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\newtheorem{theorem}{Theorem}
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\title{The Task Frontier Race\\Horizon, Forgiveness, and Dynamic AI Competition}
\author{}
\date{June 2026}

\begin{document}
\maketitle

\begin{abstract}
\noindent
This paper develops a theory of AI progress in which the primitive object is the value
of raising a task frontier. The frontier has two economically distinct bottlenecks:
horizon, the serial depth of the required action path, and forgiveness, the extent to
which mistakes can be observed, corrected, retried, or absorbed before they create
irreversible loss. A system therefore has two capabilities: one for long dependent
tasks and one for fragile tasks. The model yields three sufficient statistics: the value
of relaxing the horizon boundary, the value of relaxing the fragility boundary, and the
value of relaxing both at once. These statistics explain why AI firms race on different
technologies in different task markets. Runtime, tools, search, and test-time compute
are most valuable where tasks are long but forgiving. Base-model reliability,
verification, and monitoring are most valuable where tasks are fragile. This race
inversion is the paper's central result: a scalar quality ladder can fit either region,
but it cannot discipline the reversal without adding region-specific wedges. The dynamic
application separates productive data flywheels from pure lock-in. Deployment scale
generates durable advantage only when user traces move the scarce task boundary; when
they do not, scale mainly reallocates rents. The model explains why low-cost imitation
can be powerful in forgiving workflows while frontier premia persist in fragile
deployment domains.\\[1ex]
\textbf{JEL:} O31, O33, L13, L40, D24.
\end{abstract}

\section{Introduction}

Frontier AI progress is usually described as movement along a scalar quality ladder:
larger models, lower loss, higher benchmark scores, or more capable agents. That language
is useful but incomplete. It cannot explain why two tasks can look equally difficult on a
benchmark and yet have very different economic adoption thresholds. Coding agents can
make mistakes, run tests, inspect logs, and try again. Product recommendation is often
short and forgiving. Self-driving may involve simple local controls, but a local error
can cause large irreversible harm. Scientific discovery, drug development, and autonomous
organizations can require both long sequences of dependent steps and extremely low
tolerance for hidden error.

The paper's first claim is that the economically relevant task frontier has two
dimensions. The first is horizon: how much a task depends on a long serial chain of
state-dependent actions. The second is forgiveness: how much the task environment allows
mistakes to be retried, monitored, corrected, sandboxed, or reversed. I write the second
coordinate as fragility \(f\), the inverse of forgiveness. A high-\(f\) task is not simply
a task with many steps. It is a task with a narrow acceptable path.

This primitive keeps the model small. Retryability, reversibility, stakes, human review,
monitoring, and required error rates are not extra dimensions. They are mechanisms that
move a task's effective fragility. Tests and rollback make coding more forgiving. Real-time
externalities make driving less forgiving. Clinical harm, unobserved errors, and one-shot
decisions make some medical and scientific tasks less forgiving. The two primitive task
coordinates are horizon and forgiveness.

An AI system has capability vector \(q=(q_H,q_F)\). Horizon capability shifts the
frontier over long serial tasks. Forgiveness capability shifts the frontier over fragile
tasks that require reliability, discipline, monitoring, and low irreversible error. Task
\((h,f)\) succeeds with probability
\[
p(q,h,f)=G_H(q_H-h)G_F(q_F-f).
\]
The multiplicative form says that a task must clear both gates. Long context and planning
do not solve an unforgiving task if reliability is too low. Low local error rates do not
solve a task that is beyond the system's serial depth.

The coordinates are not only labels. In a sequential task, longer chains mechanically
raise the required local reliability, while observability, retries, verification,
rollback, and sandboxing lower it. This gives a formal reason why coding can be long but
forgiving and why driving can be short but fragile. The task frontier is a reduced form
for serial error, recovery, and irreversibility, not just another benchmark score.

Let \(a(h,f)\) be the value density of tasks. Total task surplus is
\[
W(q)=\iint a(h,f)G_H(q_H-h)G_F(q_F-f)\,dh\,df.
\]
The finite-step value of raising the frontier is
\[
A(q,\Delta)=W(q+\Delta)-W(q),\qquad \Delta=(\Delta_H,\Delta_F)\ge0.
\]
The central sufficient statistics are
\[
B_H(q)=\iint a(h,f)g_H(q_H-h)G_F(q_F-f)\,dh\,df,
\]
\[
B_F(q)=\iint a(h,f)G_H(q_H-h)g_F(q_F-f)\,dh\,df,
\]
and
\[
C(q)=\iint a(h,f)g_H(q_H-h)g_F(q_F-f)\,dh\,df.
\]
\(B_H\) is the value of relaxing the horizon boundary. \(B_F\) is the value of relaxing
the fragility boundary. \(C\) is the value mass at the corner where both constraints bind.
The exact interaction surplus from improving both dimensions is
\[
\int_0^{\Delta_H}\int_0^{\Delta_F}C(q_H+u,q_F+v)\,dv\,du.
\]
Thus horizon and reliability are complements exactly when valuable task mass sits near
both boundaries.

The paper's second claim is that firms race over technologies, not directly over \(q_H\)
and \(q_F\). Base-model scaling \(b\) is the foundation-model race: model size,
pretraining, post-training, data mixture, accelerator clusters, and distributed training
systems. Runtime/harnessing \(r\) is the deployed-agent-stack race: test-time compute,
search, tools, memory, verifiers, retrieval, retry loops, agent-computer interfaces, and
workflow integration. Deployment data \(d\) and distillation \(m\) make runtime
cumulative. More users create more task traces, corrections, and successful
trajectories; distillation compresses those trajectories into cheaper prompts, policies,
verifiers, smaller models, and task routines.

This distinction changes the race. Base scaling mostly raises core competence and
per-step reliability. Runtime/harnessing mostly extends effective horizon and can raise
reliability when errors are observable and recoverable. It is not a generic substitute
for a better base model: retries and search help most when there is feedback, a verifier,
or a safe sandbox. Deployment data matters most for runtime because it tells the firm
which tool calls, retries, verifiers, retrieval routines, and workflows actually work.
Distillation is the bridge: runtime discovers expensive successful trajectories, and
distillation turns them into cheaper future capability.

The paper's central result is therefore not that AI quality is two-dimensional in the
abstract. It is that task regions induce different races. Where the scarce boundary is
serial depth and errors are recoverable, firms should race on search, tools, workflows,
and test-time compute. Where the scarce boundary is fragile reliability, firms should
race on base-model quality, verification, monitoring, and rollback. A scalar quality
ladder can fit either pattern in isolation, but it cannot discipline the reversal
without adding region-specific technology or adoption wedges.

The resulting theory speaks to several current AI patterns. Low-cost followers can be
highly valuable where task environments are forgiving because runtime, tools, retries,
and distillation can close much of the gap. Frontier systems retain value where small
base-reliability gaps are multiplied by fragile, high-value deployment environments.
Full-stack tipping is most likely in tasks that are both long and unforgiving, because
the leader's base model makes its harness better, its deployment gives it more task
traces, and distillation lowers future runtime cost. Private racing can also persist when
productive frontier value is small, because firms value market-share shifts over already
installed task value.

The paper is organized around three results. Sections \ref{sec:primitives}--\ref{sec:suffstats}
define the frontier and derive the sufficient statistics \(B_H\), \(B_F\), and \(C\).
Sections \ref{sec:classification}--\ref{sec:technology} use those statistics to derive
race inversion across task regions and the scalar benchmark. Sections \ref{sec:dynamic}
and \ref{sec:data-tipping} apply the same objects to data-distillation tipping and pure
lock-in. The remaining sections collect supporting implications for imitation, stopping,
and frontier-surplus wedges. The appendix verifies each derivation step by step.

\section{Task Primitives}\label{sec:primitives}

The task space is \(\mathcal T\subseteq\mathbb R^2\). A task is indexed by \((h,f)\).
The coordinate \(h\) is horizon: the serial depth of the required action path. The
coordinate \(f\) is fragility, the inverse of forgiveness. Higher \(f\) means that a
task has less slack for retry, correction, observation, or reversible error.

Let an AI system have capability vector
\[
q=(q_H,q_F)\in\mathbb R^2.
\]
The task success probability is
\begin{equation}\label{eq:success}
p(q,h,f)=G_H(q_H-h)G_F(q_F-f),
\end{equation}
where \(G_H,G_F:\mathbb R\to[0,1]\) are continuously differentiable increasing functions
with densities \(g_H=G_H'\) and \(g_F=G_F'\). The shifted form says that capability moves
a boundary through task space.

The multiplicative specification is the baseline separable case. It is useful because it
lets the sufficient statistics be written exactly and transparently. It should not be
read as a claim that engineering systems always separate horizon from reliability. In
applications, long trajectories can amplify error, verifiers can trade latency for
reliability, and scaffolds can introduce new failure modes. Those forces can be modeled
as task-specific shifts in effective \(h\) and \(f\), or as an additional interaction
term around the separable benchmark. The baseline result below says what the boundary
statistics are before adding those engineering interactions.

Let \(a(h,f)\ge0\) be an integrable task-value density. Total task surplus is
\begin{equation}\label{eq:W}
W(q)=\iint_{\mathcal T}a(h,f)G_H(q_H-h)G_F(q_F-f)\,dh\,df.
\end{equation}
For a finite project step \(\Delta=(\Delta_H,\Delta_F)\ge0\), define
\begin{equation}\label{eq:A}
A(q,\Delta)=W(q+\Delta)-W(q).
\end{equation}
This is the agentic raising statistic. It is finite-step by construction because
frontier training, infrastructure, data, evaluation, and deployment projects are lumpy.

\paragraph{Operational content.}
The primitives are meant to be measurable. Horizon \(h\) can be proxied by trajectory
length, dependency depth, number of state-contingent subtasks, context span, and the
number of external actions that must be coordinated before payoff is realized. Fragility
\(f\) can be proxied by required error rate, cost of a bad action, observability of
mistakes, reversibility, rollback availability, verification coverage, and whether
failure creates harm before correction is possible. Capability \(q_H\) is measured by
success on long-horizon tasks holding the error environment fixed. Capability \(q_F\) is
measured by success in low-tolerance tasks holding horizon fixed. This makes the theory
falsifiable: the same model should predict different adoption and race patterns for
coding, product recommendation, driving, clinical support, and scientific discovery
because these tasks occupy different regions of the \((h,f)\) space.

\paragraph{Identification protocol.}
The primitives can be disciplined without turning the paper into an empirical study.
Construct benchmark or deployment cells \(r\) with known value weights \(a_r\) and
observed success rates \(\hat p_{ir}\). Two controlled task families identify the
coordinates. In the horizon family, chain length or dependency depth varies while the
verifier, retry budget, and failure tolerance are held fixed; this shifts \(h_r\) but
not \(f_r\). In the fragility family, retryability, observability, or tolerated failure
probability varies at fixed chain length; this shifts \(f_r\) but not \(h_r\).
Normalizations set one baseline task to \(h=f=0\) and one log-odds unit of difficulty to
one unit of each coordinate. The measurement equation is the success equation
\[
\hat p_{ir}=G_H(q_{iH}-h_r)G_F(q_{iF}-f_r)+\varepsilon_{ir},
\]
with cell error \(\varepsilon_{ir}\). Given the fitted kernels and task values, the
regional scarcity ratio is recovered by plug-in moments,
\[
\widehat\Theta_R
=
\frac{\sum_{r\in R} a_r g_H(\hat q_H-\hat h_r)G_F(\hat q_F-\hat f_r)}
{\sum_{r\in R} a_r G_H(\hat q_H-\hat h_r)g_F(\hat q_F-\hat f_r)}.
\]
Equivalently, \(B_H^R\) and \(B_F^R\) can be recovered from finite differences in
regional value after projects known to move mostly \(H\) or mostly \(F\). The theory is
falsified, not merely renamed, if race direction does not move with
\(\widehat\Theta_R\), or if a scalar model with region-invariant project productivities
matches the same adoption and investment rankings.

\subsection{A Sequential Microfoundation}\label{sec:microfoundation}

The two task coordinates can be derived from a simple sequential production problem.
Consider a task with \(n\) essential stages. Each stage requires one correct local action.
The local probability of a correct action is
\[
\pi(q_F,\varphi)=G_F(q_F-\varphi),
\]
where \(\varphi\) is the underlying local reliability burden. If a mistake is observable
and recoverable, the system can make \(\ell\ge0\) additional attempts before the stage is
lost. The stage success probability is therefore
\[
S_\ell(q_F,\varphi)=1-\{1-G_F(q_F-\varphi)\}^{1+\ell}.
\]
The task also requires enough coordination capacity to handle the serial depth \(n\).
Write this horizon gate as \(G_H(q_H-\log n)\). Conditional on clearing the horizon gate,
the probability that all stages succeed is \(S_\ell(q_F,\varphi)^n\). Thus the exact
sequential success probability is
\[
P(q,n,\ell,\varphi)
=G_H(q_H-\log n)\left[1-\{1-G_F(q_F-\varphi)\}^{1+\ell}\right]^n.
\]
Writing \(h=\log n\), this is
\[
P(q,n,\ell,\varphi)=G_H(q_H-h)\mathcal G_F(q_F-\varphi;h,\ell),
\]
where
\[
\mathcal G_F(u;h,\ell)=\left[1-\{1-G_F(u)\}^{1+\ell}\right]^{\exp(h)}.
\]
The baseline specification in \eqref{eq:success} is the common-kernel version of this
exact product gate. In the exact sequential model, the reliability gate can vary with
task horizon and retry technology.

\begin{theorem}[Sequential reduction to horizon and fragility]\label{t:micro}
Fix a required task-level failure probability \(\varepsilon\in(0,1)\), and suppose
\(G_F\) is strictly increasing. Define
\[
p^*(n,\ell,\varepsilon)
=1-\{1-(1-\varepsilon)^{1/n}\}^{1/(1+\ell)}
\]
and
\[
f(n,\ell,\varepsilon,\varphi)=\varphi+G_F^{-1}(p^*(n,\ell,\varepsilon)).
\]
Then the reliability condition
\[
S_\ell(q_F,\varphi)^n\ge 1-\varepsilon
\]
holds if and only if \(q_F\ge f(n,\ell,\varepsilon,\varphi)\). Moreover \(f\) is
increasing in \(n\) and \(\varphi\), decreasing in the retry budget \(\ell\), and
increasing as the tolerated failure probability \(\varepsilon\) falls.
\end{theorem}

The theorem gives the primitives economic content. Horizon is serial depth,
\(h=\log n\). Fragility is the reliability threshold \(f\). The two coordinates need not
be statistically independent: long tasks can also require higher local reliability. This
is not double counting because \(h\) is the coordination gate and \(f\) is the reliability
gate. Long tasks can still be forgiving when errors are observable and retries are cheap.
Short tasks can be fragile when the tolerated failure probability is extremely low or
mistakes are irreversible. Verification and rollback lower effective fragility by raising
\(\ell\) or by making failures observable; unsafe deployment raises it.

\section{Sufficient Statistics}\label{sec:suffstats}

Define the horizon boundary value
\begin{equation}\label{eq:BH}
B_H(q)=\iint_{\mathcal T}a(h,f)g_H(q_H-h)G_F(q_F-f)\,dh\,df,
\end{equation}
the forgiveness boundary value
\begin{equation}\label{eq:BF}
B_F(q)=\iint_{\mathcal T}a(h,f)G_H(q_H-h)g_F(q_F-f)\,dh\,df,
\end{equation}
and the cross-boundary statistic
\begin{equation}\label{eq:C}
C(q)=\iint_{\mathcal T}a(h,f)g_H(q_H-h)g_F(q_F-f)\,dh\,df.
\end{equation}

\begin{theorem}[Task-frontier sufficient statistics]\label{t:suffstats}
Suppose \(a\in L^1_+(\mathcal T)\), \(G_H\) and \(G_F\) are continuously differentiable,
and \(g_H,g_F\) are bounded and continuous. Then \(W\) is continuously differentiable and
\[
\frac{\partial W(q)}{\partial q_H}=B_H(q),\qquad
\frac{\partial W(q)}{\partial q_F}=B_F(q).
\]
Moreover \(B_H\) is differentiable in \(q_F\), \(B_F\) is differentiable in \(q_H\), and
\[
\frac{\partial B_H(q)}{\partial q_F}
=
\frac{\partial B_F(q)}{\partial q_H}
=C(q)\ge0.
\]
Thus forgiveness capability raises the marginal value of horizon capability by exactly
\(C(q)\), and horizon capability raises the marginal value of forgiveness capability by
the same statistic.
\end{theorem}

\begin{theorem}[Finite-step value and cross-boundary complementarity]\label{t:finite}
Under the assumptions of Theorem \ref{t:suffstats}, for every
\(\Delta=(\Delta_H,\Delta_F)\ge0\),
\begin{align}
A(q,\Delta)
&=\int_0^{\Delta_H}B_H(q_H+u,q_F)\,du
+\int_0^{\Delta_F}B_F(q_H+\Delta_H,q_F+v)\,dv \label{eq:pathHF}\\
&=\int_0^{\Delta_F}B_F(q_H,q_F+v)\,dv
+\int_0^{\Delta_H}B_H(q_H+u,q_F+\Delta_F)\,du. \label{eq:pathFH}
\end{align}
The extra value from doing both improvements rather than adding the two stand-alone
improvements is
\begin{align}
I(q,\Delta)
&=W(q_H+\Delta_H,q_F+\Delta_F)-W(q_H+\Delta_H,q_F) \notag\\
&\quad -W(q_H,q_F+\Delta_F)+W(q_H,q_F) \label{eq:interaction}\\
&=\int_0^{\Delta_H}\int_0^{\Delta_F}C(q_H+u,q_F+v)\,dv\,du\ge0. \notag
\end{align}
As \(\|\Delta\|\downarrow0\),
\[
A(q,\Delta)=\Delta_HB_H(q)+\Delta_FB_F(q)+o(\|\Delta\|).
\]
\end{theorem}

These two theorems are the paper's core. They replace a vague claim that a model is
``better'' with three sufficient statistics. \(B_H\) says whether the scarce dimension is
serial depth. \(B_F\) says whether the scarce dimension is reliability in unforgiving
environments. \(C\) says whether the economy is waiting for systems that can do both.

\section{Classification and the One-Dimensional Special Case}\label{sec:classification}

The two primitives classify tasks without adding more state variables. Relative to the
current frontier, the four regions are:

\begin{center}
\begin{tabular}{>{\raggedright\arraybackslash}p{0.18\linewidth}
                >{\raggedright\arraybackslash}p{0.34\linewidth}
                >{\raggedright\arraybackslash}p{0.34\linewidth}}
\toprule
 & Low fragility / high forgiveness & High fragility / low forgiveness\\
\midrule
Low horizon & Product recommendation, ad copy, search help
& Driving maneuvers, medical dosage, payments control\\
High horizon & Coding, debugging, data analysis, research assistance
& Drug discovery, critical infrastructure, autonomous organizations\\
\bottomrule
\end{tabular}
\end{center}

The classification is not fixed by nature. Task design moves tasks in this space. Tests,
sandboxing, human review, monitoring, rollback, and staged deployment lower effective
fragility. Real-time externalities, hidden errors, one-shot decisions, and delayed
feedback raise it.

The same boundary logic gives a task-level interpretation of the sufficient statistics.
A task matters most for horizon progress when it lies near the horizon boundary and has
already cleared the fragility gate. It matters most for reliability progress when it has
cleared the horizon gate and lies near the fragility boundary. It matters most for
interaction when it sits near both boundaries. This is why the model treats tasks near
the corner differently from tasks that are difficult along only one dimension.

The one-dimensional agentic frontier is a limiting case of this model. If the
forgiveness margin is fixed, already solved, or economically invariant, define
\[
\tilde a(h)=\int_{\mathcal T_f}a(h,f)G_F(q_F-f)\,df.
\]
The relevant surplus for horizon progress becomes
\[
\tilde W(q_H)=\int \tilde a(h)G_H(q_H-h)\,dh,
\]
with boundary statistic
\[
\tilde B(q_H)=\int \tilde a(h)g_H(q_H-h)\,dh.
\]
Thus the original agentic racing model is not wrong. It is the right reduced form when
the scarce margin is horizon alone. The merged paper uses the richer primitive because
frontier AI deployment varies along both margins. The appendix gives the corresponding
task-level derivative and reduction formally.

\section{Race Technologies: Base, Runtime, Data, and Distillation}\label{sec:technology}

The task frontier is described by \(q=(q_H,q_F)\), but firms do not purchase \(q\)
directly. They choose technologies that produce it. Let firm \(i\) have technology vector
\[
x_i=(b_i,s_i,\tau_i,v_i,d_i,m_i).
\]
The component \(b_i\) is base-model stock. The runtime components are search or
test-time compute \(s_i\), tools, retrieval, memory, and workflow integration
\(\tau_i\), and verification, monitoring, and rollback \(v_i\). The component \(d_i\) is
deployment-data stock, and \(m_i\) is distillation stock. Search expands the number of
candidate paths. Tools and workflows expand the action space and make long tasks
executable. Verification and monitoring determine whether runtime effort is reliable
rather than merely longer. Deployment data improves these runtime components because the
firm observes real tasks, failures, corrections, and successful workflows. Distillation
makes expensive runtime behavior cheaper and more automatic, but only when the traces
being distilled are informative and sufficiently clean.

Task capability is produced by
\[
q_{iH}=H(x_i),\qquad q_{iF}=F(x_i).
\]
Base scaling is the foundation-model race: model size, pretraining, post-training, data
mixture, accelerator clusters, and distributed training systems. Runtime/harnessing is
the deployed-agent-stack race: test-time compute, search, tools, memory, verifiers,
retry loops, retrieval, agent-computer interfaces, and workflow integration. The
maintained technology restrictions are
\[
H_b\ge0,\quad F_b>0,\quad H_s,H_\tau>0,\quad F_v>0,\quad
H_d,F_d,H_m,F_m\ge0.
\]
Search and tools are especially important for horizon, so \(H_s\) and \(H_\tau\) are
large. Base scaling and verification are especially important for reliability, so
\(F_b\) and \(F_v\) are large. Unverified search and tool use mainly raise \(H\) and can
even create engineering failure modes outside the baseline monotone case. The
technologies may be complements:
\[
H_{b s},H_{b\tau},H_{b v}\ge0,\qquad F_{b s},F_{b\tau},F_{b v}\ge0.
\]

The economic value of each technology is determined by the boundary it moves. Let
\(q_i=(H(x_i),F(x_i))\). Ignoring strategic rent shifting to isolate task-frontier
creation, the marginal value of any technology component \(y\in\{b,s,\tau,v,d\}\) is
\[
V_y=B_H(q_i)H_y(x_i)+B_F(q_i)F_y(x_i).
\]
If firm \(i\) serves task mass \(M_i\) and distillation also reduces per-task runtime
cost \(c_R(m_i)\), with \(c_R'(m_i)\le0\), then the marginal value of distillation is
\[
V_m=B_H(q_i)H_m(x_i)+B_F(q_i)F_m(x_i)-M_i c_R'(m_i).
\]

This chain rule connects the CS mechanisms to the economic task frontier. Base scaling
is valuable when it moves the scarce task boundary through \(H_b\) or \(F_b\). Runtime is
valuable when search, tools, or verification move the boundary through their \(H\) or
\(F\) derivatives. Deployment data is valuable because it raises the productivity of
runtime. Distillation is valuable because it both raises effective capability and lowers
future runtime cost. The same formula also explains why the object of the race can shift
across markets: the technology with the highest derivative is not the technology with
the highest economic value unless it moves the boundary that is scarce in that task
region.

This technology layer is deliberately close to computer-science language. Scaling laws
and compute-optimal training motivate \(b\) \cite{hoffmann2022}. Test-time compute and
verifier-guided search motivate \(s\) and \(v\) \cite{snell2024}. Tool use, retrieval,
memory, reflection, and agent interfaces motivate \(\tau\) and \(v\)
\cite{schick2023,yang2024}. Software-agent benchmarks show that interface design changes
measured capability \cite{jimenez2023}.

Distillation transfers expensive runtime behavior into cheaper systems.

Market capture matters because it generates data. Let \(M_i\) be the task mass served by
firm \(i\). A simple data law of motion is
\[
d_{i,t+1}=(1-\delta_d)d_{i,t}+\lambda_d M_{i,t}.
\]
Let successful or corrected runtime traces be
\[
T_{i,t}=\iint_{\mathcal R_{i,t}}a(h,f)p(q_i,h,f)\omega(h,f)\,dh\,df,
\]
where \(\mathcal R_{i,t}\) is the region of tasks served by firm \(i\), and
\(\omega(h,f)\) measures how informative a trace is for training, debugging, or
distillation. This weight is high when outcomes are observed, failures are recoverable,
and corrections identify the source of error; it is low when mistakes are hidden,
catastrophic, or legally unusable. Distillation evolves as
\[
m_{i,t+1}=(1-\delta_m)m_{i,t}+\lambda_m T_{i,t}.
\]
Thus runtime/harnessing creates current capability and a future data asset. Distillation
turns that asset into lower cost and more durable task-specific capability. The data
flywheel is strongest for forgiving tasks, where failures are observable, recoverable,
and informative rather than catastrophic.

The model therefore has forward prediction power. First, open or low-cost systems should
catch up fastest on high-horizon but forgiving tasks where tools, tests, retrieval, and
retry loops generate useful feedback. Second, frontier base models should retain larger
premia on fragile tasks unless a reliable external verifier can turn the task into a
forgiving one. Third, deployment scale should matter most when user interactions produce
high-quality traces, not merely when they produce many tokens. Fourth, distillation
should first compress the cost of successful workflows before it fully erases frontier
reliability gaps. Fifth, full-stack tipping should appear where base quality, runtime
harnessing, deployment traces, and distillation are complements.

The sharp prediction is a race inversion across task regions. Let \(R\subseteq\mathcal T\)
be a task region and define regional boundary values
\[
B_H^R(q)=\iint_R a(h,f)g_H(q_H-h)G_F(q_F-f)\,dh\,df,
\]
\[
B_F^R(q)=\iint_R a(h,f)G_H(q_H-h)g_F(q_F-f)\,dh\,df.
\]
The ratio \(\Theta_R(q)=B_H^R(q)/B_F^R(q)\) is the local scarcity of horizon relative to
fragile reliability in that region.

\begin{theorem}[Task-region race inversion and scalar failure]\label{t:inversion}
Consider two same-cost unit projects \(y\) and \(z\). Project \(y\) is more
horizon-intensive and project \(z\) is more reliability-intensive:
\[
d_{yH}>d_{zH},\qquad d_{yF}<d_{zF},
\]
where \(d_y=(d_{yH},d_{yF})\) and \(d_z=(d_{zH},d_{zF})\) are the induced local capability
steps. Suppose \(B_H^R(q),B_F^R(q)>0\). Ignoring strategic rent shifting, project \(y\)
has higher local frontier value than project \(z\) on region \(R\) if and only if
\[
\Theta_R(q)>\frac{d_{zF}-d_{yF}}{d_{yH}-d_{zH}}.
\]
Project \(z\) has higher local frontier value if the inequality is reversed. Hence, if
two task regions lie on opposite sides of this threshold, the privately efficient race
direction reverses across regions.

No scalar capability model with region-invariant same-cost project increments can
generate this reversal at a fixed state. In a scalar model, the local ranking is
proportional to a single regional boundary value times \(\gamma_y-\gamma_z\), whose sign
does not change across regions. This is a restricted scalar benchmark: if the scalar
model is allowed to choose region-specific project productivities or adoption wedges, it
can mimic the reversal, but only by putting the missing task heterogeneity back into the
technology or cost side of the model.
\end{theorem}

This theorem is the disciplining content of the two-dimensional primitive. A runtime
project such as search, tools, and workflow integration can dominate in long forgiving
work because it mainly relaxes the horizon boundary. A base-model or verification
project can dominate in short fragile work because it mainly relaxes the reliability
boundary. A scalar quality ladder can fit either ranking, but not the reversal without
making the technology or costs task-region-specific. That move is exactly the
two-dimensional task frontier in disguised form.

The same inversion can be stated for actual private incentives rather than frontier
value alone. Let regional current revenue be \(\sigma(r)W_R(q)\), where
\[
W_R(q)=\iint_R a(h,f)p(q,h,f)\,dh\,df,
\]
and consider small same-cost projects \(y\) and \(z\). Write
\[
D=(D_H,D_F)=d_y-d_z,\qquad D_H>0>D_F.
\]

\begin{theorem}[Private race inversion with rent shifting]\label{t:private-inversion}
At a fixed rival state, the first-order private value difference between project \(y\)
and project \(z\) on region \(R\) is
\[
\sigma(r)\{B_H^R(q)D_H+B_F^R(q)D_F\}
+\sigma'(r)W_R(q)\,\beta\cdot D.
\]
Hence project \(y\) is privately preferred to project \(z\) on region \(R\) if and only
if
\[
\Theta_R(q)>
\frac{-D_F}{D_H}
-\frac{\sigma'(r)W_R(q)}{\sigma(r)B_F^R(q)}
\frac{\beta\cdot D}{D_H}.
\]
Project \(z\) is privately preferred if the inequality is reversed. Thus rent shifting
does not eliminate the task-region inversion; it shifts the threshold by an observable
market-share term. If two regions lie on opposite sides of their adjusted thresholds,
private investment rankings reverse across regions.
\end{theorem}

The economic intuition is that private competition adds a business-stealing motive on
top of frontier creation. A project can be attractive because it creates new feasible
tasks, because it protects installed revenue, or both. But the business-stealing term
does not erase the task frontier. It only tilts the cutoff. A runtime project still wins
where the scarce boundary is horizon, and a reliability project still wins where the
scarce boundary is fragility, unless market-share incentives are strong enough to move
the region across the cutoff.

\begin{corollary}[Race direction]\label{c:race-direction}
Holding strategic rent shifting fixed, firms race on base scaling when \(V_b\) is high
relative to the marginal cost of base investment. They race on runtime/harnessing when
\(V_x\) is high for some \(x\in\{s,\tau,v\}\) relative to runtime cost. They race for
deployment and market capture when \(V_d\) is high. They distill when \(V_m\) is high.
Full-stack racing is most
valuable when base and runtime are technologically complementary and task mass lies near
both boundaries, so that \(C(q)\) is large.
\end{corollary}

The corollary is the investment interpretation of the model. It maps the same sufficient
statistics into observable races: GPU-intensive base scaling, inference-time harnessing,
market capture for data, and distillation.

\section{Low-Cost Imitation and Frontier Reliability}\label{sec:imitation}

Let \(L\) be a frontier system and \(M\) a lower-cost follower. Suppose
\[
q^L\ge q^M
\]
component by component. For a task region \(R\subseteq\mathcal T\), define delivered
task value
\[
V_i(R)=\iint_Ra(h,f)p(q^i,h,f)\,dh\,df,\qquad i\in\{L,M\}.
\]
Let \(c_i(R)\) be the cost of serving the region and
\[
\kappa(R)=c_L(R)-c_M(R)
\]
be the follower's cost advantage.

For any task region \(R\), the lower-cost follower is preferred on delivered net value
exactly when its cost advantage exceeds the frontier system's performance advantage:
\begin{equation}\label{eq:sorting}
\kappa(R)\ge
\iint_Ra(h,f)\{p(q^L,h,f)-p(q^M,h,f)\}\,dh\,df.
\end{equation}
If the performance gap on \(R\) is uniformly bounded above by \(\varepsilon\), then the
follower is preferred whenever
\[
\kappa(R)\ge\varepsilon\iint_Ra(h,f)\,dh\,df.
\]
If the performance gap on \(R\) is uniformly bounded below by \(\eta>0\), then the
frontier system is preferred whenever
\[
\eta\iint_Ra(h,f)\,dh\,df>\kappa(R).
\]

This is the economics of imitation. Cheap followers win where the remaining performance
gap is small relative to cost savings. That is most likely on forgiving tasks or tasks
well behind both boundaries, because runtime tools, retries, and distillation can
substitute for some base-model gap. Frontier systems retain value where the performance
gap lies near a valuable fragile boundary, because runtime cannot fully repair errors
that are irreversible, hidden, or costly. The appendix verifies the sorting condition
step by step.

Current AI markets illustrate the distinction without being necessary for the theorem.
Open or lower-cost models can reproduce workflows and interfaces and perform strongly on
many agentic and coding benchmarks \cite{glm2025,zai2026}.

Runtime work shows that inference procedures and harness design can change task
performance without changing the base model \cite{wei2022}. Self-consistency,
reasoning-and-acting agents, and agent-computer interfaces push the same point in more
agentic settings \cite{wang2022,yao2022,yang2024}. Distillation can compress expensive
runtime traces into cheaper future systems \cite{hinton2015,hsieh2023}. Self-generated
reasoning and instruction data provide a related feedback channel. The model's point is
not to rank specific systems. It is to distinguish the GPU-cluster base race from the
deployment-data runtime race.

\section{A Two-Period Race Decomposition}\label{sec:twoperiod}

There are two firms. Firm \(i\) has capability \(q_i\), and firm \(j\) has capability
\(q_j\). Relative market appeal is
\[
r_i=\beta\cdot(q_i-q_j),\qquad \beta=(\beta_H,\beta_F)\ge0.
\]
Let \(\sigma(r)\in(0,1)\) be firm \(i\)'s market share, increasing in \(r\). Current
revenue is
\[
\pi_i(q_i,q_j)=\sigma(r_i)W(q_i).
\]
Firm \(i\)'s project is \(\Delta_i\ge0\). The rival either invests in project
\(\Delta_j\) or does not, \(a_j\in\{0,1\}\). The gross private benefit from investing
rather than not investing, holding \(a_j\) fixed, is
\[
\begin{aligned}
P_i
&=\sigma(r_i+\beta\cdot\Delta_i-a_j\beta\cdot\Delta_j)W(q_i+\Delta_i)\\
&\quad-\sigma(r_i-a_j\beta\cdot\Delta_j)W(q_i).
\end{aligned}
\]

For every \(q_i,q_j,\Delta_i,\Delta_j\), and \(a_j\in\{0,1\}\), the gross private
benefit decomposes into task-frontier creation and rent shifting:
\begin{align}
P_i
&=\sigma(r_i+\beta\cdot\Delta_i-a_j\beta\cdot\Delta_j)A(q_i,\Delta_i) \notag\\
&\quad+\{\sigma(r_i+\beta\cdot\Delta_i-a_j\beta\cdot\Delta_j)
-\sigma(r_i-a_j\beta\cdot\Delta_j)\}W(q_i). \label{eq:race-decomp}
\end{align}

The first term is task-frontier creation. The second term is rent shifting over the
installed task stock. Locally, the productive component is governed by
\[
\Delta_{iH}B_H(q_i)+\Delta_{iF}B_F(q_i).
\]
The joint value of a project that raises both dimensions includes the cross-boundary
interaction \(I(q_i,\Delta_i)\).

Let \(\alpha_i=\beta\cdot\Delta_i\) and \(\alpha_j=\beta\cdot\Delta_j\). The current
payoff incentive for firm \(i\) to invest is higher when the rival invests than when the
rival does not invest exactly when
\begin{align}
&\{\sigma(r_i+\alpha_i-\alpha_j)-\sigma(r_i+\alpha_i)
-\sigma(r_i-\alpha_j)+\sigma(r_i)\}W(q_i) \notag\\
&\quad\ge
\{\sigma(r_i+\alpha_i)-\sigma(r_i+\alpha_i-\alpha_j)\}A(q_i,\Delta_i).
\label{eq:strategic}
\end{align}

The left side is the change in defensive rent preservation over \(W(q_i)\). The right
side is the lost preemption value from no longer being the only firm to move. Defensive
racing is most likely when the installed task stock is large relative to the new
frontier increment.

\section{The Infinite-Horizon Markov Race}\label{sec:dynamic}

The formal race is a finite-state discounted stochastic game. The state is
\[
s=(b_1,r_1,d_1,m_1,b_2,r_2,d_2,m_2,z),
\]
where \(z\) is a cost state. The induced capability vector is
\[
q_i(s)=\{H(b_i,s_i,\tau_i,v_i,d_i,m_i),F(b_i,s_i,\tau_i,v_i,d_i,m_i)\}.
\]
Here \(r_i=(s_i,\tau_i,v_i)\). Each firm chooses a project \(a\in\mathcal M\). Project
\(0\) is no investment. A project
can be a base-scaling project, runtime/harness project, deployment/data project, or
distillation project. It changes the firm's technology state and therefore induces a
capability step
\[
\Delta_{i,a}(s)=q_i(T_i(s,a))-q_i(s),
\]
where \(T_i(s,a)\) denotes firm \(i\)'s own post-project technology state before rival
updates. Project \(a\) costs \(K_a(z)\). Current payoff is
\[
u_i(s,a,n)
=
\sigma\!\left(\beta\cdot[q_i+\Delta_{i,a}(s)-q_j-\Delta_{j,n}(s)]\right)
W(q_i+\Delta_{i,a}(s))
-K_a(z),
\]
where \(n\) is the rival's project and \(q_i,\Delta_{i,a},\Delta_{j,n}\) are evaluated
at state \(s\). A transition rule \(T(s,a,n)\) updates base stocks, runtime stocks, data stocks,
distillation stocks, and the cost state. The discount factor is \(\delta\in(0,1)\).

\begin{theorem}[Dynamic race pressure]\label{t:dynamic}
On any finite state grid and finite project menu, the discounted stochastic game has a
stationary mixed Markov-perfect equilibrium. For a continuation value \(V_i\), the payoff
gain from project \(a\) rather than no investment against rival project \(n\) is
\[
\Phi_i^V(s,a,n)-K_a(z),
\]
where
\begin{align}
\Phi_i^V(s,a,n)
&=
\sigma\!\left(\beta\cdot[q_i+\Delta_{i,a}-q_j-\Delta_{j,n}]\right)A(q_i,\Delta_{i,a}) \notag\\
&\quad+
\left\{\sigma\!\left(\beta\cdot[q_i+\Delta_{i,a}-q_j-\Delta_{j,n}]\right)
-\sigma\!\left(\beta\cdot[q_i-q_j-\Delta_{j,n}]\right)\right\}W(q_i) \notag\\
&\quad+
\delta\{V_i(T(s,a,n))-V_i(T(s,0,n))\}. \label{eq:dynamic-pressure}
\end{align}
In equilibrium, every project used with positive probability maximizes the expected net
gain over the rival's mixed project choice.
\end{theorem}

This theorem is the Markov version of the two-period decomposition. It keeps the
economic objects visible. The first line is frontier creation. The second line is rent
shifting. The third line is continuation value. The result matters because it shows that
the same sufficient statistics still organize a fully dynamic race: dynamics add
continuation values, but they do not change which task boundary a project moves.

For binary investment, write \(\Phi_L(a_F)\) and \(\Phi_F(a_L)\) for the race pressure
of the leader and follower against the rival action. Let \(K\) be the project cost. At
any state, holding continuation values fixed, the pure-action continuation game has the
following equilibrium regimes:
\begin{align*}
(0,0)\text{ is an equilibrium}
&\iff
\Phi_L(0)\le K \text{ and } \Phi_F(0)\le K,\\
(1,0)\text{ is an equilibrium}
&\iff
\Phi_L(0)\ge K \text{ and } \Phi_F(1)\le K,\\
(0,1)\text{ is an equilibrium}
&\iff
\Phi_L(1)\le K \text{ and } \Phi_F(0)\ge K,\\
(1,1)\text{ is an equilibrium}
&\iff
\Phi_L(1)\ge K \text{ and } \Phi_F(1)\ge K.
\end{align*}
The preemption region in which a firm wants to invest only if the other does not is
characterized by \(\Phi_i(0)\ge K\ge\Phi_i(1)\).

The economic point is that race intensity is not monotone in technological opportunity
alone. It also depends on whether investment is defensive, preemptive, or productive at
the current state.

\subsection{A Data-Distillation Tipping Threshold}\label{sec:data-tipping}

The Markov formulation above is general. A simple one-dimensional restriction gives the
core comparative static. Consider a symmetric market with total task mass \(M\). Let
\(x_t=\beta\cdot(q_{1t}-q_{2t})\) be the market-relevant appeal gap and let firm 1 serve
share \(\sigma(x_t)\), where \(\sigma(0)=1/2\), \(\sigma'(0)>0\), and
\(\sigma(-x)=1-\sigma(x)\). Suppose ordinary catch-up shrinks gaps at rate
\(\rho\in[0,1]\). At the symmetric state, a marginal unit of served task mass raises next
period data by \(\lambda_d\) and raises next period distillation stock by
\(\lambda_m\bar\omega(q)\), where
\[
\bar\omega(q)=\frac{1}{M}\iint_{\mathcal R}a(h,f)p(q,h,f)\omega(h,f)\,dh\,df
\]
is the average usable trace intensity in the served task region. Let
\[
q_d=(H_d,F_d),\qquad q_m=(H_m,F_m)
\]
be the capability derivatives with respect to deployment data and distillation at the
symmetric technology state. One unit of served mass therefore changes next-period appeal
by
\begin{equation}\label{eq:chi-structural}
\chi(q)=\lambda_d\,\beta\cdot q_d+\lambda_m\bar\omega(q)\,\beta\cdot q_m.
\end{equation}
The gap evolves as
\begin{equation}\label{eq:data-tipping-map}
x_{t+1}=(1-\rho)x_t+\chi(q)M\{2\sigma(x_t)-1\}.
\end{equation}

\begin{theorem}[Data-distillation tipping threshold]\label{t:data-tipping}
The symmetric state \(x=0\) is locally stable if
\[
2\chi(q) M\sigma'(0)<\rho
\]
and locally unstable if
\[
2\chi(q) M\sigma'(0)>\rho.
\]
For logit demand \(\sigma(x)=1/(1+\exp(-\kappa x))\), the tipping threshold is
\[
\chi(q) M>\frac{2\rho}{\kappa}.
\]
\end{theorem}

This is the structural data flywheel. Market scale alone does not guarantee tipping.
Tipping requires market scale \(M\), trace productivity \(\bar\omega(q)\), data and
distillation productivities \((\lambda_d,\lambda_m)\), and demand sensitivity
\(\sigma'(0)\) to overcome ordinary catch-up \(\rho\). The condition predicts tipping in
tasks where deployment produces useful corrected trajectories and distillation can
compress them into future capability. It predicts weaker tipping in fragile tasks where
failures are hidden, catastrophic, or legally unavailable for learning, even if the task
market is large.

The same condition also distinguishes productive learning from pure share lock-in. On a
served region \(R\), one marginal unit of served task mass creates local task-frontier
value
\[
\zeta_R(q)=
\lambda_d\{B_H^R(q)H_d+B_F^R(q)F_d\}
+\lambda_m\bar\omega_R(q)\{B_H^R(q)H_m+B_F^R(q)F_m\},
\]
where \(\bar\omega_R\) is the average usable trace intensity on \(R\). If the tipping
inequality holds but \(\zeta_R(q)\) is below the opportunity cost of frontier learning,
scale mainly reallocates market share. If both the tipping inequality and the productive
learning condition below hold, the leader's market share also creates frontier value.
For finite data or distillation steps, the extra joint value is given by the
cross-boundary statistic \(C\) in Theorem \ref{t:finite}.

Formally, let
\[
\chi_R(q)=\lambda_d\,\beta\cdot q_d+\lambda_m\bar\omega_R(q)\,\beta\cdot q_m
\]
and define the region-level lock-in index
\[
L_R(q)=2\chi_R(q)M_R\sigma'(0)-\rho.
\]
Let \(\nu\ge0\) be the opportunity cost of one marginal unit of frontier learning on
region \(R\).

\begin{proposition}[Productive tipping versus pure lock-in]\label{p:productive-tipping}
On region \(R\), the symmetric state is locally stable if \(L_R(q)<0\) and locally
unstable if \(L_R(q)>0\). Conditional on local instability, scale creates productive
tipping if \(\zeta_R(q)>\nu\) and pure share lock-in if \(\zeta_R(q)\le\nu\). If
\(L_R(q)<0\) but \(\zeta_R(q)>\nu\), deployment creates productive learning without
local market tipping.
\end{proposition}

This proposition separates two mechanisms that are often conflated. A large installed
base can make a leader harder to displace without improving the frontier. Conversely, a
deployment channel can be socially productive even without local monopoly tipping if it
generates traces that raise the scarce horizon or fragility boundary.

\section{Tipping and Stopping}\label{sec:tipping}

The vector model needs a scalar notion of market-relevant lead. Let
\[
x=\beta\cdot(q_L-q_F)\ge0
\]
be the leader's appeal gap. Suppose catch-up compresses the inherited gap by fraction
\(\rho\in[0,1]\). If the leader invests, its appeal rises by
\(\alpha_L=\beta\cdot\Delta_L\). If the follower invests, its appeal rises by
\(\alpha_F=\beta\cdot\Delta_F\). On a region where labels do not switch, the next appeal
gap is
\[
x'=(1-\rho)x+\alpha_La_L-\alpha_Fa_F.
\]

On the identity-preserving region,
\[
\mathbb E[x'-x\mid s]
=-\rho x+\alpha_Lp_L(s)-\alpha_Fp_F(s),
\]
where \(p_L(s)\) and \(p_F(s)\) are the equilibrium investment probabilities. If the
selected pure regime is leader-only racing, \((a_L,a_F)=(1,0)\), the appeal gap expands
iff \(\alpha_L>\rho x\). If the selected pure regime is mutual racing and
\(\alpha_L=\alpha_F\), the gap contracts at rate \(\rho x\). If the selected pure regime
is follower-only racing, the gap contracts by \(\rho x+\alpha_F\). Leader tipping is
therefore possible only when the leader-only regime conditions hold and the leader's
market-relevant project step exceeds catch-up.

This expression gives the economic content of tipping. A lead does not grow simply
because a leader is ahead. It grows when the equilibrium race selects leader-only
investment and the leader's market-relevant step is large enough to dominate ordinary
catch-up.

Stopping has a different logic. Suppose \(\sigma\) is Lipschitz with constant
\(L_\sigma\). Consider a sequence of states
and projects with project step \(\Delta_n\), costs \(K_n\ge\underline K>0\), and
continuation differences bounded by
\[
|V_i(T(s_n,m,n_j))-V_i(T(s_n,0,n_j))|\le L_V\|\Delta_n\|_1.
\]
Let \(\bar B_n\) bound \(B_H+B_F\) on all own-capability rectangles crossed by the
project, and let \(\bar W_n\) bound current task surplus. If
\[
\|\Delta_n\|_1\{\bar B_n+L_\sigma\|\beta\|_\infty\bar W_n+\delta L_V\}\to0,
\]
then the maximum dynamic race pressure from the project converges to zero. Hence, for
all sufficiently large \(n\), no positive project is played in any best response.

Conversely, productive frontier value can vanish while private racing persists. If
\(A(q,\Delta)\) is arbitrarily small but the rent-shifting term in
\eqref{eq:race-decomp} exceeds the project cost, investment is privately profitable even
though productive frontier creation is negligible.

Stopping is not caused by scaling slowdown alone. It occurs when the finite-step
frontier value, the rent-shifting value, and the continuation value are all small
relative to lumpy costs. Private racing can persist after productive frontier value has
become small if the installed stock \(W(q)\) is large and market share is sensitive to
relative capability.

\section{Private and Social Race Incentives}\label{sec:welfare}

In a two-dimensional capability space, the best model need not dominate in every
coordinate. A social planner can route each task to the model that gives the highest
success probability. Define frontier task surplus for two systems by
\begin{equation}\label{eq:social-frontier}
S(q_1,q_2)=\iint_{\mathcal T}a(h,f)\max\{p(q_1,h,f),p(q_2,h,f)\}\,dh\,df.
\end{equation}
This isolates task-frontier surplus. It omits consumer surplus from competition, safety
externalities, data externalities, and compute-market incidence; those can be added
around the frontier component.

The result in this section is deliberately narrow. It is not a complete welfare theorem
for AI competition. It separates the frontier-surplus channel from other channels that
may make follower entry socially valuable, including price competition, diffusion,
resilience, and safety externalities.

Suppose the leader dominates the follower component by component, \(q_L\ge q_F\), and
the leader project is \(\Delta\ge0\). Then the social frontier gain from a leader project
is
\[
A(q_L,\Delta).
\]
If the follower project \(\Delta_F\) leaves the follower dominated,
\[
q_F+\Delta_F\le q_L,
\]
then its social frontier gain, as measured by \(S\), is zero. Yet the follower's private
current gain can be positive:
\[
\sigma(\beta\cdot[q_F+\Delta_F-q_L])A(q_F,\Delta_F)
+\{\sigma(\beta\cdot[q_F+\Delta_F-q_L])-\sigma(\beta\cdot[q_F-q_L])\}W(q_F)>0.
\]
Thus non-frontier catch-up can be privately profitable even when it does not raise the
best available task frontier.

This wedge does not say that follower catch-up is socially worthless. Diffusion,
resilience, price competition, and consumer surplus may be valuable. It says that those
benefits are distinct from raising the task frontier. The decomposition separates
frontier creation, business stealing, and socially valuable diffusion.

\section{Conclusion}

AI progress is not a scalar quality ladder. The economically relevant frontier moves
through a two-dimensional task space: horizon and forgiveness. Those coordinates can be
microfounded by serial tasks with error accumulation, retries, monitoring, rollback, and
irreversibility. The sufficient statistics for frontier movement are \(B_H\), \(B_F\),
and \(C\). They say whether the next valuable step is longer agency, higher reliability
in unforgiving environments, or a joint advance at the corner.

The central economic implication is race inversion. The same economy can rationally race
on runtime, tools, and test-time compute in one task region and on base reliability or
verification in another. Market scale then has two distinct effects: it can create pure
share lock-in, or it can create productive learning when served tasks produce usable
traces that move the scarce boundary. The one-dimensional agentic-racing model is the
special case in which horizon is the only scarce task coordinate and the technology
layer can be collapsed into a single capability step.

\appendix
\section{Step-by-Step Mathematical Checks}\label{app:proofs}

\subsection{Sequential Microfoundation}

Let \(\pi=G_F(q_F-\varphi)\). With \(\ell\) retries, a stage fails only if all
\(1+\ell\) attempts fail, so
\[
S_\ell(q_F,\varphi)=1-(1-\pi)^{1+\ell}.
\]
The reliability requirement is
\[
S_\ell(q_F,\varphi)^n\ge 1-\varepsilon.
\]
Taking the \(n\)th root and rearranging,
\[
1-(1-\pi)^{1+\ell}\ge (1-\varepsilon)^{1/n},
\]
\[
(1-\pi)^{1+\ell}\le 1-(1-\varepsilon)^{1/n},
\]
and therefore
\[
\pi\ge 1-\{1-(1-\varepsilon)^{1/n}\}^{1/(1+\ell)}
=p^*(n,\ell,\varepsilon).
\]
Since \(G_F\) is strictly increasing, this is equivalent to
\[
q_F\ge \varphi+G_F^{-1}(p^*(n,\ell,\varepsilon)).
\]
The monotonicity follows from the expression for \(p^*\). Increasing \(n\) lowers
\(1-(1-\varepsilon)^{1/n}\), which raises \(p^*\). Increasing \(\ell\) lowers \(p^*\).
Lowering the tolerated failure probability \(\varepsilon\) also raises \(p^*\). The
effect of \(\varphi\) is direct.

\subsection{Sufficient Statistics}

Start with
\[
W(q)=\iint a(h,f)G_H(q_H-h)G_F(q_F-f)\,dh\,df.
\]
The assumptions dominate the integrand and the derivative integrands. Differentiating
under the integral sign,
\[
\frac{\partial}{\partial q_H}\{G_H(q_H-h)G_F(q_F-f)\}
=g_H(q_H-h)G_F(q_F-f),
\]
so
\[
\frac{\partial W(q)}{\partial q_H}=B_H(q).
\]
Similarly,
\[
\frac{\partial W(q)}{\partial q_F}=B_F(q).
\]
Differentiating \(B_H\) in \(q_F\) gives
\[
\frac{\partial B_H(q)}{\partial q_F}
=\iint a(h,f)g_H(q_H-h)g_F(q_F-f)\,dh\,df=C(q).
\]
Differentiating \(B_F\) in \(q_H\) gives the same expression. Since all terms are
nonnegative, \(C(q)\ge0\).

\subsection{Finite-Step Value}

Move first in the horizon direction:
\[
W(q_H+\Delta_H,q_F)-W(q_H,q_F)
=\int_0^{\Delta_H}B_H(q_H+u,q_F)\,du.
\]
Then move in the forgiveness direction:
\[
W(q_H+\Delta_H,q_F+\Delta_F)-W(q_H+\Delta_H,q_F)
=\int_0^{\Delta_F}B_F(q_H+\Delta_H,q_F+v)\,dv.
\]
Adding gives \eqref{eq:pathHF}. Reversing the path gives \eqref{eq:pathFH}.

For the interaction term,
\[
\begin{aligned}
I(q,\Delta)
&=\{W(q_H+\Delta_H,q_F+\Delta_F)-W(q_H,q_F+\Delta_F)\}\\
&\quad-\{W(q_H+\Delta_H,q_F)-W(q_H,q_F)\}.
\end{aligned}
\]
Using the horizon derivative,
\[
I(q,\Delta)
=\int_0^{\Delta_H}\{B_H(q_H+u,q_F+\Delta_F)-B_H(q_H+u,q_F)\}\,du.
\]
Using the forgiveness derivative of \(B_H\),
\[
B_H(q_H+u,q_F+\Delta_F)-B_H(q_H+u,q_F)
=\int_0^{\Delta_F}C(q_H+u,q_F+v)\,dv.
\]
Substituting gives \eqref{eq:interaction}. The local expansion follows from
differentiability of \(W\).

\subsection{Local Salience}

The task-level integrand is
\[
a(h,f)G_H(q_H-h)G_F(q_F-f).
\]
Differentiating with respect to \(q_H\) and \(q_F\) gives the two first-order terms in
the local salience expression in Section \ref{sec:classification}. The cross derivative
of the integrand is
\[
a(h,f)g_H(q_H-h)g_F(q_F-f),
\]
which gives the small-rectangle interaction.

\subsection{One-Dimensional Reduction}

If \(q_F\) is fixed, then
\[
W(q_H,q_F)=\int\left[\int a(h,f)G_F(q_F-f)\,df\right]G_H(q_H-h)\,dh.
\]
The bracketed term is \(\tilde a(h)\). Differentiating this one-dimensional surplus with
respect to \(q_H\) gives \(\tilde B(q_H)\). This proves the one-dimensional reduction in
Section \ref{sec:classification}.

\subsection{Technology Race Values}

The induced capability vector is
\[
q_i(b_i,s_i,\tau_i,v_i,d_i,m_i)
=\{H(b_i,s_i,\tau_i,v_i,d_i,m_i),F(b_i,s_i,\tau_i,v_i,d_i,m_i)\}.
\]
By Theorem \ref{t:suffstats}, the gradient of \(W\) with respect to \(q\) is
\[
\nabla_q W(q_i)=(B_H(q_i),B_F(q_i)).
\]
For any technology component \(y\in\{b,s,\tau,v,d\}\), the chain rule gives
\[
\frac{\partial W(q_i)}{\partial y_i}
=B_H(q_i)H_y(x_i)+B_F(q_i)F_y(x_i).
\]
For distillation,
\[
\frac{\partial}{\partial m_i}\{W(q_i)-M_i c_R(m_i)\}
=B_H(q_i)H_m(x_i)+B_F(q_i)F_m(x_i)-M_i c_R'(m_i).
\]
This proves the technology value formulas in Section \ref{sec:technology}.
Corollary \ref{c:race-direction} follows by comparing each marginal value with the
corresponding marginal project cost. The statement about full-stack racing follows
because technological complementarity raises the relevant derivatives of \(H\) and
\(F\), while Theorem \ref{t:finite} shows that task-side complementarity is governed by
the cross-boundary statistic \(C(q)\).

\subsection{Task-Region Race Inversion}

On region \(R\), the local frontier value of project \(y\) is
\[
B_H^R(q)d_{yH}+B_F^R(q)d_{yF},
\]
and the corresponding value of project \(z\) is
\[
B_H^R(q)d_{zH}+B_F^R(q)d_{zF}.
\]
Project \(y\) is preferred if and only if
\[
B_H^R(q)(d_{yH}-d_{zH})>B_F^R(q)(d_{zF}-d_{yF}).
\]
Since \(d_{yH}>d_{zH}\) and \(B_F^R(q)>0\), division gives
\[
\frac{B_H^R(q)}{B_F^R(q)}
>\frac{d_{zF}-d_{yF}}{d_{yH}-d_{zH}}.
\]
The reverse inequality gives preference for \(z\). If two regions have ratios on
opposite sides of the threshold, the ranking reverses.

In a scalar model with same-cost region-invariant increments, the local value difference
between \(y\) and \(z\) on region \(R\) is
\[
\tilde B^R(\bar q)(\gamma_y-\gamma_z).
\]
Because \(\tilde B^R(\bar q)\ge0\), the sign is the sign of
\(\gamma_y-\gamma_z\) and cannot reverse across regions. This proves Theorem
\ref{t:inversion}.

\subsection{Private Race Inversion}

Regional private revenue after a small project \(d\) is
\[
\sigma(r+\beta\cdot d)W_R(q+d).
\]
The first-order expansion around \(d=0\) is
\[
\sigma(r)W_R(q)
+\sigma(r)\{B_H^R(q)d_H+B_F^R(q)d_F\}
+\sigma'(r)W_R(q)\beta\cdot d.
\]
Subtracting the expansion for project \(z\) from that for project \(y\) gives
\[
\sigma(r)\{B_H^R(q)D_H+B_F^R(q)D_F\}
+\sigma'(r)W_R(q)\beta\cdot D.
\]
Project \(y\) is preferred if this expression is positive. Rearranging and dividing by
\[
\sigma(r)B_F^R(q)D_H>0
\]
gives the stated threshold. The reverse inequality gives preference for \(z\). This
proves Theorem \ref{t:private-inversion}.

\subsection{Cost-Performance Sorting}

The follower is preferred iff
\[
V_M(R)-c_M(R)\ge V_L(R)-c_L(R).
\]
Rearranging gives
\[
c_L(R)-c_M(R)\ge V_L(R)-V_M(R),
\]
which is \eqref{eq:sorting}. The uniform upper and lower gap conditions follow by
bounding \(p(q^L,h,f)-p(q^M,h,f)\) on \(R\).

\subsection{Race Decomposition}

By definition,
\[
\begin{aligned}
P_i
&=\sigma(r_i+\beta\cdot\Delta_i-a_j\beta\cdot\Delta_j)W(q_i+\Delta_i)\\
&\quad-\sigma(r_i-a_j\beta\cdot\Delta_j)W(q_i).
\end{aligned}
\]
Add and subtract
\[
\sigma(r_i+\beta\cdot\Delta_i-a_j\beta\cdot\Delta_j)W(q_i).
\]
This gives \eqref{eq:race-decomp} because
\[
A(q_i,\Delta_i)=W(q_i+\Delta_i)-W(q_i).
\]

\subsection{Strategic Interaction}

Let \(\alpha_i=\beta\cdot\Delta_i\) and \(\alpha_j=\beta\cdot\Delta_j\). The incentive
when the rival invests is
\[
P_i(1)=\sigma(r_i+\alpha_i-\alpha_j)W(q_i+\Delta_i)
-\sigma(r_i-\alpha_j)W(q_i).
\]
The incentive when the rival does not invest is
\[
P_i(0)=\sigma(r_i+\alpha_i)W(q_i+\Delta_i)-\sigma(r_i)W(q_i).
\]
Subtracting and writing \(W(q_i+\Delta_i)=W(q_i)+A(q_i,\Delta_i)\) gives
\[
\begin{aligned}
P_i(1)-P_i(0)
&=\{\sigma(r_i+\alpha_i-\alpha_j)-\sigma(r_i+\alpha_i)\}A(q_i,\Delta_i)\\
&\quad+\{\sigma(r_i+\alpha_i-\alpha_j)-\sigma(r_i+\alpha_i)
-\sigma(r_i-\alpha_j)+\sigma(r_i)\}W(q_i).
\end{aligned}
\]
Rearranging gives \eqref{eq:strategic}.

\subsection{Dynamic Race Pressure and Regimes}

For fixed rival project \(n\), the current payoff difference between project \(a\) and
no investment is
\[
\begin{aligned}
&\sigma(\beta\cdot[q_i+\Delta_{i,a}-q_j-\Delta_{j,n}])W(q_i+\Delta_{i,a})-K_a\\
&\quad-\sigma(\beta\cdot[q_i-q_j-\Delta_{j,n}])W(q_i).
\end{aligned}
\]
Apply the race decomposition and add the continuation difference
\[
\delta\{V_i(T(s,a,n))-V_i(T(s,0,n))\}.
\]
This proves Theorem \ref{t:dynamic}. The regime theorem follows by applying the two
best-response inequalities to the four pure action profiles.

\subsection{Data-Distillation Tipping}

The gap map is
\[
\Psi(x)=(1-\rho)x+\chi(q) M\{2\sigma(x)-1\}.
\]
Because \(\sigma(0)=1/2\), \(x=0\) is a steady state. Differentiating,
\[
\Psi'(0)=1-\rho+2\chi(q) M\sigma'(0).
\]
Since \(\rho\in[0,1]\) and \(\chi(q)\ge0\), the derivative is nonnegative. The standard
one-dimensional local stability condition \(|\Psi'(0)|<1\) is therefore equivalent to
\(\Psi'(0)<1\), which is
\[
2\chi(q) M\sigma'(0)<\rho.
\]
If the inequality is reversed, the symmetric state is locally unstable: a small positive
lead grows rather than shrinks. For logit demand,
\[
\sigma'(0)=\frac{\kappa}{4},
\]
so instability requires \(\chi(q) M>2\rho/\kappa\).

\subsection{Productive Tipping}

The region-level map replaces \(\chi(q)\) and \(M\) by \(\chi_R(q)\) and \(M_R\). Its
derivative at the symmetric state is
\[
1-\rho+2\chi_R(q)M_R\sigma'(0)=1+L_R(q).
\]
Thus the symmetric state is locally stable when \(L_R(q)<0\) and locally unstable when
\(L_R(q)>0\). The local frontier value created by one marginal unit of served mass is
\(\zeta_R(q)\). Comparing this value with the opportunity cost \(\nu\) gives the
productive-learning classification. This proves Proposition
\ref{p:productive-tipping}.

\subsection{Tipping}

On the identity-preserving region,
\[
x'=(1-\rho)x+\alpha_La_L-\alpha_Fa_F.
\]
Taking expectations gives
\[
\mathbb E[x'-x\mid s]=-\rho x+\alpha_Lp_L(s)-\alpha_Fp_F(s).
\]
Substituting pure regimes gives the cases stated in Section \ref{sec:tipping}.

\subsection{Stopping}

The finite-step frontier value satisfies
\[
A(q,\Delta_n)\le \|\Delta_n\|_1\bar B_n.
\]
The market-share change is bounded by
\[
L_\sigma\|\beta\|_\infty\|\Delta_n\|_1.
\]
Thus the rent-shifting term is bounded by
\[
L_\sigma\|\beta\|_\infty\|\Delta_n\|_1\bar W_n.
\]
The continuation term is bounded by \(\delta L_V\|\Delta_n\|_1\). Adding the three
bounds proves that race pressure converges to zero under the stopping condition stated
in Section \ref{sec:tipping}.
Positive costs then rule out positive projects in best response. Private persistence
follows immediately if the rent-shifting term alone exceeds cost while \(A(q,\Delta)\)
is arbitrarily small.

\subsection{Private-Social Frontier Wedge}

If \(q_L\ge q_F\), then monotonicity of \(G_H\) and \(G_F\) implies
\[
p(q_L,h,f)\ge p(q_F,h,f)
\]
for every task. Hence \(S(q_L,q_F)=W(q_L)\). A leader project changes social frontier
surplus by \(W(q_L+\Delta)-W(q_L)=A(q_L,\Delta)\). If the follower remains dominated
after its project, then \(S(q_L,q_F+\Delta_F)=W(q_L)\), so its frontier social gain is
zero. The private gain expression is the race decomposition with \(q_i=q_F\),
\(q_j=q_L\), and no rival investment.

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\end{document}
