 to 4 bits). The cited anchor is the 27B-NVFP4 production checkpoint at 89.6\% overall on the 800-case BFCL-style suite (2026-09-06), with argument-value-dominant failure composition, matching its prediction P1.

In our notation the off-row cell inputs are exactly
\[
\xi_{\text{27B},\text{NVFP4}}^{(c)}
\;=\;
\frac{Q_{\text{27B},\text{BF16}}^{(c)} - Q_{\text{27B},\text{NVFP4}}^{(c)}}{Q_{\text{27B},\text{BF16}}^{(c)}},
\]
a per-cell protocol output (only the overall row is cited), and $D_c(\text{27B}, \text{8B})$, likewise a per-cell protocol output (scenario values in Section~\ref{sec:illustration}). No other per-cell quality number is used below.



\subsection{Byte parity: why the throughput ratio is near one}
\label{sec:byteparity}

In decode-bound single-node serving, sustained throughput is governed by the per-token weight bytes read (memory structure per \citep{kwon2023vllm}; KV-cache considerations per \citep{shen2026triaxialkv}). For the two variants at the crossing, the weight bytes are
\[
\text{8B BF16}: \quad 8 \times 10^{9} \times 2\ \text{B} \;=\; 16.0\ \text{GB}; \qquad
\text{27B NVFP4}: \quad 27 \times 10^{9} \times \tfrac{4.5}{8}\ \text{B} \;=\; 15.2\ \text{GB},
\]
where the 4.5 effective bits per weight of NVFP4 is the 4-bit E2M1 element plus one FP8 block scale per 16 elements. The ratio is $15.2 / 16.0 = 0.95$: the 27B-parameter model at 4.5 bits per weight reads about 5\% \emph{fewer} bytes per token than the 8B model at BF16. In the weights-dominated regime $\rho^{t} \in [0.9, 1.05]$ is therefore a reasonable prior for the throughput ratio before second-order effects.

Two second-order terms, stated as caveats rather than corrections:
(i) the 27B KV cache is larger per token than the 8B's, which lowers $\rho_c^{t}$ on long-context (agentic) categories---this is why the protocol measures $\rho_c^{t}$ per category (throughput probe, Section~\ref{sec:protocol}) rather than assuming it;
(ii) $\alpha$-inflation (A3) raises $\rho_c^{\ell}$ on the low-bit variant on reasoning-heavy categories \citep{thaki2026verbositytax, lian2026quantization}, moving the crossing threshold \eqref{eq:threshold} upward.

\subsection{Numerical illustration (scenario, not a measurement)}
\label{sec:illustration}

\textbf{Illustration.} The following uses assumed scenario values, not data; each assumption is stated inline and the arithmetic is shown. We work in the shared regime ($\rho^{\ell} = 1$; the per-true-task overlay takes $\rho^{r} = 1$); take $\rho^{t} = 0.95$ (byte parity, Section~\ref{sec:byteparity}); set $Q_{\text{27B},\text{NVFP4}}^{(c)} = 0.896$ on all categories (the cited anchor assumed category-invariant; the per-category split is a protocol output); and use the scenario values $Q_{\text{8B},\text{BF16}}^{(\text{simple})} = 0.95$ and $Q_{\text{8B},\text{BF16}}^{(\text{agentic})} = 0.80$ (assumed for illustration only). By the per-passed-task form of Theorem~\ref{thm:crossing} with $\rho^{\ell} = 1$ (shared regime), the crossing condition is $\rho^{t}\rho^{Q} > 1$:
\begin{itemize}
\item \textbf{simple:} $\rho^{Q} = 0.896 / 0.95 = 0.943$, so $\rho^{t}\,\rho^{Q} = 0.95 \times 0.943 = 0.896 < 1$---no crossing: $(\text{8B}, \text{BF16})$ is cheaper per passed task, because the small full-precision model saturates the easy category while the 27B carries the cliff tax (the net quality ratio is below one).
\item \textbf{agentic:} $\rho^{Q} = 0.896 / 0.80 = 1.120$, so $\rho^{t}\,\rho^{Q} = 0.95 \times 1.120 = 1.064 > 1$---crossing: $(\text{27B}, \text{NVFP4})$ is cheaper per passed task (cost ratio $1/1.064 \approx 0.94$, i.e., about 6\% lower cost per passed task), because the tier dividend on the structurally hard category outweighs the 5\% throughput gap.
\end{itemize}
The ladder is thus non-degenerate (prediction P1), crosses the model-only cascade (P2), and inverts by category (P3; all three pre-registered in Section~\ref{sec:protocol}). The sign flip of $\rho_c^{Q}$ across categories---equivalently, the dividend-versus-cliff balance of Corollary~\ref{cor:cliffdiv}---is the whole mechanism.

\section{Measured Feasibility Layer and Pre-Registered Protocol}
\label{sec:measured}

\subsection{Platform and workload}
\label{sec:platform}

The measured node is a single-node NVIDIA H100 NVL (93.6 GB) serving node in the development fleet; the deployment target is the H200 fleet (141 GB, same Hopper architecture; cited on-demand price ratio $p_{h'}/p_{h_0} = 3.50/2.50 = 1.4$, Proposition~\ref{prop:hardware}). The workload is the BFCL-style code-graded suite (AST and state equality, judge-free \citep{bfcl2024leaderboard}) of 12 simple, 8 parallel, and 4 agentic tasks (24 in total), with per-draw token generation budgets simple $= 96$, parallel $= 192$, agentic $= 128$ tokens --- fixed in the single-shot categories (EOS disabled, so every row generates exactly its budget) and a per-turn cap in the agentic category --- under the fixed quality gate $g = 0.85$. The model grid is the Qwen3 dense 4B/8B/14B checkpoints \citep{yang2025qwen3} in BF16, plus the 27B NVFP4 production checkpoint; the low-bit axis is delivered by framework-native quantized checkpoints (Section~\ref{sec:measured:feasibility}).

\subsection{Measured feasibility}
\label{sec:measured:feasibility}

\begin{table}[ht]
\centering
\begin{tabular}{lccc}
\toprule
Checkpoint & Files & Size & Download wall (concurrent) \\
\midrule
4B  & 13 & 8.1 GB  & 171.2 s \\
8B  & 15 & 16.4 GB & 171.1 s \\
14B & 18 & 29.6 GB & 171.0 s \\
\bottomrule
\end{tabular}
\caption{Checkpoint downloads (concurrent fetch) on the single-node NVIDIA H100 NVL (93.6 GB).}
\label{tab:downloads}
\end{table}

\begin{table}[ht]
\centering
\begin{tabular}{lccc}
\toprule
Checkpoint & BF16 load & Peak VRAM & Fraction of 93.6 GB \\
\midrule
4B  & 2.2 s & 7.5 GB  & 8.0\% \\
8B  & 2.9 s & 15.3 GB & 16.4\% \\
14B & 5.0 s & 27.5 GB & 29.4\% \\
\bottomrule
\end{tabular}
\caption{BF16 load time and peak VRAM on the 93.6 GB node; fractions computed from the measured peaks. Total measured wall for the feasibility pass: 189.8 s.}
\label{tab:load}
\end{table}

Two conclusions, with the arithmetic shown.
\textbf{(i) Memory.} The largest measured peak is 27.5 GB of 93.6 GB (29.4\%), leaving $93.6 - 27.5 = 66.1$ GB $= 70.6\%$ of capacity: all measured BF16 variants fit on a single node with at least 70\% headroom. The weights-only arithmetic of Section~\ref{sec:byteparity} extends feasibility to the 27B row: 27B BF16 $= 27 \times 10^{9} \times 2\ \text{B} = 54.0$ GB, 27B FP8 $= 27 \times 10^{9} \times 1\ \text{B} = 27.0$ GB, 27B NVFP4 $= 15.2$ GB; all are below 93.6 GB, so the entire $4 \times 3$ grid is memory-feasible on the H100 NVL and a fortiori on the H200 (Proposition~\ref{prop:hardware}(iii)).
\textbf{(ii) Stack compatibility.} The bitsandbytes 8-bit and 4-bit-nf4 load path is unsupported in the deployed serving stack: all six load attempts (three sizes $\times$ two bitsandbytes precisions) failed to load. Consequence: the feasible low-bit axis on this stack is the framework-native quantized checkpoints---NVFP4 via the production serving stack, W4A16-class re-encodings, measured for MoE checkpoints in \citep{thaki2026precisiontier} and outside the dense grid studied here---so the feasible set $\mathcal{F}$ is determined by stack support, not by memory. The stack status of FP8 re-encodings is a protocol load check; the bitsandbytes path is the only low-bit load path measured unsupported.



\subsection{Cited inputs}
\label{sec:cited}

Two input families enter the frontier by citation rather than metering. \emph{Prices:} public on-demand list prices per GPU-hour---H100 $= \$2.50$, H200 $= \$3.50$, B200 $= \$5.00$, RTX 5070 $= \$0.60$ (H200/H100 ratio $= 1.4$; feeds A1 and Proposition~\ref{prop:hardware}). \emph{Quality anchor:} 27B-NVFP4 $= 89.6\%$ overall on the 800-case BFCL-style suite (2026-09-06; \citep{thaki2026toolcallcliff}); the official BFCL suite and leaderboard \citep{bfcl2024leaderboard} supply the code-graded, judge-free harness.

\subsection{Pre-registered protocol and refutation criteria}
\label{sec:protocol}

\emph{Measurement modes.} (i) Single-shot categories (simple, parallel): lockstep batch, fixed budget, EOS disabled, batch wall attributed evenly (whole-GPU cost). (ii) Agentic category: sequential, concurrency $C = 1$, natural EOS, per-task wall $=$ exclusive GPU seconds. (iii) Throughput probe: concurrency $C = 8$ over $N = 64$ sequences, EOS disabled, aggregate rate $= C \times N$ / wall.

\emph{Falsifiable predictions (pre-registered).}
\textbf{(P1) Ladder non-degeneracy:} the routing entry $(m_c^*, q_c^*)$---equivalently, the argmin of $C_{m,q}^{(c)}$ at the gate $g$---differs across at least two of the three categories; refuted if one variant is optimal for all three.
\textbf{(P2) Cascade crossing:} for at least one category, the optimal point has $q \ne \text{BF16}$ (a low-bit column point is strictly cheaper than the BF16-row optimum); refuted if the BF16 row dominates every category, in which case the frontier reduces to the FrugalGPT cascade \citep{cheng2023frugalgpt}.
\textbf{(P3) Category inversion:} the 27B-NVFP4 versus 8B-BF16 crossing holds where the tier dividend is large (agentic) and breaks where it is small (simple), operationalized by the product form of Theorem~\ref{thm:crossing}; refuted if the crossing set is empty ($S = \varnothing$) or full ($S = \mathscr{C}$).

\emph{Refutation criteria.}
\textbf{(R1)} If the measured per-category $Q$ of 8B-BF16 is at least $Q_{\mathrm{27B},\mathrm{NVFP4}}^{(c)}\,\rho_c^{t}$ in every category --- with the cited overall anchor $Q_{\mathrm{27B},\mathrm{NVFP4}}^{(c)} \approx 0.896$ used as the category-invariant proxy until the protocol splits it --- then the headline 27B-NVFP4 over 8B-BF16 crossing is void on every category; other off-row points (FP8, W4A16-class, smaller low-bit variants) remain protocol outputs, so P2 is refuted only for the headline pair.
\textbf{(R2)} If the argmax effective-passing-throughput point is precision-invariant (BF16) for all categories, the precision-tier axis is redundant under the gate---the regime of \citep{thaki2026precisiontier}.
\textbf{(R3)} If the measured 8-to-4-bit tax jump does not exceed the BF16-to-8-bit tax increment (the 1x superlinearity margin), the superlinear-cliff prediction P2 of \citep{thaki2026toolcallcliff} is refuted and the ladder threshold $(1 + D_c)(1 - \xi_{m',q'}^{(c)})$ of Corollary~\ref{cor:divtax} must be re-derived with a linear cliff.
\textbf{(R4)} If the measured H200 effective passing throughput ratios deviate from the H100 NVL model by more than 20\% in any category, the hardware-transfer bound of Proposition~\ref{prop:hardware} ($\beta_c$ versus the price ratio 1.4) is refuted.



\subsection{Positioning}
\label{sec:positioning}

This is an analytical and positional study with a measured infrastructure-feasibility layer. The measured layer (Sections~\ref{sec:platform}--\ref{sec:cited}) establishes memory feasibility, load behavior, stack compatibility, and the price inputs; the per-cell quality-throughput grid---the actual frontier cells---is a pre-registered protocol output anchored by the cited 27B-NVFP4 row (89.6\% on 800 cases). No new per-cell quality or throughput measurements are reported in this paper. The results of Section~4 hold by construction under A1--A5, and the protocol of Section~\ref{sec:protocol} is designed to identify the regime of Proposition~\ref{prop:regimes} and to settle R1--R4.

\section{Deployable Routing Table and Composition}

The routing table is the object to be deployed: the frontier is defined in closed form, and its cells are populated by the pre-registered protocol. This section gives the routing table, the token-factory composition that keeps it correct when any single lever moves, and the reading that connects it back to the FrugalGPT cascade.

\subsection{The routing rule}

We deploy one routing entry per category. For $c \in \mathscr{C}$,
\begin{equation}\label{eq:routing-entry}
(m_c^*, q_c^*) \;=\; \arg\max_{(m,q)\,\in\, \mathcal{F}}\; E_{m,q}^{(c)}
\qquad \text{subject to} \qquad
Q_{m,q}^{(c)} \ge g,
\end{equation}
with gate $g = 0.85$ (A2) and the tie-break \emph{smaller memory footprint first, then smaller model}: among $(m,q)$ attaining the same $E_{m,q}^{(c)}$ we prefer the lower peak-VRAM checkpoint, and at equal footprint the smaller model.

We compute the routing entry not by re-solving \eqref{eq:routing-entry} from scratch but by \emph{testing the incumbent against each off-row challenger} with the product form of Theorem~2,
\begin{equation}\label{eq:crossingtest}
\rho_c^{\,t}\;\rho_c^{\,Q} \;>\; \rho_c^{\,\ell}
\end{equation}
per passed task; the per-true-task test multiplies the left-hand side by $\rho_c^{\,r}$. For a challenger $(m', q')$ with $m' > m$ and $q' \neq \mathrm{BF16}$ against the incumbent BF16 rung $(m, \mathrm{BF16})$, the ratios are $\rho_c^{\,t} = t_{m',q'}^{(c)}/t_{m,\mathrm{BF16}}^{(c)}$, $\rho_c^{\,Q} = Q_{m',q'}^{(c)}/Q_{m,\mathrm{BF16}}^{(c)}$, $\rho_c^{\,r} = r_{m',q'}^{(c)}/r_{m,\mathrm{BF16}}^{(c)}$, and $\rho_c^{\,\ell} = \ell_{m',q'}^{(c)}/\ell_{m,\mathrm{BF16}}^{(c)}$; $\rho_c^{\,\ell}$ is the cross-variant length ratio; by A3 it factors as the low-bit inflation $\alpha_{m',q'}^{(c)}$ of \citep{thaki2026verbositytax, lian2026quantization} times the within-BF16 length ratio $\ell_{m',\mathrm{BF16}}^{(c)}/\ell_{m,\mathrm{BF16}}^{(c)}$. The code-graded gate is the judge-free BFCL-style pass fraction of \citep{bfcl2024leaderboard}. The incumbent is initialized to the gated BF16-row argmax $m_c^{1D}$ --- the FrugalGPT cascade \citep{cheng2023frugalgpt} is a subroutine of the ladder; at each step every untested off-row point $(m', q')$ is tested --- with $m'$ larger or smaller than the incumbent, since the crossing test $t'Q'/\ell' > tQ/\ell$ is defined for any pair; the best admissible challenger (passing the test and clearing the gate) is promoted as the new incumbent; by finiteness (Remark~\ref{rem:finite}) the process terminates at the argmax of \eqref{eq:routing-entry}.

\begin{table}[t]
\centering
\small
\caption{Deployable routing table (symbolic template). The gate is fixed at $g=0.85$ (A2). Each cell instantiates as the pre-registered protocol measures it; today the only off-row point with a cited quality anchor is $(27\mathrm{B}, \mathrm{NVFP4})$ at $89.6\% \ge g$, pinning the first rung of the ladder.}
\label{tab:routing-table}
\begin{tabular}{l c l l l}
\toprule
Category $c$ & Gate $g$ & Route $(m_c^*, q_c^*)$ & $E_{m,q}^{(c)}$ (tasks/GPU-s) & $\$/\text{passed task} = p_h/E$ \\
\midrule
simple & $0.85$ & $(m_{\mathrm{simple}}^*, q_{\mathrm{simple}}^*)$ & $E^{(\mathrm{simple})}_{m^*, q^*}$ & $p_h/E^{(\mathrm{simple})}_{m^*, q^*}$ \\
parallel & $0.85$ & $(m_{\mathrm{parallel}}^*, q_{\mathrm{parallel}}^*)$ & $E^{(\mathrm{parallel})}_{m^*, q^*}$ & $p_h/E^{(\mathrm{parallel})}_{m^*, q^*}$ \\
agentic & $0.85$ & $(m_{\mathrm{agentic}}^*, q_{\mathrm{agentic}}^*)$ & $E^{(\mathrm{agentic})}_{m^*, q^*}$ & $p_h/E^{(\mathrm{agentic})}_{m^*, q^*}$ \\
\bottomrule
\end{tabular}
\end{table}

Because the off-row cells are not yet measured, the only off-row point carrying a cited quality anchor is $(27\mathrm{B}, \mathrm{NVFP4})$ at $89.6\%$ on the 800-case BFCL-style suite \citep{thaki2026toolcallcliff}, which clears the gate ($89.6\% \ge 85\%$); we pin the first rung of the ladder there. \textbf{Deployment convention.} Until the pre-registered protocol fills the cells, category $c$ routes to $(27\mathrm{B}, \mathrm{NVFP4})$ when \eqref{eq:crossingtest} holds against the incumbent BF16 rung, and otherwise stays on the BF16 row --- the FrugalGPT choice \citep{cheng2023frugalgpt}.

\subsection{Composition with the token factory}

The routing table is one stage of a four-stage composition, and each stage owns exactly one factor of the bill.
\begin{itemize}
\item \textbf{Skill-to-category dispatch.} The zero-token skill router \citep{thaki2026ensamblerouter} dispatches a skill to a category $c \in \mathscr{C}$ \emph{before any LLM call}: it spends no generation tokens and does not enter the per-passed-task bill.
\item \textbf{Category-to-rung dispatch.} The table of Table~\ref{tab:routing-table} dispatches category $c$ to the rung $(m_c^*, q_c^*)$.
\item \textbf{Validate-then-retry, inside the endpoint.} The free deterministic validator \citep{thaki2026validatordividend} sits inside the endpoint and is bill-invariant --- expected draws per passed task is exactly $1/Q$, so the retry budget moves the abstention rate, not the per-passed-task bill --- while setting the true-success overlay $r = 1 - s/Q$; the per-true-task bill is the displayed bill divided by $r$. The per-passed-task bill therefore composes multiplicatively,
\begin{equation}\label{eq:bill}
\text{bill}_{m,q}^{(c)} \;=\; \frac{p_h\,\ell_{m,q}^{(c)}}{t_{m,q}^{(c)}\, Q_{m,q}^{(c)}},
\end{equation}
whose four factors $p_h$, $\ell_{m,q}^{(c)}$, $t_{m,q}^{(c)}$, $Q_{m,q}^{(c)}$ are set by four independent levers (hardware price, length policy, throughput, raw quality). Because each stage owns exactly one factor, a change in any single lever is absorbed in place --- the bill re-scales and the rest of the factory is untouched --- and no re-optimization of the composition is needed; the routing entry is simply re-evaluated by the same argmax \eqref{eq:routing-entry} and crossing test \eqref{eq:crossingtest}.
\item \textbf{Queue-aware tier fallback.} Under queue pressure the system moves to the \emph{next rung in $E$-order} --- the largest-$E$ feasible point under the current load --- not to the next larger model. The ladder is an ordering by effective passing throughput, not by size: a smaller low-bit rung ahead of a larger BF16 rung in $E$-order is a valid fallback destination.
\end{itemize}

\subsection{Reading the ladder}

The FrugalGPT cascade \citep{cheng2023frugalgpt} is one-dimensional: easy queries go to a small model and hard queries escalate to a large one, and the escalation boundary is set by per-token API prices. Under single-node self-hosting that price axis collapses (A1), and the same direction reappears on the model axis with the precision axis riding along: easy categories land on a \emph{small full-precision} rung and hard categories land on a \emph{large low-bit} rung.

On the hard categories the BF16 tier dividend $D_c$ is what pays for the quantization cliff tax $\xi$ of \citep{thaki2026toolcallcliff}: the capability gap between $27\mathrm{B}$ and $8\mathrm{B}$ in BF16 concentrates in the structurally harder categories (A5), and it is that gap --- not the precision drop --- that carries the crossing of Theorem~2. The mechanism is byte parity (Section~5.2): a $27\mathrm{B}$ model at 4.5 effective bits per weight reads 15.2 GB of weights per forward pass against 16.0 GB for an $8\mathrm{B}$ model at 16 bits, more than three times the parameters at essentially the same byte budget. With the bytes per forward pass matched, the throughput ratio $\rho_c^{\,t}$ stays near one, so the crossing condition of Theorem~\ref{thm:crossing} reduces to the quality balance; in the shared regime the challenger wins iff $(1+D_c)(1-\xi) > 1/\rho_c^{\,t}$, whose necessary form is the dividend-versus-cliff inequality $D_c > \xi/(1-\xi)$ of Corollary~\ref{cor:cliffdiv}.

\section{Limitations}

The following six limitations mark the boundary between what is measured here and what the pre-registered protocol (R1--R4) must establish.
\begin{itemize}
\item \textbf{No per-cell grid measurements.} No per-cell quality-throughput grid measurements are reported. The regime of Proposition~1 --- ladder, empty, or low-bit-dominated --- is undetermined until the refutation criteria R1--R4 are executed, and the illustration of Section~5.3 is a scenario, not data.
\item \textbf{Single model family.} All grid points are the Qwen3 dense family \citep{yang2025qwen3}; transfer to MoE families and across families is open, and routing-consistent quantization matters specifically for MoE \citep{park2026vsraq}.
\item \textbf{Hardware transfer.} The measured feasibility layer is on a single-node NVIDIA H100 NVL (93.6 GB) serving node in the development fleet, while the deployment target is the H200 fleet; Proposition~2 is conditional on the measured $\beta_c$ proxies, and refutation R4 guards the hardware transfer.
\item \textbf{Fixed gate.} The gate is fixed at $g = 0.85$. The frontier shape varies with $g$ and the table re-solves in closed form --- the frontier value is decreasing in $g$, and each other operating point re-solves the finite gated argmax (the feasible set is nested in $g$) but requires re-running the protocol.
\item \textbf{Shared-validator regime.} The shared-regime corollaries take $\rho_c^{\,\ell} = 1$ (no length inflation) and, for the per-true-task bill, $\rho_c^{\,r} = 1$ (comparable silent fractions across variants). If low-bit variants inflate draw length or grow silent errors --- token inflation \citep{lian2026quantization}, silent-error growth under compression \citep{jang2026flatscore} --- then $\rho_c^{\,\ell} > 1$ or $\rho_c^{\,r} < 1$ raises the required dividend and shrinks the crossing set: a conservative direction, but the corollary thresholds must be read in that light.
\item \textbf{List-price $p_h$.} Prices are cited public on-demand list prices ($2.50$ H100, $3.50$ H200, $5.00$ B200, $0.60$ RTX~5070 per GPU-hour; H200/H100 ratio $1.4$). Spot, preemptible, or internal energy pricing \citep{vellaisamy2026energy} shift $p_h$ in level but, by Lemma~1, cannot move the ladder ordering.
\end{itemize}



The rendered figures for this work (Figures~\ref{fig:fig1-bf16-vram},~\ref{fig:fig2-bf16-load},~\ref{fig:fig3-byte-parity}) are collected here.

\begin{figure}[htbp]
\centering
\input{figures/fig1-bf16-vram.tex}
\caption{Measured BF16 peak VRAM of the 4B/8B/14B Qwen3 checkpoints on the 93.6 GB node: every variant fits with at least 70 percent headroom, so memory is not the binding constraint on the feasible set.  (Measured on GPU pod)}
\label{fig:fig1-bf16-vram}
\end{figure}

\begin{figure}[htbp]
\centering
\input{figures/fig2-bf16-load.tex}
\caption{Measured BF16 load time grows smoothly from 2.2 s (4B) to 5.0 s (14B) within a 189.8 s total feasibility wall on the 93.6 GB node.  (Measured on GPU pod)}
\label{fig:fig2-bf16-load}
\end{figure}

\begin{figure}[htbp]
\centering
\input{figures/fig3-byte-parity.tex}
\caption{Weights-only arithmetic at the crossing: 27B at 4.5 effective bits per weight (15.2 GB) reads slightly fewer bytes per forward pass than 8B at 16 bits (16.0 GB), keeping the throughput ratio near one; FP8 and BF16 27B are shown for scale.  (Analytical model (not measured): weights-only byte arithmetic as stated in the paper (BF16 = 2 bytes per weight, FP8 = 1 byte per weight, NVFP4 = 4.5 effective bits per weight).)}
\label{fig:fig3-byte-parity}
\end{figure}

\section{Conclusion}

We have upgraded the one-dimensional FrugalGPT model cascade \citep{cheng2023frugalgpt} to a two-dimensional frontier over model tier and weight precision for self-hosted agentic serving, where the price axis collapses and the only free deployment choice is the $(m,q)$ point. For each category the frontier is the argmax of effective passing throughput $E_{m,q}^{(c)} = t_{m,q}^{(c)} Q_{m,q}^{(c)}/\ell_{m,q}^{(c)}$ (passed tasks per GPU-second) under a judge-free BFCL-style code gate, the routing entry is decided by the per-passed-task crossing condition \eqref{eq:crossing} (the per-true-task bill adds the success ratio $\rho_c^{\,r}$), and in the shared regime its necessary form is the cliff-versus-dividend inequality $D_c > \xi/(1-\xi)$ (Corollary~\ref{cor:cliffdiv}). Our central prediction is a category-inverted routing table --- small full-precision on easy categories, large low-bit on hard ones --- whose mechanism is byte parity: a $27\mathrm{B}$ model at 4.5 effective bits per weight carries 15.2 GB of weights against 16.0 GB for an $8\mathrm{B}$ model at 16 bits, so the throughput ratio stays near one and the crossing reduces to a quality balance in which the BF16 tier dividend pays the quantization cliff tax.

The measured feasibility layer is deliberately thin and it carries the study: on a single-node NVIDIA H100 NVL (93.6 GB) serving node the full $4 \times 3$ grid --- four model tiers $\times$ three precisions --- is memory-feasible (the measured BF16 peaks of 4B/8B/14B are 7.5/15.3/27.5 GB, all under one-third of budget, and the 27B tier's low-bit weights are 15.2 GB by byte parity), so it is stack support for each load path, not VRAM, that determines the feasible set $\mathcal{F}$. The deployable routing table composes multiplicatively with the zero-token skill router \citep{thaki2026ensamblerouter}, the free-deterministic-validator retry \citep{thaki2026validatordividend}, and a queue-aware tier fallback that descends the ladder in $E$-order rather than in model size.

The pre-registered protocol (R1--R4) is the next step: filling the per-cell quality-throughput grid determines whether the ladder is non-degenerate, whether a low-bit column point beats the BF16-row optimum, and whether the hardware transfer from the H100 NVL measurement to the H200 fleet holds within $20\%$. We plan to extend the grid to MoE families \citep{park2026vsraq} and to the KV-precision axis \citep{shen2026triaxialkv}, and to add energy accounting \citep{vellaisamy2026energy} to the per-passed-task bill. This paper is an analytical and positional study with a measured feasibility layer; no new per-cell quality or throughput measurements are reported.

\bibliography{refs}
\end{document}