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\title{The Task Frontier\\Horizon, Forgiveness, and the Direction of AI Innovation}
\author{}
\date{July 2026}

\begin{document}
\maketitle

\begin{abstract}
\noindent
Cheap AI systems commoditize existing tasks within months, yet frontier laboratories keep
raising their spending. This paper explains the coexistence with a theory in which the
state of AI is a two-dimensional task frontier and imitation is directionally asymmetric.
Tasks are O-ring production with retries: \(n\) serial stages, each permitting \(\ell\)
retries. This one primitive \emph{delivers} the two dimensions rather than assuming them:
required reliability rises log-linearly in horizon with slope \(1/(1+\ell)\), task success
converges to a Gumbel kernel, and a scalar quality ladder is exactly the special case of
homogeneous forgiveness. Price competition task by task implies that frontier profit
equals the boundary-value integral over the capability gap, so rents survive only where
followers cannot follow. Imitation is fast along the horizon dimension---a plan can be
copied from one demonstration---and slow along the reliability dimension---a failure rate
must be estimated from many. Equilibrium investment therefore rotates from horizon toward
reliability as followers catch up: the market races along the moat, not along value. The
resulting direction of innovation is distorted relative to the planner's by the imitation
asymmetry, an observable object. Liability rules and public verification infrastructure
both raise reliability investment but move market concentration in opposite directions.
Aggregate evidence---horizon doubling times, the 50\%/80\%-horizon gap, and asymmetric
open-weight catch-up lags---disciplines the model's key objects.\\[1ex]
\textbf{JEL:} O31, O33, L13, L15, D24.
\end{abstract}

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

Two facts about the market for artificial intelligence sit awkwardly together. The price
of any fixed, demonstrated capability collapses: across public benchmarks, the cost of
reaching a given score has fallen by one to two orders of magnitude per year, and
open-weight models reproduce frontier scores within months. At the same time, the
laboratories that own the frontier keep raising their investment. If artificial
intelligence were a single quality index, this pattern would be a puzzle: when imitation
is nearly immediate, the return to advancing the index should be competed away, and
frontier spending should fall, not rise.

This paper resolves the puzzle with two ingredients. First, the state of AI is not a
scalar; it is a frontier in a two-dimensional task space. One dimension is
\emph{horizon}: the serial depth of the actions a task requires. The other is
\emph{forgiveness}: how much the task permits errors to be observed, retried, and rolled
back before they become losses. Second, imitation is not a scalar hazard; it is
\emph{directional}. The horizon dimension of capability is embodied in observable
plans, scaffolds, and reasoning traces, and can be copied from demonstrations. The
reliability dimension is a tail property: certifying that a system fails less than once
in ten thousand attempts requires on the order of ten thousand independent trials, and
the unforgiving tasks where such reliability matters are exactly the tasks that generate
few observable trials. Rents are competed away along the dimension where imitation is
fast and survive along the dimension where it is slow. Frontier investment follows the
rents: as followers close the horizon gap, equilibrium investment rotates toward the
reliability boundary. The market races along the moat, not along value.

The paper's first contribution is to derive the two-dimensional frontier from one
primitive rather than to posit it. A task is O-ring production with retries: \(n\)
stages must all succeed, and each stage may be attempted \(1+\ell\) times
\citep{kremer1993}. For a system whose per-attempt failure rate declines in a reliability
capability \(q_F\), the requirement to complete the task with tolerated failure
\(\varepsilon\) reduces, exactly and transparently, to a threshold
\[
q_F\;\ge\;\varphi+\frac{h+\ln(1/\varepsilon)}{1+\ell},
\qquad h=\ln n,
\]
up to a vanishing correction (Section \ref{sec:tasks}). Horizon imposes a reliability
tax---serial depth compounds residual error---and forgiveness divides that tax by
\(1+\ell\). Task success as a function of capability converges to a Gumbel kernel whose
location is this threshold and whose scale is \(1/(1+\ell)\), which microfounds the
reduced-form success functions used in task-based models. The representation theorem
(Theorem \ref{t:representation}) then states exactly when the familiar scalar ladder is
right: the task economy admits a one-dimensional quality representation if and only if
task requirements form a chain---equivalently, if forgiveness is homogeneous across
tasks. Coding is long but forgiving; a payment authorization is short but unforgiving.
Once forgiveness is heterogeneous, no scalar index can order tasks by difficulty, and
the direction of progress becomes an economic object rather than a redundant label.

The second contribution is to derive, rather than assume, the market payoffs that
task-based theories of AI competition have treated as primitives. Firms compete in
prices task by task, as in asymmetric Bertrand competition with vertical
differentiation. The equilibrium allocation is the familiar sorting rule---a cheaper
follower wins a task exactly when its cost advantage exceeds the remaining value
gap---and the frontier firm's flow profit equals the \emph{boundary-value integral over
the capability gap} between itself and the best alternative (Lemma
\ref{l:bertrand}). Saturation is then an equilibrium state, not an assumption: as the
fringe closes the gap in a task region, markups there converge to zero mechanically,
while total surplus stays high. Regions of large current revenue and zero frontier rent
coexist with regions of small current revenue and large frontier value.

The third contribution is the imitation technology. Theorem \ref{t:distill} formalizes
the asymmetry: a follower that observes a single successful trajectory learns the plan,
so the horizon component of capability diffuses at a rate bounded away from zero; but
any procedure that certifies a failure rate below \(\varepsilon\) from observed behavior
requires \(\Omega(1/\varepsilon)\) independent observations, and the tolerated failure
rate at the fragility frontier is exponentially small in the frontier's position. The
cost of imitating reliability therefore grows exponentially as the reliability frontier
advances, while the cost of imitating demonstrated plans does not. Appropriability is
not an institutional parameter here; it is a statistical property of tasks. Forgiving,
observable, easily evaluated tasks are cheap to distill. Unforgiving tasks protect
their own rents.

These three pieces assemble into a theory of the direction of innovation. In the
dynamic model, a frontier laboratory invests against a competitive fringe that closes
the capability gap at rate \(\lambda_H\) in the horizon dimension and \(\lambda_F\) in
the reliability dimension, with \(\lambda_H>\lambda_F\) by Theorem \ref{t:distill}. The
laboratory's shadow value of a unit of capability in direction \(i\) is
\(B_i/(r+\lambda_i)\): the boundary value of tasks unlocked in that direction, priced by
the \emph{persistence} of the gap there (Theorem \ref{t:rotation}). Investment follows
persistence-adjusted boundary values. Because pushing the horizon frontier unlocks
long tasks whose binding constraint is then reliability---the cross-boundary
complementarity is positive by construction---the value ratio \(B_F/B_H\) rises along
any horizon push, and investment rotates in finite time toward the reliability
boundary. Faster imitation of the horizon dimension makes the rotation come
\emph{earlier}: imitation does not merely deter frontier investment, it steers it.
Strategic competition sharpens the result: when two laboratories choose which boundary
to race on, sufficiently strong imitation asymmetry makes racing on the moat dimension a
dominant strategy, so competitors herd onto the same unforgiving frontier even though
they split the prize there (Theorem \ref{t:herding}).

The welfare economics is transparent and, we think, the right way to pose the policy
question. A planner values a unit of direction-\(i\) capability at \(B_i/r\): capability
is permanent for society, and the fringe's catch-up is diffusion, not destruction. The
market values it at \(B_i/(r+\lambda_i)\). The direction of innovation is therefore
distorted by the factor \((r+\lambda_F)/(r+\lambda_H)\)---the market undervalues exactly
the dimension that is easy to imitate---and the sign and size of the distortion are
pinned by an observable, the asymmetry in imitation lags (Theorem \ref{t:wedge}). Two
policies that both ``promote reliable AI'' then have opposite competitive effects.
Liability for failures in unforgiving tasks raises the private demand price of
reliability, accelerates investment in it, and \emph{thickens} the moat: concentration
rises. Public verification infrastructure---shared evaluation suites, incident
registries, certification institutions---raises \(\lambda_F\), diffuses reliability, and
\emph{thins} the moat: concentration falls, at the cost of weakening the private
incentive it disciplines. Safety policy and competition policy are the same policy
lever pointed in different directions.

The paper is theoretical, and the discipline we impose is aggregate, in the style of
quantitative theories of ideas and growth \citep{bloom2020}. Three public facts anchor
the model's objects. First, the measured horizon frontier---the task duration at which
frontier systems succeed half the time---has doubled roughly every seven months
\citep{kwa2025}. Second, the same measurement contains the second dimension: horizons
measured at an 80\% success requirement are roughly five times shorter than at 50\%. A
scalar constant-hazard benchmark \citep{ord2025} makes a parameter-free prediction for
this ratio, \(\ln 0.5/\ln 0.8\approx 3.1\); the observed gap of roughly five is the
reliability tax the scalar model misses, and its evolution identifies the coupling
parameter \(1/(1+\ell)\). Third, imitation lags are measurably asymmetric: open-weight
models trail frontier capability indices by three to four months, but trail frontier
\emph{autonomy} time-horizons by six to twelve months, and the gap at high reliability
thresholds is wider still. Section \ref{sec:quant} uses these objects to place the
current race on the model's phase diagram.

\medskip
\noindent\textbf{Related literature.} The paper builds on task-based models of
technology \citep{acemogluautor2011,acemoglurestrepo2018} but moves the action from the
worker--task substitution margin to the structure of tasks themselves, and on directed
technical change \citep{acemoglu2002,acemoglu2007}, from which it inherits the question
``which direction does the market choose, and is it the right one?'' Its answer
mechanism is new: in \citet{acemoglu2002} direction is steered by market size and factor
prices; here it is steered by the statistical structure of imitation. The closest
antecedents on the direction of innovation are \citet{bryanlemus2017} and
\citet{dasguptamaskin1987}, where racing distorts which projects firms pursue, and
\citet{callander2011}, where innovators search an unknown landscape;
\citet{budish2015} document direction distortions empirically. Relative to this work,
the direction space here is derived from a task primitive, the distortion is signed by
an observable (imitation-lag asymmetry), and the dynamics deliver rotation and herding
rather than a static portfolio bias. The O-ring production structure descends from
\citet{kremer1993} and its weakest-link successors \citep{jones2011}; forgiveness
(retries) is the new margin, and it is what makes the task space two-dimensional. The
racing and persistence-of-leadership questions connect to patent races and preemption
\citep{gilbertnewbery1982,reinganum1983} and to step-by-step innovation and escape
competition \citep{aghion2001}; the rotation result is an escape-competition force
acting on the \emph{direction} rather than the rate of innovation. The appropriability
mechanism formalizes, for AI, the classic observations of \citet{arrow1962},
\citet{teece1986}, and \citet{antonyao1994,antonyao2002}: what cannot be disclosed or
copied earns the rents. Multidimensional quality competition goes back to
\citet{mussarosen1978} and \citet{shakedsutton1982}; the contribution here is not that
two dimensions differ from one, but which two dimensions, why, and what they imply for
the direction of investment. On the AI side, the paper connects the measurement of task
horizons \citep{kwa2025,ord2025}, distillation \citep{hinton2015,hsieh2023}, and the
economics of foundation-model competition \citep{korinekvipra2025,acemoglu2025} into a
single equilibrium framework; general-purpose-technology dynamics are as in
\citet{bresnahantrajtenberg1995}.

The rest of the paper proceeds as follows. Section \ref{sec:facts} states the three
aggregate facts. Section \ref{sec:tasks} develops the task technology and the
representation theorem. Section \ref{sec:market} derives market payoffs. Section
\ref{sec:imitation} derives the directional imitation technology. Section
\ref{sec:dynamics} characterizes the direction of frontier investment and the rotation
result. Section \ref{sec:strategy} adds strategic competition between laboratories.
Section \ref{sec:welfare} states the welfare wedge and the policy reversal. Section
\ref{sec:quant} confronts the model with aggregate evidence. Section \ref{sec:conclusion}
concludes. Proofs are in Appendix \ref{app:proofs}; every formal claim in the paper is
also verified line by line in a machine-checked companion script described in Appendix
\ref{app:verification}.

\section{Three Aggregate Facts}\label{sec:facts}

The theory is disciplined by three facts, each computable from public aggregate series.
None requires micro data, and none is an identified causal estimate; their role, as in
the quantitative-ideas literature \citep{bloom2020}, is to pin the model's objects and
to state in advance what would falsify it.

\begin{fact}[The horizon frontier moves exponentially, and carries a reliability tax]
\label{fact:horizon}
The task duration at which frontier systems succeed 50\% of the time has doubled roughly
every seven months since 2019 \citep{kwa2025}. Horizons measured at an 80\% success
requirement are roughly five times shorter than 50\% horizons, with the same doubling
time.
\end{fact}

The second half of Fact \ref{fact:horizon} is, to our knowledge, unexploited. A scalar
model in which task success is governed by a constant per-minute hazard
\citep{ord2025} predicts the 50\%/80\% horizon ratio \emph{without free parameters}:
if success on a task of length \(t\) is \(e^{-t/T}\) for a model-specific \(T\), then
\(t_{50}/t_{80}=\ln 0.5/\ln 0.8\approx 3.1\), independent of \(T\), and hence constant
across models and time. An observed ratio near five is a rejection of the
constant-hazard scalar benchmark in a specific direction: success decays with task
length faster than geometrically at high reliability requirements. Section
\ref{sec:tasks} shows this is exactly what task heterogeneity in \emph{forgiveness}
produces, and Section \ref{sec:quant} uses the ratio to calibrate the coupling between
the two dimensions.

\begin{fact}[Imitation is fast, and directionally asymmetric]\label{fact:lags}
Open-weight models trail the frontier by roughly three to four months on aggregate
capability indices. On autonomy time-horizon measures the lag is larger---mid-2025
open-weight systems matched the late-2024 frontier---and practitioner evidence
consistently places the widest gaps in long-horizon agentic \emph{reliability} rather
than in demonstrated capability.\footnote{Epoch AI, ``Open-weight models lag
state-of-the-art by around 3--4 months'' (data insight, 2025--2026); METR,
``Details about METR's preliminary evaluation of DeepSeek and Qwen models'' (2025);
Epoch AI, FrontierMath open-model analysis (lag of roughly seven months on the hardest
tiers).}
\end{fact}

\begin{fact}[Served tasks commoditize while frontier spending rises]\label{fact:prices}
The price of reaching a fixed benchmark score has fallen by one to two orders of
magnitude per year (median estimate roughly $50\times$ per year across benchmarks),
while frontier training expenditure has continued to grow.\footnote{Epoch AI, ``LLM
inference prices have fallen rapidly but unequally across tasks'' (2025): declines of
$9\times$--$900\times$ per year across benchmarks and thresholds.}
\end{fact}

Fact \ref{fact:prices} is the puzzle. Facts \ref{fact:horizon} and \ref{fact:lags}
contain the resolution: the frontier is two-dimensional, and imitation erodes it
asymmetrically. Any theory of the AI race should (i) generate rents that survive
near-immediate imitation of demonstrated capability, (ii) predict which tasks
commoditize and which do not, and (iii) say where frontier investment goes next. The
model below is built to those specifications.

\section{Task Technology and the Two-Dimensional Frontier}\label{sec:tasks}

\subsection{O-Ring Production with Retries}

A task consists of \(n\) essential stages in series. Executing a stage is an
\emph{attempt}; an attempt by a system with reliability capability \(q_F\in\mathbb R\)
succeeds with probability \(G_F(q_F-\varphi)\), where \(\varphi\) is the stage's local
difficulty and \(G_F\) is a strictly increasing, differentiable CDF on \(\mathbb R\).
The task's \emph{forgiveness} is the number of retries it permits: a stage may be
attempted up to \(1+\ell\) times, \(\ell\in\{0,1,2,\dots\}\), because errors are
observable, reversible, or sandboxed enough to try again. The stage succeeds if any
attempt does, so with \(\pi=G_F(q_F-\varphi)\) the stage success probability is
\(S_\ell(\pi)=1-(1-\pi)^{1+\ell}\), and the task's execution success probability is
\(S_\ell(\pi)^n\). Without retries this is Kremer's O-ring technology
\citep{kremer1993}; \(\ell\) is the new margin, and it is the economic content of
``forgiveness'': version-controlled code allows many retries, a payment authorization
or a lane change allows none.

A task also has a coordination requirement: the system must generate and hold together
the \(n\)-step plan at all. We model this margin in reduced form: the plan succeeds
with probability \(G_H(q_H-h)\), where \(h=\ln n\) is the task's \emph{horizon}, \(q_H\)
is the system's horizon capability, and \(G_H\) is a strictly increasing, differentiable
CDF. Measuring horizon in logs matches how the frontier is measured empirically
\citep{kwa2025} and makes Fact \ref{fact:horizon}'s exponential trend a linear drift in
\(q_H\). The task succeeds if the plan succeeds and every stage executes; a task is thus
a tuple \(\tau=(n,\varphi,\ell,\varepsilon)\), where \(\varepsilon\) is the failure
probability the task's user tolerates.

\subsection{The Reduced Fragility Requirement}

Execution imposes a requirement on \(q_F\) that compounds with serial depth. Requiring
\(S_\ell(\pi)^n\ge 1-\varepsilon\) is equivalent to
\begin{equation}\label{eq:threshold}
q_F\;\ge\; f(\tau)\;\equiv\;\varphi+G_F^{-1}\!\Big(1-\big\{1-(1-\varepsilon)^{1/n}\big\}^{1/(1+\ell)}\Big),
\end{equation}
and we call \(f(\tau)\) the task's \emph{fragility}: the reliability capability the task
demands. Fragility rises with serial depth \(n\) and local difficulty \(\varphi\), falls
with retries \(\ell\), and rises as the tolerated failure \(\varepsilon\) falls. Its
structure is transparent in the logistic benchmark.

\begin{lemma}[Log-linear fragility and the Gumbel kernel]\label{l:gumbel}
Let \(G_F\) be the logistic CDF, \(G_F(x)=1/(1+e^{-x})\), and \(h=\ln n\).
\begin{itemize}
\item[(i)] \emph{(Threshold.)} For \(\varepsilon\le 1/2\) and
\(n\ge 4^{1+\ell}\varepsilon\) (a task long relative to its forgiveness), the fragility
requirement satisfies
\[
f(\tau)\;=\;\varphi+\frac{h+\ln(1/\varepsilon)}{1+\ell}+e(\tau),
\qquad
|e(\tau)|\;\le\;2\varepsilon+3\Big(\frac{\varepsilon}{n}\Big)^{\!1/(1+\ell)}.
\]
The reliability requirement is linear in log-horizon with slope \(1/(1+\ell)\): serial
depth taxes reliability, and forgiveness divides the tax.
\item[(ii)] \emph{(Kernel.)} Fix \(u\in\mathbb R\) and let
\(q_F=\varphi+(\ln n+u)/(1+\ell)\). Then as \(n\to\infty\),
\[
S_\ell\big(G_F(q_F-\varphi)\big)^n\;\longrightarrow\;\exp\!\big(-e^{-u}\big),
\]
with error \(O\!\big(n^{-1/(1+\ell)}\big)\). Equivalently, execution success at
capability \(q_F\) converges to the Gumbel kernel
\(K_\ell(q_F-f)=\exp\{-e^{-(1+\ell)(q_F-f)}\}\) with location equal to the fragility
\(f\) of the task at \(\varepsilon\)-scale and scale \(1/(1+\ell)\).
\end{itemize}
\end{lemma}

Lemma \ref{l:gumbel} does two jobs. First, it replaces an assumption with a derivation:
smooth ``success curves'' \(P(q-f)\) of the kind task-based models posit are the
large-\(n\) shape of O-ring-with-retries production, and their steepness is not free---
it is \(1+\ell\), the task's forgiveness. Forgiving tasks flip from infeasible to
routine quickly; unforgiving tasks have long, shallow reliability ramps. Second, it
reduces a four-parameter task \(\tau\) to a two-dimensional requirement
\((h,f)\in\mathbb R^2\): horizon, and the reliability demanded at that horizon. These
two coordinates are the task frontier's space.

\subsection{When Is a Scalar Ladder Enough?}

The reduced requirements answer the first-order modeling question---when is ``model
quality'' one number?---as a theorem rather than a taste. Say a system \emph{serves} a
task if it meets both requirements: \(q_H\ge h\) and \(q_F\ge f\).\footnote{The
deterministic-threshold statement is for transparency; by Lemma \ref{l:gumbel}(ii) it is
the sharp-kernel limit of the probabilistic model, and the theorem's logic---nestedness
of served sets---is unchanged under strictly monotone kernels.} A \emph{scalar
representation} of a task economy \(\mathcal T=\{(h_t,f_t)\}\) is a pair of indices
\(\iota:\mathcal T\to\mathbb R\) and \(\tilde\iota:\mathbb R^2\to\mathbb R\) such that
any system \(q\) serves task \(t\) if and only if \(\tilde\iota(q)\ge\iota(t)\).

\begin{theorem}[Representation]\label{t:representation}
A task economy \(\mathcal T\) admits a scalar representation if and only if its
requirement pairs are totally ordered by the componentwise order (a chain): there is no
pair of tasks with \(h_1<h_2\) and \(f_1>f_2\). In particular:
\begin{itemize}
\item[(i)] If all tasks share the same forgiveness and local difficulty
\((\ell,\varphi)\), then \(f\) is a strictly increasing function of \(h\) along
\eqref{eq:threshold}, \(\mathcal T\) is a chain, and the scalar quality ladder is
exact.
\item[(ii)] If forgiveness is heterogeneous, the economy generically contains
unordered pairs---a long forgiving task and a short unforgiving one---and no scalar
index can rank systems: there exist systems \(q,q'\) neither of which serves a superset
of the other's tasks.
\end{itemize}
\end{theorem}

The scalar ladder is thus the special case of \emph{homogeneous forgiveness}. It fails
on precisely the comparisons that organize the AI market: multi-hour coding with tests
and version control (large \(h\), large \(\ell\)) against payment authorization,
medication dosing, or a driving maneuver (small \(h\), \(\ell=0\)). Once such pairs
carry value, ``better'' is not a number, and the direction of progress is an economic
choice. The rest of the paper is about who chooses it and whether they choose well.

\subsection{Task Value and Boundary Statistics}

Let \(a(h,f)\ge 0\) be the value density over requirements: the flow surplus from tasks
at \((h,f)\) if they are served. A system \(q=(q_H,q_F)\) generates gross surplus
\begin{equation}\label{eq:W}
W(q)\;=\;\iint a(h,f)\,P_H(q_H-h)\,P_F(q_F-f)\,dh\,df,
\end{equation}
where \(P_H,P_F\) are strictly increasing success kernels; by Lemma \ref{l:gumbel},
\(P_F=K_\ell\) is the derived Gumbel shape, and we carry general kernels for
robustness. Write \(W_R\) for the integral restricted to a region \(R\). The marginal
values of capability are the \emph{boundary values}
\begin{equation}\label{eq:boundary}
B_H(q)=\frac{\partial W}{\partial q_H}
=\iint a\,g_H\,P_F,
\qquad
B_F(q)=\frac{\partial W}{\partial q_F}
=\iint a\,P_H\,g_F,
\end{equation}
with \(g_i\) the kernel densities, and the cross statistic
\begin{equation}\label{eq:cross}
C(q)\;=\;\frac{\partial B_H}{\partial q_F}\;=\;\frac{\partial B_F}{\partial q_H}
\;=\;\iint a\,g_H\,g_F\;\ge\;0.
\end{equation}
\(B_H\) prices the value mass just beyond the horizon boundary, \(B_F\) the mass just
beyond the reliability boundary, and \(C\)---always nonnegative---the mass near the
corner where both bind. For a finite capability step \(\Delta\ge 0\),
\begin{equation}\label{eq:path}
W(q+\Delta)-W(q)
=\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 .
\end{equation}
Equation \eqref{eq:cross} is worth pausing on: it says that pushing the horizon
boundary \emph{mechanically raises} the marginal value of reliability whenever value
sits near both boundaries. Long tasks, once reachable in horizon, wait on reliability.
This derived complementarity---not an assumption---is what will drive the rotation of
investment in Section \ref{sec:dynamics}.

\begin{figure}[H]
\centering
\begin{tikzpicture}[x=1cm,y=1cm,>=Latex]
  \draw[->, thick] (0,0) -- (9.4,0) node[right] {Horizon \(h\)};
  \draw[->, thick] (0,0) -- (0,5.6) node[above] {Fragility \(f\)};
  \draw[dashed] (4.6,0) -- (4.6,5.2);
  \draw[dashed] (0,2.6) -- (8.9,2.6);
  \node[below] at (4.6,0) {\(q_H\)};
  \node[left] at (0,2.6) {\(q_F\)};
  \draw[thick, gray] (0.3,0.5) -- (8.6,4.9);
  \node[gray, rotate=27] at (7.0,3.55) {\footnotesize slope \(1/(1+\ell)\)};
  \node[align=center,text width=3.3cm] at (2.25,1.25)
    {Saturated interior\\ \footnotesize served, commoditizing};
  \node[align=center,text width=3.3cm] at (6.9,1.25)
    {Horizon-scarce\\ \footnotesize plans, tools, search};
  \node[align=center,text width=3.3cm] at (2.3,4.15)
    {Reliability-scarce\\ \footnotesize verification, rollback};
  \node[align=center,text width=3.3cm] at (6.9,4.5)
    {Corner: both bind\\ \footnotesize \(C(q)>0\)};
\end{tikzpicture}
\caption{The task frontier. Tasks are points \((h,f)\); a family with common
forgiveness \(\ell\) lies on a line of slope \(1/(1+\ell)\) (Lemma \ref{l:gumbel}).
Capability \(q=(q_H,q_F)\) serves the lower-left region; boundary values
\(B_H,B_F\) price the two frontier directions.}
\label{fig:taskspace}
\end{figure}

\section{Market Equilibrium: Profits Are Gaps}\label{sec:market}

Task-based theories of AI competition typically posit profit functions---a rent from
being ahead, an outside option that erodes with ``competitive pressure.'' Here both are
equilibrium objects, and their structure does real work later, so we derive them.

Each task \((h,f)\) is a market. The buyer values successful completion at \(a(h,f)\)
per period, so system \(k\) delivers expected gross value
\(v_k(h,f)=a(h,f)\,P_H(q^k_H-h)P_F(q^k_F-f)\) at serving cost \(c_k(h,f)\); let
\(s_k=v_k-c_k\) denote delivered net surplus. Sellers set task-specific prices;
buyers choose sellers. This is asymmetric Bertrand competition with vertically
differentiated suppliers.

\begin{lemma}[Bertrand in tasks]\label{l:bertrand}
In every task's pricing equilibrium:
\begin{itemize}
\item[(i)] \emph{(Allocation.)} The task is served by the firm with the highest
delivered net surplus \(s_k(h,f)\). A lower-cost follower \(m\) displaces the frontier
firm \(L\) exactly where its cost advantage exceeds the value gap:
\(c_L-c_m\;\ge\;v_L-v_m\).
\item[(ii)] \emph{(Pricing.)} The winner's margin on the task is the surplus gap over
the runner-up, \(s_{(1)}(h,f)-s_{(2)}(h,f)\).
\item[(iii)] \emph{(Profit is a gap integral.)} If the frontier firm has capability
\(q^L\ge q^m\) componentwise, equal serving costs, and the best alternative on every
task is the fringe system \(q^m\), its flow profit is
\[
\Pi\big(q^L,q^m\big)\;=\;W(q^L)-W(q^m)
\;=\;\int_0^{\Delta_H}\!B_H(q^m_H+u,\,q^m_F)\,du
+\int_0^{\Delta_F}\!B_F(q^L_H,\,q^m_F+v)\,dv,
\]
where \(\Delta=q^L-q^m\): boundary values integrated over the capability gap.
\item[(iv)] \emph{(Saturation.)} As the fringe closes the gap in a region
(\(q^m\to q^L\) there), the frontier firm's margins in that region converge to zero
while delivered surplus does not: the region is saturated---served, valuable, and
rentless.
\end{itemize}
\end{lemma}

Three consequences organize what follows. First, part (i) is the sorting rule that, in
earlier task-frontier drafts, appeared as a standalone observation; here it is the
equilibrium allocation, and ``cheap models win the saturated interior'' is a location
statement about where value gaps are small. Second, part (iii) says \emph{rents live in
the gap}: the frontier firm's profit is not the value of what it serves but the
boundary-value mass between itself and its best imitator. Current revenue and frontier
rent are different objects---a region can be economically huge and competitively
worthless. Third, the envelope property of (iii) gives the frontier firm's marginal
value of capability: \(\partial\Pi/\partial q^L_i=B_i(q^L)\), the boundary value at its
own frontier, \emph{so long as the gap persists}. What the fringe does to the gap over
time is therefore the entire strategic question, and it is where we go next.

\section{Imitation Technology: The Tail Cannot Be Distilled}\label{sec:imitation}

The fringe in Lemma \ref{l:bertrand} was a capability \(q^m\); we now give it a
technology. Followers learn from what the frontier deploys: successful trajectories,
tool scaffolds, reasoning traces, evaluation results---the raw material of distillation
\citep{hinton2015,hsieh2023}. The question is what observation can and cannot teach.
The answer is asymmetric in exactly the two dimensions of the task frontier.

\emph{Horizon is a blueprint good.} The plan structure that completes a long task---the
decomposition, the tool calls, the checkpoints---is embodied in any successful
trajectory. One demonstration reveals it, in Arrow's sense that the information is
disclosed by the product itself \citep{arrow1962}. We model this as: access to any
positive flow of demonstrations moves \(q^m_H\) toward \(q^L_H\) at rate at least
\(\underline\lambda_H>0\), a constant of access, not of statistics.

\emph{Reliability is a tail statistic.} A follower cannot serve an unforgiving task by
exhibiting a plan; it must \emph{be} reliable at the task's tolerated failure rate
\(\varepsilon\), and---because unforgiving tasks are precisely those where failures are
costly---its system must be certified at that rate before deployment, by itself, its
customers, or a regulator. Certification is a hypothesis-testing problem, and it has a
sample-complexity floor.

\begin{theorem}[The tail cannot be distilled]\label{t:distill}
\begin{itemize}
\item[(i)] \emph{(Sample floor.)} Any certification procedure that, with error
probabilities at most \(\delta<1/2\), distinguishes a system with failure rate
\(\le\varepsilon/2\) from one with failure rate \(\ge\varepsilon\) requires
\[
N(\varepsilon)\;\ge\;\frac{2(1-2\delta)^2}{\mathrm{KL}\big(\mathrm{Bern}(\varepsilon/2)\,\|\,\mathrm{Bern}(\varepsilon)\big)}
\;\ge\;\frac{4(1-2\delta)^2}{\varepsilon}
\]
independent task-level observations for \(\varepsilon\le1/2\), since
\(\mathrm{KL}\le\chi^2=\varepsilon/\{4(1-\varepsilon)\}\le\varepsilon/2\) there; the
sharp small-\(\varepsilon\) coefficient is \(4/(1-\ln 2)\approx 13\) times
\((1-2\delta)^2\).
\item[(ii)] \emph{(Exponential wall.)} At the fragility frontier, the tolerated failure
rate of the marginal task is \(\varepsilon(q_F)\asymp n\,e^{-(1+\ell)(q_F-\varphi)}\)
(Lemma \ref{l:gumbel}). If demonstrations of frontier behavior on such tasks arrive at
a bounded flow \(D\) per period, the time for a follower to certify at the frontier's
reliability level satisfies
\[
T_{\mathrm{imitate}}(q_F)\;\ge\;\frac{N(\varepsilon(q_F))}{D}
\;\gtrsim\;\frac{e^{(1+\ell)(q_F-\varphi)}}{D\,n},
\]
growing exponentially in the position of the reliability frontier. The implied gap-decay
rate \(\lambda_F(q_F)\) falls toward zero as the frontier advances, while
\(\lambda_H\ge\underline\lambda_H>0\).
\item[(iii)] \emph{(Directional appropriability.)} Consequently, for any discount rate
\(r>0\), the fraction of a capability lead's value that survives imitation,
\(\lambda\mapsto r/(r+\lambda)\) per Section \ref{sec:dynamics}, is strictly larger in
the reliability direction than in the horizon direction, and the asymmetry widens as
the reliability frontier advances.
\end{itemize}
\end{theorem}

The one-line version: \emph{a plan can be copied from one observation; a failure rate
must be estimated from many}. Appropriability here is not a patent, a secret, or a
complementary asset \citep{teece1986,antonyao1994}---it is a statistical property of
the task being served. Forgiving, observable, high-volume tasks teach imitators
everything and protect nothing. Unforgiving tasks are self-protecting: the same
scarcity of tolerated failures that makes them hard makes them slow to copy. Two
remarks bound the claim honestly. First, the floor in (i) binds \emph{certification
from behavior}; a follower could instead re-derive reliability from first
principles---the theorem prices imitation, not independent invention, which is what
the investment technology in the next section is. Second, public verification infrastructure---shared
incident registries, standardized high-reliability evaluations, certification
authorities---directly attacks the supply side of the bound by raising the usable
observation flow \(D\): it is, in this precise sense, imitation policy for the
reliability dimension. Section \ref{sec:welfare} returns to this.

\section{The Direction of Frontier Investment}\label{sec:dynamics}

We now put the three derived objects---gap profits, directional imitation, and the
cross-boundary complementarity---into motion. To characterize direction cleanly we
study a single frontier laboratory against the imitative fringe; strategic interaction
between laboratories is Section \ref{sec:strategy}.

\subsection{The Leader--Fringe Problem}

Time is continuous, the discount rate is \(r>0\). The state is the capability gap
\(\Delta=(\Delta_H,\Delta_F)=q^L-q^m\ge 0\). The laboratory's investment \(x_i\ge0\)
raises its own capability in direction \(i\); the fringe closes each gap at its
imitation rate from Theorem \ref{t:distill}:
\begin{equation}\label{eq:gapdyn}
\dot\Delta_i \;=\; x_i-\lambda_i\Delta_i,
\qquad i\in\{H,F\},\qquad \lambda_H>\lambda_F\ge 0 .
\end{equation}
By Lemma \ref{l:bertrand}(iii), flow profit is the gap integral of boundary values. We
work in the \emph{local regime} in which the boundary values are constant over the
width of the gap,\footnote{Formally, an approximation whose error is second order in
the gap: by \eqref{eq:path}, \(|\Pi(\Delta)-B\!\cdot\!\Delta|\) is bounded by the
variation of the boundary values over the gap rectangle, i.e., by the curvature of
\(W\) (including the cross statistic \(C\)) times \(\|\Delta\|^2/2\). The regime is
the empirically relevant one:
measured frontier--fringe gaps (Fact \ref{fact:lags}) are months wide, while the value
distribution moves on the scale of years.} so
\(\Pi(\Delta)=B_H\Delta_H+B_F\Delta_F\), with \(B_i=B_i(q^L)\) evolving slowly as the
frontier itself moves. Investment costs are \(c(x_H)+c(x_F)\), increasing and
convex.

\begin{theorem}[Shadow values and rotation]\label{t:rotation}
\begin{itemize}
\item[(i)] \emph{(Shadow values.)} The value function is linear:
\[
V(\Delta)=\mu_H\Delta_H+\mu_F\Delta_F+v_0,
\qquad
\mu_i\;=\;\frac{B_i}{r+\lambda_i}:
\]
a unit of capability lead in direction \(i\) is worth its boundary value, discounted by
interest \emph{plus the rate at which imitation erases it}. Optimal investment equates
marginal cost to \(\mu_i\) (interior case \(x_i^*=c'^{-1}(\mu_i)\); with a linear
budget, all investment goes to the direction with the higher \(\mu_i/c_i\)).
\item[(ii)] \emph{(Rotation.)} Suppose the frontier starts horizon-directed,
\(\mu_H/c_H>\mu_F/c_F\), and along the induced horizon push the scarcity ratio rises,
\(\tfrac{d}{dq_H}\big(B_F/B_H\big)>0\)---which holds whenever \(C(q)>0\) and
\(\partial B_H/\partial q_H\le 0\), i.e., whenever unlocked long tasks wait on
reliability and pure-horizon value mass depletes---and eventually exceeds
\(\rho^*\equiv\dfrac{(r+\lambda_F)\,c_F}{(r+\lambda_H)\,c_H}\).
Then there is a finite rotation time \(T^*\), before which investment is
horizon-directed and after which it is reliability-directed.
\item[(iii)] \emph{(Imitation steers.)} The rotation threshold \(\rho^*\) is strictly
decreasing in \(\lambda_H\) and increasing in \(\lambda_F\): faster imitation of the
horizon dimension brings the rotation \emph{earlier}, and faster diffusion of
reliability postpones it. Under single crossing of \(B_F/B_H\), \(T^*\) inherits these
comparative statics.
\end{itemize}
\end{theorem}

Part (i) is the paper's price theory of direction: the market prices frontier
directions by \emph{persistence-adjusted} boundary values \(B_i/(r+\lambda_i)\), not by
boundary values. Part (ii) turns the derived complementarity \eqref{eq:cross} into
dynamics. A horizon push does two things at once: it depletes the stock of tasks whose
binding constraint is horizon, and it unlocks---in the horizon dimension---long tasks
that now wait on reliability. Both forces raise \(B_F/B_H\), so the direction of
investment must eventually rotate toward the unforgiving boundary. Part (iii) is the
result we regard as the model's signature. In scalar models, imitation is a tax on
innovation: it lowers the return and can only slow the race (or, in escape-competition
settings, speed it along the same axis). Here imitation has a \emph{steering} effect:
commoditization of demonstrated capability is precisely what pushes the frontier toward
reliability. The fringe does not slow the leader down so much as tell it where to go.

\begin{figure}[H]
\centering
\begin{tikzpicture}[x=1cm,y=1cm,>=Latex]
  \draw[->, thick] (0,0) -- (10.2,0) node[right] {\(q_H\) (horizon push \(\to\) time)};
  \draw[->, thick] (0,0) -- (0,5.4) node[above] {\(B_F/B_H\)};
  \draw[thick] plot[smooth] coordinates
    {(0.4,0.55) (2.0,0.75) (3.6,1.1) (5.2,1.7) (6.8,2.7) (8.4,4.1) (9.6,5.0)};
  \draw[dashed] (0,2.2) -- (9.8,2.2) node[pos=0.03, above] {\footnotesize \(\rho^*\)};
  \draw[dashed, gray] (0,1.45) -- (9.8,1.45)
    node[pos=0.06, below] {\footnotesize \(\rho^{*\prime}\) (higher \(\lambda_H\))};
  \draw[dotted] (6.05,0) -- (6.05,2.2);
  \node[below] at (6.05,0) {\(T^*\)};
  \draw[dotted, gray] (4.55,0) -- (4.55,1.45);
  \node[below, gray] at (4.55,0) {\(T^{*\prime}\)};
  \node[align=left] at (2.6,4.3) {\footnotesize invest in horizon\\[-2pt]\footnotesize left of crossing;\\[-2pt]\footnotesize reliability after};
\end{tikzpicture}
\caption{Rotation. Along a horizon push the scarcity ratio \(B_F/B_H\) rises
(depletion plus the derived complementarity \(C>0\)); investment rotates to the
reliability boundary when it crosses the persistence-adjusted threshold \(\rho^*\).
Faster horizon imitation (higher \(\lambda_H\)) lowers the threshold and brings the
rotation earlier.}
\label{fig:rotation}
\end{figure}

The rotation result reads directly onto the industry's trajectory: a phase in which
frontier competition is about demonstrated capability---longer benchmarks, agents,
scaffolds---followed, as those capabilities commoditize (Facts \ref{fact:lags} and
\ref{fact:prices}), by a phase in which it is about reliability in unforgiving
deployments: verification, monitoring, rollback, certified autonomy. Section
\ref{sec:quant} asks where on this diagram the observable aggregates place us.

\section{Strategic Competition: Racing on the Same Boundary}\label{sec:strategy}

A scalar race leaves rivals only one question: how hard to run. A two-dimensional
frontier adds a question a scalar model cannot pose: \emph{where} to run---and whether
competitors split the frontier or crowd onto the same boundary.

Let two frontier laboratories simultaneously commit investment programs to a direction
\(d_i\in\{H,F\}\). Racing alone on boundary \(i\) is worth the leader--fringe value of
that direction, \(V_i\) (by Theorem \ref{t:rotation}, \(V_i\) is increasing in
\(\mu_i=B_i/(r+\lambda_i)\)); racing head-to-head on the same boundary splits and
dissipates the prize to \(\sigma V_i\) each, \(\sigma\in(0,\tfrac{1}{2}]\). Order the
directions so \(V_F\ge V_H\)---by Theorem \ref{t:distill} this is the configuration
imitation asymmetry produces even when raw boundary values are comparable.

\begin{theorem}[Herding on the moat]\label{t:herding}
In the direction game:
\begin{itemize}
\item[(i)] If \(\sigma V_F\ge V_H\), racing on the reliability boundary is a dominant
strategy: the unique equilibrium is \emph{herding}, both laboratories on the moat
dimension, each accepting the split prize \(\sigma V_F\).
\item[(ii)] If \(\sigma V_F<V_H\le V_F\), the pure equilibria are the two
\emph{differentiation} profiles (one laboratory per boundary), and there is a mixed
equilibrium in which each races on \(F\) with probability
\(p^*=\dfrac{V_F-\sigma V_H}{(V_F-\sigma V_H)+(V_H-\sigma V_F)}\),
strictly increasing in \(V_F/V_H\).
\item[(iii)] Because \(V_F/V_H\) is increasing in \(\lambda_H\) and decreasing in
\(\lambda_F\), faster imitation of demonstrated capability expands the herding region:
sufficiently asymmetric imitation makes \emph{both} rivals race on the same unforgiving
frontier despite splitting its rents.
\end{itemize}
\end{theorem}

The result inverts the differentiation intuition. In classic quality competition,
rivals spread out to soften price competition \citep{shakedsutton1982}; a naive port to
innovation says rival laboratories should divide the frontier between them. Theorem
\ref{t:herding} says the moat is worth sharing: when the horizon dimension is copied in
months, half of a defensible reliability lead beats all of an indefensible capability
lead. The prediction is concentration of frontier R\&D portfolios on the same
reliability-heavy direction---simultaneous pivots by rival laboratories toward agentic
reliability, verification, and high-stakes deployment---rather than specialization,
exactly when open-weight catch-up is fastest.

\section{Welfare: The Direction Wedge and a Policy Reversal}\label{sec:welfare}

\subsection{Whose Direction Is Right?}

The planner and the market price the same two directions differently, and the
difference is the paper's welfare economics. Society keeps a capability advance
forever: once the frontier reaches \(q\), surplus \(W(q)\) is available regardless of
who serves the tasks, and the fringe's subsequent catch-up is diffusion---a transfer
from margins to buyers---not destruction. The planner's marginal value of a unit of
direction-\(i\) capability is therefore \(B_i/r\). The laboratory, by Theorem
\ref{t:rotation}, values it at \(B_i/(r+\lambda_i)\): it only earns the gap, and the
gap decays.

\begin{theorem}[The direction wedge is the imitation asymmetry]\label{t:wedge}
\begin{itemize}
\item[(i)] The market's valuation of direction \(i\) relative to the planner's is
\(r/(r+\lambda_i)\), and the \emph{relative direction distortion} is
\[
\frac{\mu_H/\mu_F}{(B_H/r)\big/(B_F/r)}
\;=\;\frac{r+\lambda_F}{r+\lambda_H}\;<\;1
\quad\Longleftrightarrow\quad \lambda_H>\lambda_F .
\]
The market undervalues exactly the dimension that is easy to imitate. Its sign and
magnitude are pinned by an observable: the asymmetry of imitation lags (Fact
\ref{fact:lags}).
\item[(ii)] Measured by surplus \(W\), the market therefore \emph{over}-rotates toward
the reliability boundary relative to the planner---it is drawn by persistence, not
value. If, in addition, failures on unforgiving tasks carry uninternalized social harm,
the planner's reliability boundary value scales to \(\theta_s B_F\) with
\(\theta_s>1\), and the two distortions oppose: the net direction bias has the sign of
\(\theta_s-\dfrac{r+\lambda_H}{r+\lambda_F}\).
Whether the market is too safety-directed or not safety-directed enough is not an
ideological constant; it is a horse race between an externality and an imitation
asymmetry, both in principle measurable.
\end{itemize}
\end{theorem}

Part (ii) deserves emphasis because it cuts against both popular narratives at once.
The pure appropriability logic says markets \emph{favor} reliability---it is the
moat---so laissez-faire innovation drifts toward the unforgiving frontier faster than
surplus alone justifies. The pure externality logic says markets underinvest in safety. Both
forces are real in the model, they load on the same margin with opposite signs, and the
theory's contribution is to say precisely which observables settle the race.

\subsection{Two Safety Policies, Opposite Market Structures}

Take the quadratic-cost leader--fringe economy of Theorem \ref{t:rotation}
(\(x_i^*=\mu_i\), steady-state gap \(\Delta_i^*=x_i^*/\lambda_i\)), and compare two
instruments aimed at ``more reliable AI.''

\emph{Liability.} A liability rule or regulatory standard on unforgiving tasks makes
sellers internalize failures, scaling the private reliability boundary value to
\(\theta B_F\), \(\theta>1\).

\emph{Public verification infrastructure.} Shared high-reliability evaluation suites,
incident registries, and certification institutions raise the observation flow that
Theorem \ref{t:distill} shows is the binding constraint on imitating reliability:
they raise \(\lambda_F\).

\begin{corollary}[Policy reversal]\label{c:policy}
In steady state, with frontier-rent concentration measured by the reliability component
of the gap profit, \(\Pi_F^*=B_F\Delta_F^*=B_F^2\big/\{(r+\lambda_F)\lambda_F\}\):
\begin{itemize}
\item[(i)] Liability raises reliability investment
(\(\partial x_F^*/\partial\theta>0\)) and \emph{thickens} the moat:
\(\partial\Delta_F^*/\partial\theta>0\), concentration rises.
\item[(ii)] Verification infrastructure \emph{thins} the moat and diffuses
reliability: \(\partial\Delta_F^*/\partial\lambda_F<0\), so the fringe's
reliability rises toward the frontier. The cost is the private frontier incentive
itself, \(\partial x_F^*/\partial\lambda_F<0\).
\end{itemize}
Both policies deliver ``safer AI''; they move market structure in opposite directions.
\end{corollary}

The corollary gives competition and safety authorities a common language. Liability is
safety policy that doubles as concentration policy: it pays the frontier firm in
exactly the currency---persistent, hard-to-imitate reliability---that Theorem
\ref{t:distill} says cannot be competed away. Verification infrastructure is
competition policy for the reliability dimension: it attacks the sample-complexity
wall directly, converting the moat into a commons. The model does not rank them---the
ranking depends on \(\theta_s\), \(r\), and the \(\lambda\)'s---but it insists they are
not the same policy, and that treating ``AI safety investment'' as a single quantity
conflates a frontier incentive with a diffusion externality.

\section{Quantitative Discipline}\label{sec:quant}

The model's load-bearing objects are few: the coupling slope \(1/(1+\ell)\), the
boundary values \(B_i\), and the imitation rates \(\lambda_i\). Each is disciplined by
one of the aggregate facts of Section \ref{sec:facts}. We emphasize what this section
is and is not: it is measurement that constrains a theory, in the tradition of
\citet{bloom2020}; it identifies no causal parameter, and the theory carries the causal
weight.

\subsection{The 50\%/80\% Ratio as a Specification Test}

The scalar constant-hazard benchmark makes a parameter-free prediction. If task success
is \(e^{-t/T}\) in task length \(t\) \citep{ord2025}---the continuous O-ring limit with
homogeneous stages and no retry heterogeneity---then measured horizons at reliability
requirements \(50\%\) and \(80\%\) satisfy
\[
\frac{t_{50}}{t_{80}}\;=\;\frac{\ln 0.5}{\ln 0.8}\;\approx\;3.11,
\]
for every model, at every date, independent of the hazard \(T\); the same invariance
holds for any common retry depth (the ratio is unchanged by \(\ell\)). The measured
ratio is roughly \emph{five} \citep{kwa2025}. The excess is a rejection of the
scalar benchmark in the model's direction: at high reliability requirements, feasible
horizons contract faster than a homogeneous-hazard economy allows, exactly what a task
mix heterogeneous in forgiveness produces---raising the required reliability
disproportionately excludes the unforgiving tasks. The \(50/80\) ratio is thus a
one-number specification test separating scalar from two-dimensional task economies,
and its evolution as the reliability frontier advances is a standing falsification
opportunity for the model.

\subsection{The Imitation Asymmetry and the Size of the Wedge}

Fact \ref{fact:lags} gives gap half-lives by dimension: capability indices are
reproduced in \(\tau_H\approx 3\)--\(4\) months; documented autonomy time-horizon
lags run six to twelve months, and high-reliability agentic performance sits beyond
them, so we take \(\tau_F\approx14\) months as an illustrative value for the
high-reliability margin. Reading \(\lambda_i=\ln 2/\tau_i\) and a monthly discount
rate of one percent, the relative direction distortion of Theorem \ref{t:wedge} is
\[
\frac{r+\lambda_F}{r+\lambda_H}
\;=\;\frac{0.01+\ln 2/14}{0.01+\ln 2/4}
\;\approx\;0.3 :
\]
per unit of boundary value, the market prices a horizon lead at roughly one-third of a
reliability lead. Equivalently, at equal marginal costs the rotation threshold \(\rho^*\) of
Theorem \ref{t:rotation} sits near one-third: frontier investment tips toward the unforgiving
boundary as soon as reliability-boundary value per dollar reaches a third of
horizon-boundary value per dollar. An asymmetry this large makes the model's herding
region (Theorem \ref{t:herding}) wide: rival laboratories should be observed pivoting
to the \emph{same} reliability-heavy direction, not specializing---consistent with the
simultaneous industry turn toward agentic reliability, verification, and certified
deployment as capability benchmarks commoditized.

\subsection{What Would Falsify the Theory}

The model stakes out four aggregate predictions. (1) \emph{Widening asymmetry}: as the
reliability frontier advances, the open-weight lag on high-reliability measures should
grow relative to the capability-index lag (the exponential wall of Theorem
\ref{t:distill}); convergence of the two lags would falsify the mechanism. (2)
\emph{Rotation}: frontier laboratories' investment mix should shift toward
reliability-directed projects following accelerations in capability commoditization,
with the timing ordered by Fact \ref{fact:prices}'s price-collapse episodes. (3)
\emph{Herding, not specialization}: rival laboratories' frontier portfolios should
become more similar as imitation asymmetry widens. (4) \emph{Policy cross-check}: where
public verification infrastructure expands (standardized high-stakes evaluations,
incident registries), fringe entry into fragile-task markets should follow and
frontier margins there should compress. Each prediction uses only public aggregates;
none has, to our knowledge, been assembled, and any of them could kill the model.

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

The economically relevant state of AI is a frontier in a task space with two derived
dimensions: how long a chain of actions a task requires, and how unforgiving it is of
error along the way. One O-ring-with-retries primitive delivers the geometry: serial
depth taxes reliability, forgiveness divides the tax, and the scalar quality ladder is
the knife-edge of homogeneous forgiveness. On this frontier, competition prices
capability leads by their persistence, and persistence is statistics: plans are copied
from single demonstrations, failure rates only from many. So rents are competed away
along the horizon dimension and defended along the reliability dimension; investment
rotates from the former to the latter as followers close in; rivals herd onto the moat;
and the direction of AI innovation is distorted by an observable---the asymmetry of
imitation lags---in a horse race with the safety externality that runs the other way.
Cheap models and rising frontier spending are not a paradox. They are the same force
seen from two sides: imitation erases the value of what can be demonstrated, and
thereby concentrates the race on what cannot.

\appendix

\section{Proofs}\label{app:proofs}

\subsection{Proof of Lemma \ref{l:gumbel}}

\emph{Part (i).} The requirement \(S_\ell(\pi)^n\ge1-\varepsilon\) with
\(S_\ell(\pi)=1-(1-\pi)^{1+\ell}\) is equivalent to
\(1-(1-\pi)^{1+\ell}\ge(1-\varepsilon)^{1/n}\), i.e.\ to
\(\pi\ge1-g^{1/(1+\ell)}\) with \(g\equiv1-(1-\varepsilon)^{1/n}\); since \(G_F\) is
strictly increasing this is \eqref{eq:threshold}. For the error bound, write
\(y=g^{1/(1+\ell)}\) and, with logistic \(G_F\),
\[
f(\tau)-\varphi=\operatorname{logit}(1-y)=-\ln y+\ln(1-y)
=\frac{1}{1+\ell}\ln\frac{1}{g}+\ln(1-y),
\]
so
\[
e(\tau)=\underbrace{\frac{1}{1+\ell}\,\ln\frac{\varepsilon/n}{g}}_{(a)}
+\underbrace{\ln(1-y)}_{(b)} .
\]
The integral representation \(g=\int_0^\varepsilon\frac1n(1-t)^{\frac1n-1}dt\) with
integrand between \(1/n\) and \(\frac1n(1-\varepsilon)^{\frac1n-1}\le\frac{1}{n(1-\varepsilon)}\)
gives the sandwich
\begin{equation}\label{eq:sandwich}
\frac{\varepsilon}{n}\;\le\;g\;\le\;\frac{\varepsilon}{n(1-\varepsilon)} .
\end{equation}
Hence \((a)\in[-\frac{1}{1+\ell}\ln\frac{1}{1-\varepsilon},0]\), and for
\(\varepsilon\le1/2\), \(|(a)|\le\ln\frac{1}{1-\varepsilon}\le\frac{\varepsilon}{1-\varepsilon}\le2\varepsilon\).
For \((b)\): the condition \(n\ge4^{1+\ell}\varepsilon\) means
\((\varepsilon/n)^{1/(1+\ell)}\le1/4\), and \eqref{eq:sandwich} with
\(\varepsilon\le1/2\) gives
\(y\le\big(\tfrac{2\varepsilon}{n}\big)^{1/(1+\ell)}
=2^{1/(1+\ell)}\big(\tfrac{\varepsilon}{n}\big)^{1/(1+\ell)}\le2\cdot\tfrac14=\tfrac12\). On \(y\le1/2\), \(|\ln(1-y)|\le(2\ln2)\,y\le
1.39\cdot2^{1/(1+\ell)}(\varepsilon/n)^{1/(1+\ell)}\le3(\varepsilon/n)^{1/(1+\ell)}\).
Adding the two bounds gives the claim. Monotonicities: \(g\) is increasing in
\(\varepsilon\) and decreasing in \(n\); \(y\) is increasing in \(g\) and decreasing in
\(\ell\); \(f\) is decreasing in \(y\)-threshold direction as displayed; and \(\varphi\)
enters additively. \hfill\(\square\)

\emph{Part (ii).} Let \(x_n=q_F-\varphi=(\ln n+u)/(1+\ell)\) and
\(y_n=(1-\pi_n)^{1+\ell}\) with \(\pi_n=G_F(x_n)\). For the logistic,
\(1-\pi_n=e^{-x_n}/(1+e^{-x_n})\), so
\[
n\,y_n=n\,e^{-(1+\ell)x_n}\big(1+e^{-x_n}\big)^{-(1+\ell)}
=e^{-u}\big(1+e^{-x_n}\big)^{-(1+\ell)},
\qquad e^{-x_n}=\Big(\frac{e^{-u}}{n}\Big)^{\!1/(1+\ell)}\!\!\to0 .
\]
Thus \(n y_n=e^{-u}\{1+O(n^{-1/(1+\ell)})\}\). Then
\(S_\ell(\pi_n)^n=(1-y_n)^n=\exp\{n\ln(1-y_n)\}=\exp\{-ny_n+O(ny_n^2)\}\), and
\(n y_n^2=(n y_n)\,y_n=O(1/n)\). Combining,
\(S_\ell(\pi_n)^n=\exp(-e^{-u})\{1+O(n^{-1/(1+\ell)})\}\). The kernel form follows by
setting \(u=(1+\ell)(q_F-f)\) with \(f=\varphi+(\ln n)/(1+\ell)\).
\hfill\(\square\)

\subsection{Proof of Theorem \ref{t:representation}}

Throughout, ``serves'' means \(q_H\ge h_t\) and \(q_F\ge f_t\); take \(\mathcal T\)
finite (or compact in \((h,f)\)).

\emph{(Chain \(\Rightarrow\) scalar.)} Suppose the requirement pairs form a chain. Define
\(\iota(t)=h_t+f_t\); along a chain, \(t\prec t'\) (componentwise, one strict) implies
\(\iota(t)<\iota(t')\), so \(\iota\) is a strict order embedding. The set of tasks a
system \(q\) serves is a \emph{lower set} of the chain: if \(q\) serves \(t'\) and
\(t\preceq t'\) then \(q\) serves \(t\). Hence the served set is an initial segment,
and defining \(\tilde\iota(q)=\max\{\iota(t):q\text{ serves }t\}\) (\(-\infty\) if
none; the max exists by finiteness/compactness), \(q\) serves \(t\) iff
\(\iota(t)\le\tilde\iota(q)\).

\emph{(Unordered pair \(\Rightarrow\) no scalar.)} Let \(h_1<h_2\) and \(f_1>f_2\).
System \(A=(h_2,f_2)\) serves \(t_2\) but not \(t_1\) (it fails \(f_1>f_2\)); system
\(B=(h_1,f_1)\) serves \(t_1\) but not \(t_2\) (it fails \(h_2>h_1\)). A scalar
representation would require \(\iota(t_2)\le\tilde\iota(A)<\iota(t_1)\) and
\(\iota(t_1)\le\tilde\iota(B)<\iota(t_2)\), a contradiction.

\emph{(i)} With common \((\ell,\varphi,\varepsilon)\), \eqref{eq:threshold} makes
\(f\) a strictly increasing function of \(n\), hence of \(h=\ln n\): the requirement
set lies on an increasing curve and is a chain. \emph{(ii)} With two forgiveness
classes \(\ell_1>\ell_2\), Lemma \ref{l:gumbel}(i) puts their requirement pairs on
lines of slopes \(1/(1+\ell_1)<1/(1+\ell_2)\); whenever the economy contains a long
task of the forgiving class and a short task of the unforgiving class with
\(f\)-ranking reversed---e.g.\ multi-hour tested code versus a one-shot payment
authorization---the pair is unordered and the previous paragraph applies.
\hfill\(\square\)

\subsection{Proof of Lemma \ref{l:bertrand}}

Fix a task and drop its index. Sellers simultaneously post prices; the buyer accepts
the offer maximizing \(v_k-p_k\), so seller \(k\) wins with any price
\(p_k\le v_k-(v_j-p_j)\) against best rival offer \(j\). Pricing below cost is weakly
dominated; in equilibrium the runner-up offers \(p_{(2)}=c_{(2)}\) (any higher loses
for sure against the standard undercutting argument, any lower is dominated), and the
winner posts the highest winning price
\(p_{(1)}=v_{(1)}-(v_{(2)}-c_{(2)})\), earning margin
\(s_{(1)}-s_{(2)}\ge0\). This proves (ii), and (i) since winning requires
\(s_k\ge s_j\) for all \(j\); the displacement condition is \(s_m\ge s_L\)
rearranged. For (iii), with equal serving costs the margin on each task is
\(v_L-v_m=a(h,f)\{P_H(q^L_H-h)P_F(q^L_F-f)-P_H(q^m_H-h)P_F(q^m_F-f)\}\ge0\)
(componentwise dominance and monotone kernels); integrating over tasks gives
\(\Pi=W(q^L)-W(q^m)\), and \eqref{eq:path} is the fundamental theorem of calculus
applied along the horizontal-then-vertical path, using \eqref{eq:boundary}. Part (iv)
is continuity of \(W\).\hfill\(\square\)

\subsection{Proof of Theorem \ref{t:distill}}

\emph{(i)} Let \(P=\mathrm{Bern}(\varepsilon/2)^{\otimes N}\),
\(Q=\mathrm{Bern}(\varepsilon)^{\otimes N}\). Any test \(\psi\) satisfies
\(\Pr_P(\psi=1)+\Pr_Q(\psi=0)\ge1-\mathrm{TV}(P,Q)\), and by Pinsker
\(\mathrm{TV}(P,Q)\le\sqrt{\mathrm{KL}(P\|Q)/2}=\sqrt{N\,\mathrm{KL}/2}\) with
\(\mathrm{KL}=\mathrm{KL}(\mathrm{Bern}(\varepsilon/2)\|\mathrm{Bern}(\varepsilon))\).
Error probabilities at most \(\delta\) therefore require
\(2\delta\ge1-\sqrt{N\,\mathrm{KL}/2}\), i.e.\ \(N\ge2(1-2\delta)^2/\mathrm{KL}\). The
\(\chi^2\) bound \(\mathrm{KL}(P_1\|Q_1)\le\chi^2(P_1\|Q_1)\) gives, for Bernoulli,
\[
\chi^2=\frac{(\varepsilon/2)^2}{\varepsilon}+\frac{(\varepsilon/2)^2}{1-\varepsilon}
=\frac{\varepsilon}{4(1-\varepsilon)}\;\le\;\frac{\varepsilon}{2}
\quad(\varepsilon\le1/2),
\]
hence \(N\ge4(1-2\delta)^2/\varepsilon\). As \(\varepsilon\to0\),
\(\mathrm{KL}/\varepsilon\to(1-\ln2)/2\), giving the sharp coefficient
\(4/(1-\ln2)\approx13.0\).

\emph{(ii)} By Lemma \ref{l:gumbel}(ii), the failure rate of the frontier system on a
task at its fragility frontier is
\(1-\exp\{-e^{-(1+\ell)(q_F-f)}\}\asymp n\,e^{-(1+\ell)(q_F-\varphi)}\) at the marginal
served task, so the tolerated \(\varepsilon\) a follower must certify is of this order.
With at most \(D\) usable independent observations per period,
\(T_{\mathrm{imitate}}\ge N(\varepsilon)/D\ge
4(1-2\delta)^2 e^{(1+\ell)(q_F-\varphi)}/(Dn)\) up to the constant absorbed in
\(\asymp\). The implied exponential-decay rate of the reliability gap is
\(\lambda_F(q_F)\propto1/T_{\mathrm{imitate}}\to0\). The horizon dimension needs one
successful trajectory per plan structure, so its lag is bounded by access delay:
\(\lambda_H\ge\underline\lambda_H>0\).

\emph{(iii)} Immediate from (ii) and the definition of the persistence factor.
\hfill\(\square\)

\subsection{Proof of Theorem \ref{t:rotation}}

\emph{(i)} Consider \(V(\Delta)=\mu_H\Delta_H+\mu_F\Delta_F+v_0\). The HJB equation is
\[
rV(\Delta)=\max_{x\ge0}\Big\{B_H\Delta_H+B_F\Delta_F-c(x_H)-c(x_F)
+\sum_i V_i'(\Delta)\,(x_i-\lambda_i\Delta_i)\Big\}.
\]
Matching coefficients on \(\Delta_i\): \(r\mu_i=B_i-\mu_i\lambda_i\), i.e.\
\(\mu_i=B_i/(r+\lambda_i)\); the constant \(v_0\) collects the maximized
\(\sum_i\{\mu_ix_i^*-c(x_i^*)\}\) term. The maximand is concave, so
\(x_i^*=c'^{-1}(\mu_i)\) (interior) or, with linear cost \(c_ix_i\) and a budget, the
bang-bang rule in \(\mu_i/c_i\). Since flow and dynamics are linear and the gap is
bounded under \(\lambda_i>0\), the linear \(V\) is the value function by a standard
verification argument.

\emph{(ii)} Investment is horizon-directed iff \(\mu_H/c_H>\mu_F/c_F\), i.e.\ iff
\(B_F/B_H<\rho^*=(r+\lambda_F)c_F/\{(r+\lambda_H)c_H\}\). Along a horizon push
\(q_H\) increases while \(q_F\) is constant, and
\[
\frac{d}{dq_H}\ln\frac{B_F}{B_H}
=\frac{C(q)}{B_F}-\frac{\partial B_H/\partial q_H}{B_H}\;>\;0
\]
under the stated conditions \(C>0\) and \(\partial B_H/\partial q_H\le0\), using
\eqref{eq:cross}. If the ratio eventually exceeds \(\rho^*\), it crosses it exactly
once (strict monotonicity), at a finite frontier position and hence---investment rates
being bounded below on the horizon phase---at a finite time \(T^*\); before the
crossing \(\mu_H/c_H>\mu_F/c_F\) and after it the ranking flips.

\emph{(iii)} \(\partial\rho^*/\partial\lambda_H=-(r+\lambda_F)c_F/\{(r+\lambda_H)^2c_H\}<0\)
and \(\partial\rho^*/\partial\lambda_F=c_F/\{(r+\lambda_H)c_H\}>0\). With
\(B_F/B_H\) strictly increasing along the push, a lower threshold is crossed earlier,
so \(T^*\) falls with \(\lambda_H\) and rises with \(\lambda_F\).\hfill\(\square\)

\subsection{Proof of Theorem \ref{t:herding}}

Payoffs: same direction \(k\) gives \(\sigma V_k\) each; split gives each its own
\(V_{d_i}\). Against an opponent on \(F\), playing \(F\) yields \(\sigma V_F\) and
playing \(H\) yields \(V_H\); against an opponent on \(H\), playing \(F\) yields
\(V_F\ge\sigma V_H\) always (as \(V_F\ge V_H>\sigma V_H\)). \emph{(i)} If
\(\sigma V_F\ge V_H\), \(F\) is a best reply to both opponent actions, strictly
dominant when strict, and \((F,F)\) is the unique equilibrium. \emph{(ii)} If
\(\sigma V_F<V_H\), then \(H\) is the unique best reply to \(F\) and \(F\) to \(H\):
the pure equilibria are \((F,H)\) and \((H,F)\). Indifference for the mixed
equilibrium, with \(p=\Pr(\text{opponent plays }F)\):
\(p\,\sigma V_F+(1-p)V_F=p\,V_H+(1-p)\sigma V_H\), giving
\[
p^*=\frac{V_F-\sigma V_H}{(V_F-\sigma V_H)+(V_H-\sigma V_F)}
=\frac{\rho-\sigma}{(1-\sigma)(\rho+1)},\qquad\rho=\frac{V_F}{V_H},
\]
which lies in \((0,1)\) exactly on the stated region, and
\(dp^*/d\rho=(1+\sigma)/\{(1-\sigma)(\rho+1)^2\}>0\). \emph{(iii)} The herding condition
\(\sigma\ge V_H/V_F=1/\rho\) relaxes as \(\rho\) rises; \(\rho\) is increasing in
\(\mu_F/\mu_H=\{B_F/B_H\}\{(r+\lambda_H)/(r+\lambda_F)\}\), which is increasing in
\(\lambda_H\) and decreasing in \(\lambda_F\).\hfill\(\square\)

\subsection{Proof of Theorem \ref{t:wedge} and Corollary \ref{c:policy}}

The planner's flow surplus is \(W(q^L(t))\): served value depends on the best
available capability, and reallocation of margins between frontier and fringe is a
transfer. A unit of direction-\(i\) capability raises the flow by \(B_i\) permanently,
so its present value is \(B_i/r\). The market values it at \(\mu_i=B_i/(r+\lambda_i)\)
by Theorem \ref{t:rotation}(i). The ratio of relative valuations is
\[
\frac{\mu_H/\mu_F}{(B_H/r)/(B_F/r)}
=\frac{r+\lambda_F}{r+\lambda_H},
\]
which is below one iff \(\lambda_H>\lambda_F\), proving (i); (ii) follows by replacing
the planner's \(B_F\) with \(\theta_sB_F\) and comparing
\(\theta_s\) with \((r+\lambda_H)/(r+\lambda_F)\). For the corollary, quadratic costs
give \(x_F^*=\mu_F\); under liability the private reliability boundary value is
\(\theta B_F\), so
\[
x_F^*=\frac{\theta B_F}{r+\lambda_F},\qquad
\Delta_F^*=\frac{x_F^*}{\lambda_F}=\frac{\theta B_F}{(r+\lambda_F)\lambda_F},
\]
whence \(\partial x_F^*/\partial\theta>0\) and \(\partial\Delta_F^*/\partial\theta>0\).
Differentiating in \(\lambda_F\):
\(\partial\Delta_F^*/\partial\lambda_F
=-\theta B_F(r+2\lambda_F)/\{(r+\lambda_F)^2\lambda_F^2\}<0\) and
\(\partial x_F^*/\partial\lambda_F<0\); the fringe's steady-state reliability is
\(q_F^L-\Delta_F^*\), which rises toward the frontier as \(\lambda_F\) rises.
\hfill\(\square\)

\section{Machine Verification}\label{app:verification}

Every formal claim in the paper is checked in a companion script,
\texttt{math\_check\_v3.py}, which verifies symbolically (SymPy) and numerically:
the threshold identity and sandwich \eqref{eq:sandwich}; the two-term error bound and
the \(1/(1+\ell)\) slope of Lemma \ref{l:gumbel}(i); the Gumbel limit and its
\(O(n^{-1/(1+\ell)})\) rate; the parameter-free \(\ln0.5/\ln0.8\approx3.106\) ratio and
its invariance to per-stage success and retries; the representation logic of Theorem
\ref{t:representation} on explicit task sets; the derivative, cross-derivative, and
path identities \eqref{eq:boundary}--\eqref{eq:path}; the Bertrand margin and
profit--gap identity and saturation limit of Lemma \ref{l:bertrand}; the
\(\chi^2\)/KL bounds and sample-floor constants of Theorem \ref{t:distill}; the HJB
coefficients, rotation single-crossing, and threshold comparative statics of Theorem
\ref{t:rotation} on a fully specified two-family task economy; the equilibrium regions,
mixed-strategy formula, and monotonicity of Theorem \ref{t:herding}; the wedge algebra
of Theorem \ref{t:wedge}; the steady-state policy derivatives of Corollary
\ref{c:policy}; and the calibration arithmetic of Section \ref{sec:quant}. The script
reports \(0\) failures.

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