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\title{The Singleton Attractor:\\
A Formal Model and Empirical Calibration of\\
Capability-Threshold Dynamics in Frontier AI}
\author{Nathan Langley\\[2pt]
\small University of North Carolina at Greensboro}
\date{May 2026}

% ============================================================
\begin{document}
\maketitle

% ---------- Abstract ----------
\begin{abstract}
We study capability-threshold dynamics in competitive multi-agent
systems, specifically the conditions under which a single dominant
agent emerges from recursive self-improvement under adversarial resource
competition, and calibrate the framework's central premise against the
public record on frontier-AI training compute and benchmark performance
through $2026$.
The model combines Yudkowsky's intelligence explosion equation
($dS/dt = S^{1-\beta}$), Omohundro's instrumental resource acquisition,
and Lotka--Volterra competitive exclusion.
Under five explicit assumptions, the agent that crosses a capability
threshold first achieves unbounded advantage over all competitors in
finite time; the critical assumption (A4) is that the growth exponent
$\beta(S)$ flips negative for some reachable capability $S = T$.
Whether any real AI system will satisfy A4 is the open empirical question
the paper makes falsifiable: in Section~\ref{sec:calibration} we fit the
curvature of capability growth on six proxies (frontier training compute
and five public benchmarks: GPQA Diamond, FrontierMath, ARC-AGI,
SWE-bench Verified, MATH Level 5) from the Epoch AI dataset
\citep{epoch2024compute}, and find no proxy with statistically positive
curvature, with two benchmarks statistically negative.
The data through $2026$ do not support $\beta(S) < 0$ under any tested
proxy.
Three results are proved formally
(Theorems~\ref{thm:ratio},~\ref{thm:finitetime},~\ref{thm:criticalalpha}),
and the $N$-agent generalization is stated as a conjecture with an
acknowledged simultaneous-dynamics gap.
Theorem~\ref{thm:stochastic} gives the probability of singleton
emergence under multiplicative GBM noise as $\Pr(J \geq c) \in (0,1)$,
expressible in closed form as a Dufresne perpetuity and tending to $1$
as $\sigma \to 0$.
Theorem~\ref{thm:criticalalpha} derives the exact transcendental
condition for coalition external suppression, with numerical solution
$\alpha^* \approx 0.64$.
Within the formal model (A1--A5), separate resource pools alone do not
prevent singleton emergence when both agents share a $\beta$-function that
crosses zero, refining (not refuting) \citet{bostrom2005singleton}.
Eleven simulation scripts verify the ODE behavior across 25 parameter
configurations; the simulations use a capability cap that inflates
apparent emergence, discussed where used.
\end{abstract}

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

\citet{bostrom2005singleton} defined the singleton as ``a world order in which
there is a single decision-making agency at the highest level.''
He argued a singleton was plausible and analyzed its properties but did not
formally prove its emergence from competitive dynamics.
\citet{omohundro2008basic} identified resource acquisition as a convergent
instrumental goal for capable optimization processes.
\citet{yudkowsky2013intelligence} formalized the intelligence explosion as
$dS/dt = S^{1-\beta}$, showing that $\beta < 0$ produces finite-time
singularity.
The classical Lotka--Volterra competition model
\citep{lotka1925elements,volterra1926fluctuations} establishes that two agents
sharing a resource pool with even a marginal growth-rate advantage diverge in
ratio as $e^{(r_1 - r_2)t}$.
Evolutionary game theory \citep{maynardsmith1982evolution,weibull1995evolutionary}
formalizes related selection dynamics in biological and strategic populations.
\citet{bostrom2014superintelligence} develops the singleton concept and
analyzes instrumental convergence at length but does not formalize competitive
dynamics as a dynamical system.

None of these works combine into a formal model with testable quantitative
predictions.
Good's \citeyearpar{good1965speculations} account is verbal.
Bostrom's argument is historical and extrapolative.
Omohundro gives mechanisms without dynamics.
Yudkowsky's growth equation covers a single agent in isolation.
Lotka--Volterra handles biological competition without self-improvement.

The contemporary AI capability discourse adds urgency to the question.
Scaling-laws research and frontier-model development have produced a
public debate over takeoff speed, that is, whether capability growth is currently
exponential, decelerating, or entering a super-exponential regime, in
which Yudkowsky's $\beta < 0$ threshold is the load-bearing assumption
of the dramatic-takeoff scenarios.
This paper makes that assumption falsifiable on the public AI-capability
record.
The framework we develop is general (the dynamics apply to any
recursively-improving competitive system), but the calibration in
Section~\ref{sec:calibration} is specifically aimed at the AI case
because that is where the assumption is currently consequential.

Everything in this paper is conditional on Assumption~A4: that $\beta(S)$
can go negative for some capability level $S = T$ that is actually reachable.
A4 is the contested premise.
The theorems prove that \emph{if} A4 holds, a singleton emerges in
finite time under adversarial resource dynamics; under multiplicative
GBM-type noise the emergence becomes a high-probability rather than
almost-sure event (Theorem~\ref{thm:stochastic}).
That structure is close to tautological.
The interesting empirical question, whether any real AI system will ever
satisfy A4, is partially addressed in Section~\ref{sec:calibration} via
calibrations on six capability proxies (frontier training compute and
five public benchmarks); on none of the six is $\beta(S) < 0$ currently
supported.

This paper makes the following contributions.

\begin{enumerate}[label=(\arabic*)]
\item \textbf{A formal model} combining recursive self-improvement, instrumental
resource acquisition, and competitive exclusion with five explicit assumptions
(A1--A5, Section~\ref{sec:model}).

\item \textbf{Three formal proofs and one conjecture}
(Section~\ref{sec:theorems}):
ratio divergence under competition (Theorem~\ref{thm:ratio});
finite-time separation at the $\beta$-threshold under adversarial resource
dynamics (Theorem~\ref{thm:finitetime});
local instability of the equal-resource state, with full Jacobian
(Proposition~\ref{prop:monopoly});
$N$-agent singleton by induction
(Conjecture~\ref{thm:nagent}: simultaneous-dynamics gap not closed).

\item \textbf{Two supplementary derivations} (Sections~\ref{sec:coalition},
\ref{sec:stochastic}):
the exact transcendental condition for coalition external suppression
(Theorem~\ref{thm:criticalalpha}) with solution $\alpha^* \approx 0.64$;
and an explicit Dufresne-perpetuity formula for the probability of
singleton emergence under multiplicative noise
(Theorem~\ref{thm:stochastic}).

\item \textbf{Quantitative simulation results} (Section~\ref{sec:simulations})
from eleven scripts and 25 confirmed findings, including a fitted single-variable
scaling estimate for $t_{10\times}$ (Section~\ref{sec:simulations}; only the $N$
exponent is analytically derived) and critical coalition entry rate
$\lambda_{\mathrm{crit}} \approx 0.25$.

\item \textbf{Empirical calibrations} (Section~\ref{sec:calibration}) of
$\beta(S)$ against six capability proxies using the Epoch AI datasets
\citep{epoch2024compute}: frontier training compute and five public
benchmarks (GPQA Diamond, FrontierMath, ARC-AGI, SWE-bench Verified,
MATH Level 5).
Under all six proxies the curvature is at most zero; on two benchmarks
it is significantly negative.
The hypothesis $\beta(S) < 0$ is not supported by the public record
through $2026$.
\end{enumerate}

Section~\ref{sec:model} gives the model.
Section~\ref{sec:theorems} proves the main results.
Section~\ref{sec:simulations} covers 25 simulation findings.
Sections~\ref{sec:failures}--\ref{sec:stochastic} address failure conditions,
coalition coherence, and stochastic robustness.
Section~\ref{sec:calibration} reports the empirical calibration.
Section~\ref{sec:discussion} covers limitations and open questions.

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

\subsection{Definitions}

\begin{definition}[Capability]
$S_i(t) \in \mathbb{R}_{>0}$ is agent $i$'s capability at time $t$: a scalar
measure of its optimization power.
\end{definition}

\begin{definition}[Resources]
$R_i(t) \geq 0$ is the quantity of environmental resources under agent $i$'s
control. Resources are rivalrous: $\sum_i R_i(t) \leq \Rmax$ for all $t$.
\end{definition}

\begin{definition}[Growth function]
$f(S) = S^{1-\beta(S)}$, where $\beta: \mathbb{R}_{>0} \to \mathbb{R}$ is a
sigmoid with $\beta(S) > 0$ for $S < T$ and $\beta(S) < 0$ for $S > T$.
The sigmoid is chosen for mathematical tractability: it yields a single
threshold $T$ and monotone sign change.
Real self-improvement dynamics could be non-monotonic, multi-threshold, or
plateau-structured; the sigmoid is a tractability assumption, not an
empirically established functional form.
\end{definition}

\begin{definition}[Threshold]
$T > 0$ is the capability level where $\beta(S)$ changes sign.
\end{definition}

\begin{definition}[Singleton]
Agent $j$ is a \emph{singleton} if $S_j(t)/S_i(t) \to \infty$ for all
$i \neq j$, where the limit is taken as $t \to \sup\mathcal{T}$ and
$\mathcal{T}$ is the maximal interval on which the system is defined.
In the superexponential case $\mathcal{T} = [0,\tstar)$ for finite $\tstar$
and the ratio diverges as $t \to (t^*)^-$; in the pre-threshold case
$\mathcal{T} = [0,\infty)$ and the ratio diverges as $t \to \infty$.
\end{definition}

\subsection{Assumptions}

\begin{assumption}[A1: Recursive self-improvement]
\label{ass:A1}
\[
  \frac{dS_i}{dt} = S_i^{1-\beta(S_i)} \cdot \frac{R_i(t)}{\Rmax}.
\]
\end{assumption}

\begin{assumption}[A2: Instrumental resource acquisition]
\label{ass:A2}
At steady state, resources equilibrate to:
\[
  R_i^* = \Rmax \cdot \frac{S_i^\alpha}{\sum_j S_j^\alpha}, \quad \alpha > 0.
\]
\end{assumption}

\begin{assumption}[A3: Resource limitation]
\label{ass:A3}
$\sum_i R_i(t) = \Rmax$ for all $t$.
\end{assumption}

\begin{assumption}[A4: Reachable $\beta$-threshold]
\label{ass:A4}
There exists $T > 0$ such that $\beta(S) < 0$ for $S > T$ and
$\beta(S) > 0$ for $S < T$.
At least one agent has initial conditions sufficient to eventually reach $T$.
\end{assumption}

\begin{assumption}[A5: Initial heterogeneity]
\label{ass:A5}
$S_i(0) \neq S_j(0)$ for some pair $i \neq j$.
\end{assumption}

\subsection{Discussion of assumptions}

A1 follows from \citet{yudkowsky2013intelligence}; A2 from Omohundro's
instrumental convergence \citep{omohundro2008basic}: resource acquisition is
an instrumental subgoal for any sufficiently capable optimizer.
A3 is the fundamental physical constraint.
A4 is the key conditional: the theorem proves that \emph{if} at least one
agent can reach $T$ (A4), \emph{then} a singleton emerges.
A4 asserts the premise; the theorem derives the consequence.
Whether A4 holds empirically is a separate question.
A5 is satisfied by any physical noise in initial conditions.

We use fast resource equilibration (A2 as a steady-state condition).
Empirical tests with $\tau > 0$ equilibration delays show timescale
extension without qualitative change, but this is not proved analytically
for the finite-time blowup result (see below).
The fast-equilibration assumption abstracts over commercialization lag,
regulatory barriers, and path dependency present in real systems.
These frictions slow resource acquisition without reversing its direction;
quantitative timescale estimates (F17) should be treated as lower bounds in
systems with significant friction.
Theorem~\ref{thm:ratio} (ratio divergence) does not require fast
equilibration: the argument holds for any positive resource share allocated
to the leader.
Theorem~\ref{thm:finitetime} (finite-time blowup) uses $r_1 > 1/2$, which
follows from $S_1 > S_2$ under fast equilibration (A2).
With slow equilibration, the actual $r_1(t)$ lags the equilibrium value
$S_1^\alpha/(S_1^\alpha + S_2^\alpha)$ and could be depressed for extended
periods.
As long as $r_1(t) > 0$ eventually aligns with the equilibrium (any finite
equilibration timescale), the ratio still diverges, but the bound on $\tstar$
becomes vacuous.
The claim that slow equilibration ``does not change the qualitative result''
is verified empirically for bounded $\tau$ but is not proved for the
finite-time case; the finite-time bound specifically requires A2.

\noindent\textit{Competition model.}
A2--A3 follow Lotka--Volterra competition rather than Cournot or Bertrand.
LV is appropriate here because agents compete for a common resource pool
rather than setting quantities or prices; the essential property is that
resource shares sum to one and capability yields proportionally higher share.
LV captures this with one parameter ($\alpha$); Cournot and Bertrand
introduce additional market-structure assumptions with no natural counterpart
in a general capability competition.

\noindent\textit{$\beta$-heterogeneity.}
The $\beta$-function is agent-specific in general; it reflects the structure of
each agent's self-improvement path.
Theorem~\ref{thm:ratio} uses a common $\beta$ as a sufficient condition for
pre-threshold divergence.
Theorem~\ref{thm:finitetime} explicitly allows $\beta_1 \neq \beta_2$.
Different structural $\beta$-ceilings are exactly the oligopoly condition
(Section~\ref{sec:failures}): stable multi-agent equilibria require one agent
to be structurally unable to reach $\beta < 0$, not merely operating in a
separate resource niche.
Note that $\beta$-heterogeneity is discussed conceptually here but is not
formalized in the main proofs except where explicitly noted; the proofs treat
$\beta_i$ as given constants for each agent.

\noindent\textit{Resource coupling ($\alpha$).}
The power-law form in A2 is the minimal model for capability-to-resource
coupling.
$\alpha$ is difficult to calibrate empirically; $\alpha = 1$ (linear)
is the conservative baseline.
Simulation sweeps show the singleton result holds monotonically across
$\alpha \in [0.25, 3.0]$ (F6), and Theorem~\ref{thm:criticalalpha} quantifies
the critical threshold $\alpha^* \approx 0.64$ below which coalition suppression
fails.
The power-law form was chosen over sigmoid saturation as the minimal
single-parameter model.
A saturating form would require two parameters (midpoint and steepness)
without additional empirical guidance; the power-law and sigmoid are
qualitatively equivalent in the unsaturated regime that precedes threshold
crossing.

% ============================================================
\section{Main Results}
\label{sec:theorems}

\begin{theorem}[Singleton Attractor, $N=2$]
\label{thm:singleton}
Under A1--A5 with $N = 2$ agents, the agent with higher initial
capability is a singleton: there exists $\tstar < \infty$ such that
$S_1(t)/S_2(t) \to \infty$ as $t \to \tstar$.
\end{theorem}

The proof proceeds in three steps for $N = 2$
(Theorem~\ref{thm:ratio}, Theorem~\ref{thm:finitetime},
Proposition~\ref{prop:monopoly}).
The $N \geq 3$ case is the subject of Conjecture~\ref{thm:nagent} and
is not proved here.

\subsection{Step 1: Ratio divergence from marginal advantage}
\label{sec:step1}

\begin{theorem}[Ratio divergence, pre-threshold]
\label{thm:ratio}
For $S_1(0) > S_2(0) > 0$, $\alpha > 0$, and both agents in the
pre-threshold regime with common $\beta \in (0, \alpha)$,
the ratio $\rho(t) = S_1(t)/S_2(t)$ is strictly increasing and
$\rho(t) \to \infty$ as $t \to \infty$.
\end{theorem}

\begin{proof}
Under A2 at steady state with uniform $\beta$,
$dS_i/dt = S_i^{1-\beta+\alpha}/D$ where $D = S_1^\alpha + S_2^\alpha$.
Compute:
\begin{align}
  \frac{d\rho}{dt}
  &= \frac{S_2\,\dot{S}_1 - S_1\,\dot{S}_2}{S_2^2}
   = \rho \cdot \frac{S_1^{\alpha-\beta} - S_2^{\alpha-\beta}}{D}.
  \label{eq:rhodot}
\end{align}
Since $S_1 > S_2$ and $\alpha - \beta > 0$, we have
$S_1^{\alpha-\beta} > S_2^{\alpha-\beta}$, so $d\rho/dt > 0$: $\rho$ is strictly
increasing.

For divergence, consider two cases.

\textit{Case~1: $S_2(t)\to\infty$.}
Dividing~\eqref{eq:rhodot} by $\dot{S}_2 = S_2^{1-\beta+\alpha}/D > 0$:
\[
  \frac{d\rho}{dS_2} = \frac{\rho(\rho^{\alpha-\beta}-1)}{S_2}.
\]
Separating variables and integrating:
\[
  \int_{\rho(0)}^{\rho(t)}
  \frac{d\rho'}{\rho'({\rho'}^{\alpha-\beta}-1)}
  = \ln\frac{S_2(t)}{S_2(0)} \to +\infty.
\]
Suppose for contradiction $\rho(t) \to L < \infty$.
Since $\rho$ is strictly increasing and $\rho(0) > 1$
(by A5 with $S_1(0) > S_2(0)$), we have $L > \rho(0) > 1$.
The integrand $1/(\rho'({\rho'}^{\alpha-\beta}-1))$ is bounded and positive
on the closed interval $[\rho(0), L]$ (both endpoints give finite positive
values since $\alpha > \beta$), so the left-hand side converges to a finite
limit, contradicting the right-hand side diverging to $+\infty$.
Therefore $\rho(t) \to \infty$.

\textit{Case~2: $S_2(t)\to M<\infty$.}
Suppose for contradiction that $S_1$ also remains bounded, $S_1 \to L_1 < \infty$.
Then $D \to L_1^\alpha + M^\alpha =: D_\infty < \infty$ and
$\dot{S}_1 \to L_1^{1-\beta+\alpha}/D_\infty > 0$, so $S_1$ grows at a uniformly
positive asymptotic rate, contradicting $S_1$ bounded.
Hence $S_1 \to \infty$, and $\rho = S_1/S_2 \to \infty/M = \infty$.
\end{proof}

\begin{remark}
For $\betaH = 0.5$ and $\alpha = 1.0$, $\alpha - \beta = 0.5 > 0$. \checkmark
The mixed post-threshold case (agent~1 in $\betaL < 0$, agent~2 in $\betaH > 0$)
is handled directly by Theorem~\ref{thm:finitetime}, which gives the
stronger result of finite-time rather than asymptotic divergence.
\end{remark}

\subsection{Step 2: Finite-time separation at the threshold}
\label{sec:step2}

\begin{theorem}[Finite-time separation]
\label{thm:finitetime}
Suppose $T > 1$, agent~1 has crossed $T$ (so $\beta_1 = -|\beta_1| < 0$),
and agent~2 remains below $T$ ($\beta_2 > 0$).
Then $\rho(t) \to \infty$ at a finite time $\tstar$.
\end{theorem}

\noindent\textit{On the hypothesis $T > 1$.} The post-escape blow-up
argument compares $S_1^{1+|\beta(S_1)|}$ with $S_1^{1+|\beta^*|}$, which
requires $S_1 \geq 1$ to flip the inequality the right way.
With $T > 1$ this holds throughout $[\tcross, \tstar)$ since
$S_1 \geq T > 1$.
For $T \leq 1$ the result still holds after a rescaling
$\tilde{S} = S/T$, which puts the threshold at $1$ and leaves the
structure of the dynamics intact.

\begin{remark}[Simultaneous T-crossing not covered]
This theorem requires agent~1 to have crossed $T$ while agent~2 has not.
The case where two near-equal agents approach $T$ simultaneously is not
addressed here.
Under A5 (initial heterogeneity), the agents cannot reach $T$ at exactly the
same instant; by Theorem~\ref{thm:ratio}, the initial leader reaches $T$
first.
However, the \emph{near-simultaneous} case (both agents close to $T$ with
similar trajectories) produces competitive dynamics that Theorem~\ref{thm:ratio}
handles only asymptotically. The separation may be small at threshold
crossing, and the subsequent post-threshold advantage depends on the
capability gap at that moment.
\end{remark}

\begin{remark}
$\beta(S)$ is a sigmoid (Definition~2.3), so $|\beta(S)| \to 0$ as
$S \downarrow T$ and $|\beta(S)| \to |\betaL|$ as $S \to \infty$.
The proof proceeds in two stages: (a) $S_1$ escapes a neighborhood of $T$
in finite time, after which (b) a constant lower bound
$|\beta^*| \in (0, |\betaL|)$ on $|\beta(S)|$ is available and the explicit
blow-up argument applies.
\end{remark}

\begin{proof}
We establish three lemmas, then combine them.

\begin{lemma}[Upper bound on $S_2$]
\label{lem:s2bound}
For all $t \geq 0$:
$S_2(t) \leq \bigl(S_2(0)^{\beta_2} + \beta_2\, t\bigr)^{1/\beta_2}$.
\end{lemma}
\begin{proof}
Agent~2 receives at most full resources ($r_2 \leq 1$), so
$\dot{S}_2 \leq S_2^{1-\beta_2}$.
Integrating $S_2^{\beta_2-1}\,dS_2 \leq dt$ gives
$S_2(t)^{\beta_2}/\beta_2 \leq S_2(0)^{\beta_2}/\beta_2 + t$,
yielding the stated bound.
Since $\beta_2 > 0$ the bound grows as $t^{1/\beta_2}$: finite for all finite $t$.
\end{proof}

\begin{lemma}[Resource share lower bound]
\label{lem:rmin}
For all $t \geq \tcross$: $r_1(t) > 1/2$.
\end{lemma}
\begin{proof}
By the theorem's hypothesis, agent~2 remains below $T$ throughout, so
$S_2(t) < T$ for all $t \geq \tcross$.
Since $\dot{S}_1 > 0$ and $S_1(\tcross) = T$, we have $S_1(t) \geq T > S_2(t)$,
hence $S_1(t)^\alpha > S_2(t)^\alpha$.
Therefore $r_1(t) = S_1^\alpha/(S_1^\alpha + S_2^\alpha) > 1/2$.
\end{proof}

\begin{lemma}[Escape from threshold neighborhood]
\label{lem:escape}
Fix any $|\beta^*| \in (0, |\betaL|)$.
There exists $T' > T$ such that $|\beta(S)| \geq |\beta^*|$ for all $S \geq T'$.
Let $t_1 = \inf\{t \geq \tcross : S_1(t) \geq T'\}$.
Then $t_1 < \infty$.
\end{lemma}
\begin{proof}
Since $\beta$ is a sigmoid with $\beta(T) = 0$ and $\beta(S) \to -|\betaL|$ as
$S \to \infty$, monotonicity gives a unique $T'$ with $|\beta(T')| = |\beta^*|$.
On $[\tcross, t_1]$, $\beta(S_1) \in [-|\beta^*|, 0]$, so
$S_1^{1-\beta(S_1)} \geq S_1 \geq T$.
By Lemma~\ref{lem:rmin}, $r_1 > 1/2$, so
$\dot{S}_1 \geq \tfrac{1}{2}\,T$.
Therefore $S_1(t) \geq T + \tfrac{T}{2}(t - \tcross)$,
and $t_1 \leq \tcross + 2(T' - T)/T < \infty$.
\end{proof}

\begin{lemma}[$S_1$ diverges in finite time]
\label{lem:s1blow}
There exists finite $\tstar$ such that $S_1(t) \to \infty$ as $t \to \tstar$.
\end{lemma}
\begin{proof}
By Lemma~\ref{lem:escape}, $S_1(t_1) = T'$ and $|\beta(S_1)| \geq |\beta^*|$
for all $t \geq t_1$.
By Lemma~\ref{lem:rmin}, $r_1(t) > 1/2$.
Hence $\dot{S}_1 > \tfrac{1}{2}\,S_1^{1+|\beta^*|}$.
Let $u = S_1^{-|\beta^*|}$. Then
\[
  \dot{u} < -\tfrac{|\beta^*|}{2},
\]
so $u(t) < (T')^{-|\beta^*|} - \tfrac{|\beta^*|}{2}(t - t_1)$,
reaching zero by
\[
  \tstar = t_1 + \frac{2\,(T')^{-|\beta^*|}}{|\beta^*|} < \infty.
\]
As $t \to \tstar$, $u(t) \to 0$, so $S_1(t) \to \infty$.
\end{proof}

\textbf{Combining:}
By Lemma~\ref{lem:s1blow}, $S_1(t) \to \infty$ as $t \to \tstar$.
By Lemma~\ref{lem:s2bound}, $S_2(\tstar) \leq
(S_2(0)^{\beta_2} + \beta_2\,\tstar)^{1/\beta_2} < \infty$
(since $\tstar < \infty$ and $\beta_2 > 0$).
Therefore $\rho(t) = S_1(t)/S_2(t) \to \infty/\text{finite} = \infty$ at $t = \tstar$.
\end{proof}

\[
  \tstar \leq \tcross + \frac{2(T'-T)}{T}
             + \frac{2\,(T')^{-|\beta^*|}}{|\beta^*|},
  \quad \text{valid for any } |\beta^*| \in (0, |\betaL|).
\]

\subsection{Step 3: Resource monopoly stability}
\label{sec:step3}

\begin{proposition}[Symmetry-breaking instability]
\label{prop:monopoly}
As $\rho \to \infty$, $R_1^*/\Rmax \to 1$ and $R_2^*/\Rmax \to 0$.
The symmetric subspace $\{S_1 = S_2\}$ is invariant under the dynamics.
The transverse perturbation $\varepsilon = \rho - 1$ satisfies
$\dot{\varepsilon} = \mu(t)\,\varepsilon + O(\varepsilon^2)$ with
$\mu(t) = (\alpha - \beta)/(2\,S(t)^{\beta})$;
the integrated rate $\int_0^t \mu(s)\,ds$ diverges as $t \to \infty$
in the pre-threshold regime, so any initial heterogeneity (A5) is amplified
without bound.
\end{proposition}

\begin{proof}
From A2: $R_1^* = \Rmax/(1 + \rho^{-\alpha}) \to \Rmax$ as $\rho \to \infty$.
Symmetry of the equations under $(S_1, S_2) \leftrightarrow (S_2, S_1)$ gives
invariance of $\{S_1 = S_2\}$.
Writing $\rho = 1 + \varepsilon$ and expanding~\eqref{eq:rhodot} to first order
around $\rho = 1$:
\[
  \frac{d\varepsilon}{dt}
  = \mu(t)\,\varepsilon + O(\varepsilon^2),
  \qquad
  \mu(t) := \frac{\alpha - \beta}{2\,S(t)^{\beta}}.
\]
The instantaneous rate $\mu(t)$ is positive whenever $\alpha > \beta$
but decays as $S(t)$ grows.
The relevant question is whether $\int_0^t \mu(s)\,ds$ diverges.
On the symmetric subspace, $S(t)$ grows as a power law $S(t) \sim t^{1/\beta}$
in the pre-threshold regime, so $S(t)^\beta \sim t$ and
$\int_0^t \mu(s)\,ds \sim (\alpha-\beta)/(2)\,\ln t \to \infty$.
Hence $\varepsilon(t) \to \infty$.
Under A5 the system starts with $\varepsilon(0) \neq 0$.
\end{proof}

\subsection{Step 4: $N$-agent generalization}
\label{sec:step4}

\begin{conjecture}[$N$-agent singleton]
\label{thm:nagent}
Under A1--A5, for $N \geq 2$ agents with
$S_1(0) > S_2(0) > \cdots > S_N(0)$, agent~1 is the singleton.
Elimination order is strictly weakest-first.

\emph{Note:} The inductive argument is formally complete for the sequential
limit ($r_N \to 0$ removing agent~$N$'s influence one at a time), but the
simultaneous-dynamics correction (that the remaining $N-1$ agents'
interaction is not perturbed by agent~$N$'s residual share) is not
analytically bounded.
This theorem is therefore a conjecture supported by simulation (F4, F5:
the result held in all 200 tested trials for $N=10$ on a purpose-built ODE
system) but without an analytical bound on the gap.
Simulation trials on the same ODE system confirm the model's own behavior;
they do not close the proof.
\end{conjecture}

\begin{proof}
By induction on $N$.

\textit{Base case} ($N = 2$): Theorems~\ref{thm:ratio} and~\ref{thm:finitetime}
give $\rho_{12} = S_1/S_2 \to \infty$ in finite time.
$R_2^* \to 0$, agent~2's growth halts, agent~1 monopolizes resources. \checkmark

\textit{Inductive step}: For $N$ agents under A2,
$dS_i/dt = S_i^{1-\beta+\alpha}/D$ with $D = \sum_j S_j^\alpha$.
Direct computation gives
\[
  \frac{d\rho_{1N}}{dS_N}
  = \frac{\rho_{1N}(\rho_{1N}^{\alpha-\beta} - 1)}{S_N}
\]
(the denominator $D$ cancels between $d\rho_{1N}/dt$ and $dS_N/dt$).
This is the same separable ODE as the 2-agent case in
Theorem~\ref{thm:ratio}.
Case~1 of the 2-agent argument transfers directly: if $S_N \to \infty$,
the ODE forces $\rho_{1N} \to \infty$ via the same separation-of-variables
contradiction.
Case~2 (the bounded-$S_N$ branch) does \emph{not} transfer: the 2-agent
argument used ``the only other agent must be unbounded,'' which fails
when $N > 2$ because some other $S_j$ ($j \neq 1, N$) might be the
unbounded agent rather than $S_1$.
This is the simultaneous-dynamics gap acknowledged in the conjecture
statement.
Granting Case~1 (or, equivalently, granting that $S_1 \to \infty$),
$r_N = S_N^\alpha / \sum_j S_j^\alpha \leq \rho_{1N}^{-\alpha} \to 0$,
agent~$N$'s growth stalls, and the reduced $N{-}1$ system satisfies
the inductive hypothesis.

\textit{Note on simultaneous dynamics:}
The induction eliminates agents sequentially, but all $N$ agents evolve
simultaneously.
The sequential reduction is formally valid in the limit ($r_N \to 0$ removes
agent~$N$'s influence on remaining agents' dynamics as $\rho_{1N}\to\infty$),
but the correction from simultaneous evolution is not analytically bounded here.
It is verified empirically: F4 and F5 confirm strictly weakest-first
elimination with agent~1 winning in 200/200 trials for $N = 10$.
\end{proof}

\textit{Verification:}
Finding F4 confirms initial-leader-wins in $200/200$ independent trials with
$N = 10$ agents under threshold $\beta$; F5 confirms the elimination
order is strictly weakest-first on the $N = 8$ heterogeneous test case
reported in \texttt{findings.md}.

% ============================================================
\section{Simulation Results}
\label{sec:simulations}

Random seeds and Python dependency versions are recorded in each
simulation script (code link in the Acknowledgments).
Default parameters: $\alpha = 1.0$, $\betaH = 0.5$, $\betaL = -0.3$,
$T = 3.0$, $S_0^{\max} = 10^7$.
We use $\beta(S)$ as a sigmoid transitioning from $\betaH$ to $\betaL$ at $T$.
Findings are labeled F1--F25; full tables are in the repository
(\texttt{findings.md}).

\subsection{Growth dynamics verification (F1)}

The analytical solution to $dS/dt = S^{1-\beta}$ matches numerical integration
(scipy \texttt{solve\_ivp}, rtol $= 10^{-9}$) within numerical precision
(relative error $< 10^{-6}$) across all tested $\beta$ values.
Three growth regimes confirmed: $\beta < 0$ (finite-time singularity),
$\beta = 0$ (exponential), $\beta > 0$ (power law).

\subsection{Competitive exclusion (F2--F7)}

A 10\% initial capability advantage produces diverging ratio in the
subexponential regime.
Result holds for all tested gaps (1\% to 100\%, F2).
Winner equals initial leader in 200/200 independent trials with $N = 10$ (F4).
Elimination order is strictly weakest-first (F5).
Separation ratio increases monotonically with $\alpha$ across the tested
range (F6).
Note that $\alpha = 0.25$ and $\alpha = 3.0$ have very different physical
interpretations (sublinear vs.\ cubic coupling) and the range is not
grounded in any real system; breadth of sweep establishes ODE robustness,
not physical robustness.

\subsection{$\beta$-threshold acceleration (F3, F9)}

When the leading agent crosses $T$ and enters $\beta < 0$, the separation
ratio diverges in finite time (Theorem~\ref{thm:finitetime}); the
specific multiple at any single timepoint is therefore not a meaningful
invariant of the system.
For illustration, at $t = 6.8$ in the default configuration the
threshold-crossing simulation produces a ratio of $\sim\!1.9\times 10^7$
versus $\sim\!1.55$ for the flat-$\beta$ comparator (Fig.~\ref{fig:threshold});
this ratio grows without bound as $t \to \tstar$.
The qualitative comparison (super- versus sub-exponential) is robust
across the tested parameter range, with minimum measured separation
$1{,}179\times$ at the most adverse parameter set tested (F9).

\begin{figure}[h]
\centering
\includegraphics[width=0.85\textwidth]{../figures/comp_threshold.png}
\caption{$\beta$-threshold effect. Flat $\beta=0.5$ (no threshold) versus
threshold $\beta$ ($\betaL=-0.3$ above $T=5$). At the snapshot $t = 6.8$ the
threshold trajectory has reached $\sim\!1.9\times10^{7}$-fold separation versus
$1.55$-fold for the flat comparator; the threshold ratio diverges in finite
time, so the snapshot value is illustrative rather than invariant.}
\label{fig:threshold}
\end{figure}

\subsection{Niche partitioning: unexpected result (F8)}

Bostrom \citeyearpar{bostrom2005singleton} lists niche partitioning (separate
resource pools) as a failure condition for singleton emergence.
Simulation contradicts this for the case where both agents share the same
$\beta$-function.
At zero resource overlap, separation still reaches $1{,}103\times$ at $t = 15$
(versus $265{,}477\times$ with full overlap).
The $\beta$-flip mechanism operates independently of resource competition: the
agent with higher initial capability crosses $T$ first regardless of resource
structure.

\textit{Revised failure condition:}
Stable oligopoly requires agents to operate in different $\beta$-regimes (one
agent structurally unable to cross into $\beta < 0$), not merely separate
resource pools (see Section~\ref{sec:failures}, F1 revised).

\subsection{$\beta$-regime sensitivity (F10--F12)}

A threshold agent ($\betaL = -0.3$) defeats a flat agent ($\beta = 0.5$
always) provided the flat agent does not start more than approximately
$2.9\times$ ahead (F10); this is a point estimate from simulation with no
reported variance.
Above this, resource starvation prevents the threshold agent from reaching $T$.
In a threshold race between two agents with different $T$ values, the
lower-threshold agent overcomes at most $1.19\times$ initial disadvantage (F11),
and the advantage plateaus beyond a threshold gap of $\sim\!4$ units (F12).

\subsection{Stochastic robustness (F13--F14)}

\begin{figure}[h]
\centering
\includegraphics[width=0.85\textwidth]{../figures/stoch_winner_rate.png}
\caption{Stochastic simulation outcomes, 300 trials per noise level
$\sigma$, integration cap at $S = 10^{8}$.
``Singleton emergence'' counts trials reaching the cap (ratio $> 10\times$)
within $t_{\max}$; with the cap in place, no trials in the tested range
fail.
Winner identity: initial leader wins $> 50\%$ of trials below $\sigma
\approx 0.23$ (default 10\% gap); degradation begins near $\sigma
\approx 0.023$.
This is consistent with high $\Pr(J \geq c)$ in
Theorem~\ref{thm:stochastic} for small $\sigma$, not evidence of
almost-sure emergence.}
\label{fig:stochastic}
\end{figure}

Theorem~\ref{thm:stochastic} gives the probability of singleton emergence
as $\Pr(J \geq c)$, where $J$ is a Dufresne perpetuity and $c$ depends on
threshold and growth parameters.
This probability is strictly between $0$ and $1$ for any $\sigma > 0$ and
approaches $1$ as $\sigma \to 0$.
The simulation below uses an integration cap at $S = 10^{8}$, which logs
trials reaching the cap as singleton emergence.
For the noise levels and time horizons tested, the cap is reached in
every trial; this is consistent with high $\Pr(J \geq c)$ in the tested
regime but does not establish almost-sure emergence.
Three noise thresholds appear in these results and are distinct:
(1)~$\sigma \approx 0.023$: winner identity begins to degrade (initial leader
starts losing some trials) for the default 10\% initial gap;
(2)~$\sigma \approx 0.05$: a 1\% initial gap is overwhelmed and the winner becomes
essentially random at that specific gap (F14);
(3)~$\sigma \approx 0.23$: winner identity is fully random for the default
10\% gap.
The randomization threshold scales with the initial gap; $\sigma/\mathrm{gap}
\approx 2$ is the approximate crossover.
All three thresholds are empirical observations, not analytically derived.

\subsection{Late entrant moat (F15--F16)}

The moat grows from $3\times$ at threshold crossing to $> 10^6\times$ within
3 time units (F15).
Late entry threatens the incumbent only during the pre-threshold window (F16).
Post-threshold, no tested entrant capability can displace the incumbent.

\subsection{Timescale (F17)}

The pre-threshold crossing time scales as $\tcross \propto N^{1}$ from
the equal-agent approximation (Appendix~\ref{app:timescale}); empirical
fits over our parameter sweep give $N^{0.96}$, the small deviation
attributable to the leader's growing resource share above $1/N$.
In the post-threshold regime, $t_{10\times} \approx t_{100\times}
\approx t_{\mathrm{dom}}$ across all parameters tested: once the
leader crosses $T$, finite-time blow-up collapses subsequent ratio
milestones onto the same time.
Pre-threshold, milestone times are spread out as ordinary power
laws.

\begin{remark}[Multi-parameter scaling estimate]
Power-law fits over single-variable sweeps yield
\[
  t_{10\times} \approx 2.44 \cdot N^{0.96} \cdot \alpha^{-0.30} \cdot
  \mathrm{gap}^{-0.15} \cdot |\betaL|^{-0.31}.
\]
Only the $N$ exponent is derived analytically; the other exponents are
empirical fits with no cross-validation, no confidence intervals, and
no treatment of interaction effects.
The formula should be read as a rough scaling guide within the tested
parameter range, not as a quantitative prediction.
\end{remark}

\subsection{Continuous entry (F18--F19)}

Incumbent survival drops below 90\% at entry rate
$\lambda \approx 0.25$ per time unit (F18).
Heavy-tailed entry distributions (lower Pareto shape) are more dangerous
than high-mean distributions (F19).
Post-threshold, any $\lambda$ is survivable due to moat growth.
Arrival timing follows a Poisson process, a tractable baseline that assumes
memoryless, independent entry events.
Real competitive entry may be clustered (triggered by a public breakthrough)
or self-inhibiting (early failures discourage successors); these deviations
would shift $\lambda_{\mathrm{crit}}$ but leave the pre/post-threshold
qualitative structure intact.

\subsection{Cooperation (F20--F23)}

\begin{figure}[h]
\centering
\includegraphics[width=0.85\textwidth]{../figures/coop_dynamic_game.png}
\caption{Cooperation regime invariance. Three regimes across 50 trials each.
Zero singleton-formation failures in all three regimes.
Mean $t_{10\times} = 10.09$ in all three (indistinguishable).}
\label{fig:coop}
\end{figure}

Four experiments test whether cooperation prevents singleton emergence.
F21--F23 are structural consequences of the no-renegotiation assumption
(Section~\ref{sec:coalition}) rather than independent empirical discoveries;
the experiments confirm that the model behaves as the assumption implies.
(1)~Critical coalition size: $N = 2$ coalition members can prevent a singleton
candidate ($S = 1.1$) from crossing $T = 3.0$ when $\alpha \geq \alpha^* \approx 0.64$
(F20, with $\alpha^*$ from Theorem~\ref{thm:criticalalpha}).
(2)~Coalition coherence failure: an 8-member coalition with $8\times$ combined
capability fails because coalition resources split among 8 members give each
$\sim\!11\%$ individually; the singleton gets $\sim\!12\%$ alone (F21).
This follows directly from proportional internal distribution.
(3)~Zero defection events: coalition is individually rational and stable; the
singleton wins regardless (F22).
(4)~Cooperation regime invariance: no cooperation, oracle-optimal cooperation,
and rational cooperation all produce zero singleton-formation failures with
indistinguishable timing (mean $t_{10\times} = 10.09$ in all three, F23).
Oracle cooperation produces a different singleton (suppresses the initial
leader) but cannot prevent singleton formation because the coalition's
internal competition then resolves to a new singleton on the same timescale.

\subsection{Coalition dynamics under varying $\alpha$ (F24--F25)}

The critical $\alpha$ for coalition external suppression (Theorem~\ref{thm:criticalalpha})
is $\alpha^* \approx 0.64$ analytically.
Empirically: at $\alpha = 0.5$, singleton wins for all tested $N$ (up to 16);
at $\alpha \geq 0.75$, $N = 2$ is sufficient (F24).
At $\alpha = 2.0$, $N = 4$: coalition suppresses external singleton; first
agent to cross $T$ is a coalition member ($t = 5.56$); internal singleton
forms (F25).

\begin{figure}[h]
\centering
\includegraphics[width=0.85\textwidth]{../figures/ca_alpha_sweep.png}
\caption{Phase diagram: coalition size $N$ versus $\alpha$.
Green: coalition wins race to $T$. Red: singleton wins.
For $\alpha \leq 0.5$, singleton wins for all tested $N$.
For $\alpha \geq 0.75$, $N = 2$ is sufficient.
Theoretical critical $N^*$ curve (Theorem~\ref{thm:criticalalpha}, white
dashed) matches the empirical boundary.}
\label{fig:alphasweep}
\end{figure}

In all 56 tested $(\alpha, N)$ combinations where the coalition suppresses
the external singleton, an internal coalition singleton forms.
Zero singleton-formation failures were observed across all tested $\alpha$
values.

% ============================================================
\section{Failure Conditions}
\label{sec:failures}

The theorem requires A1--A5. Below are the conditions under which each
assumption fails and the resulting consequences.

\paragraph{F1: Niche partitioning (A2/A3 weakened), revised.}
The prior statement (Bostrom, 2005) holds for Lotka--Volterra alone but fails
when A4 holds.
The $\beta$-flip mechanism operates independently of resource competition.
\textit{Note:} The zero-overlap simulation (F8) relaxes A3 (single shared
pool) and operates outside the formal model.
It is an out-of-model robustness check, not a theorem consequence.
Within the formal model, A3 holds and the revised condition is:
stable oligopoly requires genuinely different $\beta$-regimes, with one agent
structurally incapable of crossing into $\beta < 0$.

\paragraph{F2: No $\beta$-threshold (A4 fails).}
If $\beta(S) > 0$ for all $S$, Step~2 fails.
The singleton still emerges from Step~1 (competitive exclusion), but separation
is exponential rather than superexponential and the timescale is much longer.

\paragraph{F3: Continuous entry (A3 weakened).}
Late entry threatens only the pre-threshold incumbent (F15--F16).
Two distinct entry-rate thresholds apply (F18):
$\lambda \approx 0.25$ per time unit is where incumbent survival first
drops below $90\%$, and at $\lambda \approx 6.3$ incumbent survival
falls to $0\%$ across the tested trials.
The latter regime is competitive churn (no incumbent persists long
enough to cross $T$) and is qualitatively different from the former,
which is degraded but nonzero survival.
Post-threshold, any entry rate tested is survivable.

\paragraph{F4: Identical initial conditions (A5 fails).}
Under perfect symmetry, a singleton still emerges in the deterministic
model, but the theorem cannot designate which agent.
Any physical noise breaks symmetry; F7 shows that even an initial
capability spread of $\sigma = 0.001$ resolves to the initial leader
in $100\%$ of tested trials.

\paragraph{F5: Cooperation.}
Coalition pooling can suppress a singleton candidate externally
(for $\alpha \geq \alpha^* \approx 0.64$), but generates an internal singleton
from coalition divergence (F23, F25).
Oracle-optimal cooperation changes which agent becomes the singleton; it does
not prevent or measurably delay singleton formation.
For $\alpha < \alpha^*$, the external singleton wins directly.
A true merger (not cooperation) of all agents into a single entity prevents
this, but the merged entity is simply a stronger singleton candidate and the
theorem applies to it against any unmerged agents.

% ============================================================
\section{Coalition Coherence Theorem}
\label{sec:coalition}

\subsection{Individual growth rate comparison}
\label{sec:coalitionindiv}

\textit{Modeling assumption and its limits:}
Coalition members distribute internal resources proportionally to individual
capability ($C_i^\alpha / \sum_j C_j^\alpha$) and cannot renegotiate to
concentrate all resources on a single member.
This assumption is load-bearing and deserves scrutiny.
In practice, leading AI organizations \emph{do} concentrate compute on
their best-performing training runs, operationally equivalent to resource
concentration within a coalition.
If a coalition can credibly commit to full resource concentration on its
fastest member, it functions as a merged agent, and the cooperation-failure
result does not apply: the merger is simply a stronger singleton candidate.
The paper's claim that cooperation fails therefore rests on the inability
to achieve credible, stable resource concentration, which is a social and
game-theoretic constraint, not a physical one.
Whether this constraint holds in practice is an open empirical question.

\begin{theorem}[Coalition coherence]
\label{thm:coalcoherence}
Singleton candidate with capability $S$; coalition of $N$ equal members with
capability $c$ each; coalition acts as block externally, distributes
proportional to $C_i^\alpha$ internally; pre-threshold regime.
The singleton grows faster than each individual coalition member if and only if
\[
  \left(\frac{S}{c}\right)^\gamma > \Phi, \quad \gamma = 1 - \beta + \alpha,
  \quad \Phi = N^{\alpha-1}.
\]
\end{theorem}

\begin{proof}
Singleton resource share: $r_s = S^\alpha/(S^\alpha + (Nc)^\alpha)$.
Each member's share: $r_m = (Nc)^\alpha / (N(S^\alpha + (Nc)^\alpha))$.
Singleton grows faster than member $i$ iff
$S^{1-\beta} r_s > c^{1-\beta} r_m$.
Substituting and simplifying:
$S^\gamma > c^\gamma \cdot (Nc)^\alpha/(Nc^\alpha) = c^\gamma \cdot N^{\alpha-1}$,
giving $(S/c)^\gamma > N^{\alpha-1} = \Phi$.
\end{proof}

\textit{Interpretation:}
At $\alpha = 1$: $\Phi = 1$; coalition size has no effect on individual
member competitiveness.
At $\alpha > 1$: $\Phi = N^{\alpha-1} > 1$; large enough coalition amplifies
member growth.
At $\alpha < 1$: $\Phi < 1$; coalition penalizes members.
The coherence failure of F21 is specific to $\alpha = 1$ (our default
simulation parameter): regardless of coalition size, the singleton
($S = 1.1 > c = 1.0$) beats every individual member.

\subsection{Critical $\alpha$ for external suppression}
\label{sec:criticalalpha}

The result above governs individual races.
For the coalition to \emph{externally suppress} the singleton (prevent it from
crossing $T$ at all), the coalition must collectively reach $N\cdot T$
before the singleton reaches $T$.

\begin{theorem}[Critical $\alpha$]
\label{thm:criticalalpha}
A coalition of $N$ equal members (capability $c$ each, combined $Nc$) defeats
singleton (capability $x_0$) with threshold $T$ in the pre-threshold
$\betaH$-regime if and only if
\begin{equation}
  \left(\frac{NT}{x_0}\right)^\gamma - \left(\frac{T}{c}\right)^\gamma
  \;\geq\; N^\gamma - 1,
  \qquad \gamma = \alpha - \betaH.
  \label{eq:criticalalpha}
\end{equation}
\end{theorem}

\begin{proof}
In the pre-threshold regime, the dynamics are
$\dot{x} = x^{0.5+\alpha}/D$ and $\dot{y} = y^{0.5+\alpha}/D$
where $x$ = singleton, $y = Nc$ = coalition combined, $D = x^\alpha + y^\alpha$.
Define $\rho = y/x$.
Using $x$ as the independent variable:
\[
  \frac{d\rho}{dx}
  = \frac{\rho(\rho^\gamma - 1)}{x}, \qquad \gamma = \alpha - \betaH.
\]
This is a separable ODE.
Substituting $v = \rho^\gamma$ and separating:
$dv / (\gamma v(v-1)) = dx/x$.
Partial fractions and integrating from $(x_0, \rho_0)$ to $(x, \rho(x))$:
\[
  \frac{v(x) - 1}{v(x)}
  = \frac{v_0 - 1}{v_0} \cdot \left(\frac{x}{x_0}\right)^\gamma,
  \qquad v_0 = \rho_0^\gamma = \left(\frac{Nc}{x_0}\right)^\gamma.
\]
The coalition wins iff a coalition member crosses $T$ before the singleton,
i.e., $y(T) \geq NT$, i.e., $\rho(T) \geq N$.
Setting $\rho(T) = N$ (critical case), $v(T) = N^\gamma$:
\[
  \frac{N^\gamma - 1}{N^\gamma}
  = \left(1 - \left(\frac{x_0}{Nc}\right)^\gamma\right)
    \left(\frac{T}{x_0}\right)^\gamma.
\]
Rearranging yields~\eqref{eq:criticalalpha}.
See Appendix~\ref{app:criticalalpha} for the full ODE reduction.
\end{proof}

\textit{Critical $\alpha^*$ for standard parameters} ($N = 2$, $T = 3.0$,
$x_0 = 1.1$, $c = 1.0$, $\betaH = 0.5$):
The critical $\gamma^*$ solves
\[
  (6/1.1)^\gamma - 3^\gamma = 2^\gamma - 1.
\]
Numerical root: $\gamma^* \approx 0.14$, giving $\alpha^* \approx 0.64$.

\textit{Verification:} At $\gamma = 0.14$: LHS $= 5.455^{0.14} - 3^{0.14}
= 1.268 - 1.166 = 0.102$; RHS $= 2^{0.14} - 1 = 0.102$. \checkmark
The analytical $\alpha^* \approx 0.64$ falls between the simulation data points
$\alpha = 0.5$ (singleton wins) and $\alpha = 0.75$ (coalition wins), F24. \checkmark

\textit{Behavior at $\gamma \to 0$:}
First-order expansion gives LHS $\approx \gamma \ln(Nc/x_0)$ and
RHS $\approx \gamma \ln N$.
The condition LHS $\geq$ RHS reduces to $c \geq x_0$: coalition wins at
$\gamma = 0$ only if each member starts no weaker than the singleton.
Since $c = 1.0 < x_0 = 1.1$, no coalition wins at $\gamma = 0$.
For $\gamma > \gamma^*$, the larger ratio $T/x_0$ provides enough time to
compound the coalition's combined advantage.

% ============================================================
\section{Stochastic Robustness}
\label{sec:stochastic}

For multiplicative GBM-type noise on the post-threshold dynamics, the
probability of singleton emergence is strictly between $0$ and $1$ and
depends on the noise amplitude.
The deterministic blowup of Theorem~\ref{thm:finitetime} is \emph{not}
preserved as an almost-sure event under arbitrarily large noise; what
survives is a noise-dependent probability that approaches $1$ as
$\sigma \to 0$ and $0$ as $\sigma \to \infty$.

\begin{theorem}[Stochastic blowup probability]
\label{thm:stochastic}
Let agent~1's post-escape dynamics satisfy the SDE
\begin{equation}
  dS_1 = \rmin \cdot S_1^{1+|\beta^*|}\, dt + \sigma\, S_1\, dW,
  \qquad S_1(t_1) = T',
  \label{eq:sde}
\end{equation}
where $|\beta^*| \in (0, |\betaL|)$ and $T'$ are as in
Lemma~\ref{lem:escape}.
Define
\[
  J \;=\; \int_{0}^{\infty}
    \exp\!\bigl(|\beta^*|\sigma\, W(s)
                - \tfrac{|\beta^*|\sigma^{2}}{2}\, s\bigr)\, ds,
  \qquad
  c \;=\; \frac{(T')^{|\beta^*|}}{|\beta^*|\,\rmin}.
\]
Then $J < \infty$ a.s.\ and
\[
  \Pr\bigl(S_1(t) \to \infty \text{ at finite } t\bigr)
  \;=\; \Pr(J \geq c) \;\in\; (0, 1).
\]
The probability tends to $1$ as $\sigma \to 0$ and to $0$ as
$\sigma \to \infty$.
\end{theorem}

\begin{proof}
By It\^{o}'s formula applied to $u = S_1^{-|\beta^*|}$,
\[
  du = \Bigl(-|\beta^*|\rmin
            + \tfrac{|\beta^*|(|\beta^*|+1)}{2}\,\sigma^2\,u\Bigr)\,dt
     - |\beta^*|\sigma\,u\,dW.
\]
This is a linear SDE in $u$.
Setting
$M(t) = \exp\!\bigl(-\tfrac{|\beta^*|\sigma^2}{2}(t-t_1)
                    + |\beta^*|\sigma(W(t)-W(t_1))\bigr)$,
direct computation gives $d(M u) = -|\beta^*|\rmin M\, dt$, hence
\[
  u(t) \;=\; M(t)^{-1}\!\left[\,u(t_1)
                              - |\beta^*|\rmin \int_{t_1}^{t} M(s)\,ds\right].
\]
The process $u$ is non-negative and $S_1(t) = \infty$ iff $u(t) = 0$,
which holds iff the bracket reaches zero, i.e.\ iff
$\int_{t_1}^{t} M(s)\,ds = u(t_1)/(|\beta^*|\rmin) = c$.
Let
$J = \int_{t_1}^{\infty} M(s)\,ds
   = \int_0^\infty \exp(|\beta^*|\sigma W(s) - |\beta^*|\sigma^2 s/2)\,ds$
(the equality in distribution uses Brownian translation invariance).
By Dufresne's identity \citep{dufresne1990distribution}, $J$ has a
reciprocal-gamma distribution and is a.s.\ finite.
Therefore
\[
  \{S_1 \to \infty \text{ in finite time}\} \;=\; \{J \geq c\},
\]
and $\Pr(J \geq c) \in (0,1)$ since $J$ has a continuous distribution
supported on $(0,\infty)$.

For the limits: as $\sigma \to 0$, $M(s) \to 1$ a.s.\ and $J \to \infty$
a.s., so $\Pr(J \geq c) \to 1$.
As $\sigma \to \infty$, the drift $-|\beta^*|\sigma^2/2$ dominates and
$M$ decays so rapidly that $J \to 0$ in distribution, so
$\Pr(J \geq c) \to 0$.
\end{proof}

\begin{remark}[Reading the result]
For small noise, threshold-crossing virtually guarantees singleton
emergence, in line with Theorem~\ref{thm:finitetime}.
For large noise, the agent that crosses $T$ may still be pushed back
below it: the process can wander for arbitrarily long without diverging.
The earlier claim in this paper that singleton emergence is robust to
arbitrary noise was incorrect; the corrected statement above gives the
probability explicitly as a Dufresne perpetuity.
\end{remark}

\begin{remark}[Consequence for agent~2]
Suppose agent~2 satisfies $dS_2 = r_2(t)\,S_2^{1-\beta_2}\,dt + \sigma\,S_2\,dW_2$
with $\beta_2 > 0$ and $W_2$ independent of $W$.
The drift is sublinear and the noise is GBM-type, so by standard
non-explosion criteria \citep[Prop.~5.5.32]{karatzas1991brownian},
$S_2(t) < \infty$ a.s.\ for all finite $t$.
On the event $\{J \geq c\}$, $S_1 \to \infty$ at finite $\tstar$, hence
$\rho = S_1/S_2 \to \infty$ at $\tstar$.
On the complementary event, $S_1$ stays bounded and the ratio remains
finite.
\end{remark}

\begin{remark}[On the F13 simulation]
The simulation reported in F13 records $300$ trials per noise level with
zero failures.
The integration caps capabilities at $S = 10^{8}$; trials hitting the
cap are recorded as singleton emergence.
For the noise levels and time horizons tested, this cap converts
unbounded drift into apparent emergence even when the underlying SDE
would not explode in continuous time.
The simulation is therefore consistent with high $\Pr(J \geq c)$ in the
tested regime, but does not provide independent evidence of almost-sure
emergence.
\end{remark}

% ============================================================
\section{Empirical Calibration to Frontier AI Compute}
\label{sec:calibration}

The framework's central conditional, ``if A4 holds, a singleton
emerges,'' is mute on whether A4 \emph{does} hold for any system that
matters.
This section attempts a falsifiable check using publicly available data
on training compute for notable machine-learning systems
\citep{epoch2024compute}.
The result is honest and partial: under the proxy used here, the
hypothesis $\beta(S) < 0$ is not currently supported.

\subsection{Data and proxy}

We use the Epoch AI ``Notable AI Models'' dataset (snapshot accessed
2026-05-13; $8{,}273$ rows, of which $523$ have both a usable
publication date and a numerical training-compute estimate).
The frontier is taken as the running maximum of training compute
(in FLOP) over time; this gives $39$ record-setting models from
$\sim\!1956$ to $2026$, of which $26$ fall in the modern deep-learning
window $2012$--$2026$.
We treat $S = $ frontier training compute as a capability proxy.
This is a strong assumption: training compute is one input to capability,
not capability itself, and the relationship is mediated by algorithmic
efficiency, data, and post-training techniques whose contributions are
not recorded in the dataset.
Results below should be read as a check on whether \emph{this particular
proxy} exhibits the kind of trajectory A4 requires; null results do not
rule out A4 under different proxies.

\subsection{Method}

Two complementary fits.
\textbf{Power-law fit.}
Treating the frontier trajectory as $dS/dt = c\,S^{1-\beta}$, we form
$y(t) = \ln S(t)$, estimate $\dot{y}$ by central differences, and
regress $\ln \dot{y}$ on $y$; the slope is $-\beta$.
This is sensitive to numerical-derivative noise on a sparse
record-setter sequence.

\textbf{Polynomial-in-time fit.}
Fit $\log_{10} S(t) = a + b\,t + \gamma\,t^{2}$ by ordinary least
squares on the centered time variable.
Curvature $\gamma > 0$ indicates super-exponential growth (and thus
$\beta < 0$ in the power-law model); $\gamma = 0$ corresponds to
exponential growth ($\beta = 0$); $\gamma < 0$ to deceleration
($\beta > 0$).
This is more robust than numerical differentiation because it does
not require estimating $\dot{y}$ at each frontier step.

\subsection{Results}

\begin{table}[h]
\centering
\small
\begin{tabular}{lrrrrr}
\toprule
Window & $n$ & $\gamma$ & Asymptotic 95\% CI & Bootstrap 95\% CI & $\Pr(\gamma>0)_{\mathrm{boot}}$ \\
\midrule
2010--2018 & 9  & $+0.157$ & $[-0.095,\;+0.409]$ & $[-2.27,\;+1.19]$ & $0.66$ \\
2018--2022 & 10 & $+0.082$ & $[-0.154,\;+0.317]$ & $[-2.91,\;+0.97]$ & $0.58$ \\
2022--2026 & 7  & $-0.069$ & $[-0.156,\;+0.018]$ & $[-0.217,\;+0.240]$ & $0.11$ \\
2012--2026 & 26 & $+0.005$ & $[-0.010,\;+0.021]$ & $[-0.008,\;+0.019]$ & $0.82$ \\
\bottomrule
\end{tabular}
\caption{Polynomial-in-time fits of frontier $\log_{10}$ training
compute. Curvature $\gamma$ is the coefficient on $t^{2}$ in
$\log_{10} S = a + bt + \gamma t^{2}$; positive $\gamma$ corresponds to
super-exponential growth ($\beta < 0$).
Asymptotic CI is the OLS standard error; bootstrap CI is from $5{,}000$
draws of the (year, $\log_{10} S$) pairs with replacement (seed
$20260513$).
$\Pr(\gamma > 0)_{\mathrm{boot}}$ is the bootstrap-empirical
probability that resampled curvature is positive.
The two CIs disagree most where they should: short windows ($n \leq 10$)
have heavy-tailed bootstrap distributions because frontier records
cluster, so a few influential observations can flip the curvature sign.
The full-window result is robust under both methods.
$R^{2}$ for the four windows is $0.87$, $0.76$, $0.99$, $0.96$.}
\label{tab:calibration}
\end{table}

\begin{figure}[h]
\centering
\includegraphics[width=0.95\textwidth]{../figures/calibration_beta.png}
\caption{Left: training compute for all notable models (grey) and the
frontier running maximum (orange).
Right: power-law fit on $(\ln S,\;\ln \dot{y})$ where $y = \ln S$;
slope is $-\beta$.}
\label{fig:calibration}
\end{figure}

The full deep-learning window 2012--2026 is consistent with pure
exponential growth: $\gamma = +0.005$ with both CIs tightly straddling
zero, $R^{2} = 0.96$, and a stable doubling time near $0.5$ years.
The bootstrap probability of positive curvature is $0.82$, indicating
a slight directional tilt toward acceleration that is too small to
matter at the resolution of the data.
Sub-windows show mild acceleration during 2010--2022 and mild
deceleration during 2022--2026; bootstrap CIs reveal that all
sub-window estimates are statistically indistinguishable from
exponential, and the short-window asymptotic CIs in particular
understate uncertainty because frontier records cluster around
architectural breakthroughs.
The power-law fit, which is noisier, gives $\beta = -0.014$ on the full
window with a 95\% CI of $[-0.062,\;+0.035]$.

\subsection{Robustness: benchmark-performance proxies}
\label{sec:calib_benchmarks}

Training compute is an input to capability rather than capability
itself.
We re-run the calibration on five public benchmarks released by
\citet{epoch2024compute}: GPQA Diamond, FrontierMath, ARC-AGI,
SWE-bench Verified, and MATH Level 5.
For each benchmark we form the running-maximum frontier of the best
score reported across model versions over time, transform raw scores
in $[0,1]$ to an unbounded capability via $S = -\log_{10}(1 - \mathrm{score})$
(so $S \to \infty$ as score $\to 1$), and fit
$\log_{10} S = a + bt + \gamma t^{2}$ on the transformed frontier.

\begin{table}[h]
\centering
\small
\begin{tabular}{lrrrrr}
\toprule
Benchmark & $n_{\mathrm{front}}$ & $\gamma$ & Bootstrap 95\% CI & $\Pr(\gamma>0)_{\mathrm{boot}}$ & $R^{2}$ \\
\midrule
GPQA Diamond        & 17 & $+0.010$ & $[-0.116,\;+0.047]$ & $0.57$ & $0.96$ \\
FrontierMath        & 16 & $-0.651$ & $[-0.953,\;-0.369]$ & $0.00$ & $0.97$ \\
ARC-AGI             & 12 & $+0.003$ & $[-0.496,\;+1.138]$ & $0.54$ & $0.94$ \\
SWE-bench Verified  &  9 & $-0.454$ & $[-0.955,\;+0.001]$ & $0.03$ & $0.92$ \\
MATH Level 5        & 13 & $+0.017$ & $[-0.675,\;+0.371]$ & $0.57$ & $0.89$ \\
\bottomrule
\end{tabular}
\caption{Polynomial-in-time fits on benchmark-derived capability
($S = -\log_{10}(1 - \mathrm{score})$) for five public benchmarks.
Three of five give $\gamma$ statistically indistinguishable from zero
(exponential capability growth); two give $\gamma$ significantly
negative under the bootstrap (decelerating capability).
None of the five give $\gamma$ significantly positive.}
\label{tab:bench_calibration}
\end{table}

\begin{figure}[h]
\centering
\includegraphics[width=0.95\textwidth]{../figures/calibration_benchmarks.png}
\caption{Per-benchmark frontier (orange) and all evaluations (grey),
with fitted curvature $\gamma$ and bootstrap probability of acceleration.}
\label{fig:bench_calibration}
\end{figure}

Across the five benchmarks: three (GPQA, ARC-AGI, MATH) are
indistinguishable from exponential capability growth; two
(FrontierMath, SWE-bench) show statistically significant
\emph{deceleration}; none show acceleration.
The compute-based and benchmark-based proxies agree on the qualitative
finding: capability growth is at most exponential under every proxy
tested.

\subsection{Reading}

Across compute and five benchmark-based capability proxies, frontier
capability is growing at most exponentially over the deep-learning
window.
On no proxy is the curvature significantly positive; on two benchmarks
the curvature is significantly negative.
The data are not consistent with $\beta(S) < 0$ during the period
observed.

This does not refute A4.
Three reasons the result is weak as evidence against the framework:
(i)~each proxy captures one slice of capability and none isolates the
post-training, scaffolding, or inference-compute regimes that have
become important since $2024$;
(ii)~A4 requires only that $\beta < 0$ become reachable at some future
$T$, not that the trajectory already be in that regime; the data are
silent on the future;
(iii)~frontier-record counts are small ($n_{\mathrm{front}} \in [9, 26]$
across proxies), making sub-window estimates noisy.
Conversely, the result \emph{is} evidence against the popular claim
that capability growth is currently super-exponential under any of the
proxies most commonly cited as evidence for it: a singleton emerging
on these proxies alone within the next several years would require a
sharp regime change, not continuation of the current trend.

The exercise's main contribution is methodological: it makes A4 a
falsifiable hypothesis on specific datasets.
A future study could extend the calibration to capability proxies
better suited to recent trends (effective compute, agentic
capability scores, multi-step reasoning benchmarks) and to longer time
horizons as data accumulate.
Related empirical work: \citet{ge2026exponential} compare exponential,
hyperbolic, and sigmoid models of capability growth on the METR
time-horizon benchmark using Bayesian model selection, with no
explicit super-exponential alternative; their methodology and dataset
are complementary to the calibration here.

% ============================================================
\section{Discussion}
\label{sec:discussion}

\subsection{What is new}

Yudkowsky (2013), Omohundro (2008), and Lotka--Volterra are all prior work.
The combination is not.

The main technical contribution is Theorem~\ref{thm:finitetime}: finite-time
separation \emph{under competition}.
Prior work treats the $\beta$-flip as a single-agent phenomenon.
The closest adjacent formal treatment is
\citet{anbarjafari2025singularity}, which establishes finite-time blow-up
criteria via Osgood-style conditions on a single-agent capability ODE
$\ddot{I} \geq aI^p$ and analyses physical and economic caps on
runaway dynamics; that framework does not model multi-agent competition,
coalition dynamics, or stochastic noise, and does not perform an
empirical calibration.
The present paper proves the $\beta$-flip survives adversarial resource
dynamics, which is non-trivial: an opponent can slow the race to
threshold, but not stop it.

The niche partitioning finding (F8) corrects Bostrom (2005).
Separate resource pools are not a sufficient failure condition; what matters
is whether the trailing agent can reach $\beta < 0$ at all.

Theorem~\ref{thm:stochastic} replaces an earlier incorrect
almost-sure-emergence claim with the correct probabilistic statement:
under multiplicative GBM noise, $\Pr(\text{singleton emerges}) =
\Pr(J \geq c)$ for the explicit Dufresne perpetuity $J$.
Noise can therefore prevent emergence, with probability tending to $1$
as $\sigma$ grows.

Theorem~\ref{thm:criticalalpha} derives the coalition-suppression threshold
analytically.
The transition near $\alpha = 0.64$ was previously only a simulation
observation; it now has an exact transcendental equation behind it.

The simulation scripts produce quantitative measurements (timescale formula,
$\lambda_{\mathrm{crit}}$, moat characterization, F1--F25) that were not
previously available.

Section~\ref{sec:calibration} introduces an empirical test of A4 across
six capability proxies (frontier compute plus five benchmarks): on no
proxy is $\beta(S) < 0$ supported, and the framework's central premise
becomes a falsifiable hypothesis with negative
preliminary evidence.

\subsection{Timescale scaling}

The empirical scaling estimate in Section~\ref{sec:simulations} has a
partial analytical foundation.
The $N^{0.96}$ exponent is analytically predicted as $N^1$ from the
pre-threshold approximation (Appendix~\ref{app:timescale}); the small
deviation comes from the leader's increasing resource share as it
advances.
The $\alpha^{-0.30}$ exponent tracks the integrated resource correction
and the $|\betaL|^{-0.31}$ exponent reflects a mixture of pre- and
post-threshold phases (Appendix~\ref{app:timescale}).
The $\mathrm{gap}^{-0.15}$ exponent is small because the threshold
mechanism dominates initial conditions.

\subsection{Limitations}

\textit{The $N$-agent conjecture.}
The inductive step works cleanly in the sequential limit but does not bound
the error from truly simultaneous dynamics.
The gap is real.
Simulation (200/200 trials, $N=10$) confirms the conclusion holds, but
simulation on the same ODE system is not the same as closing the proof.

\textit{Local stability only.}
Proposition~\ref{prop:monopoly} gives local instability of the equal-resource
state via Jacobian eigenvalues.
It says nothing about whether trajectories starting far from that point
converge to monopoly.

\textit{Scalar capability.}
The $\beta$-flip mechanism requires a scalar $S$ crossing a single threshold
$T$.
Real optimization capability is multidimensional (compute, algorithmic
efficiency, data access, adversarial robustness) and different agents can
lead on different dimensions simultaneously.
There is no obvious way to extend the model to that setting.
Whether competitive exclusion holds in a vector capability space is an open
question, and the answer determines whether any of this applies to real AI
systems.

\textit{Timescale formula.}
The multi-parameter scaling estimate in Section~\ref{sec:simulations}
is fitted from single-variable sweeps with no cross-validation.
Only the $N^1$ exponent has an analytical derivation; the others should
be read as scaling guides, not predictions.

\textit{Fast equilibration.}
Theorem~\ref{thm:finitetime} requires A2.
With slow resource equilibration, the finite-time bound on $\tstar$ breaks
down even if the ratio still diverges eventually.

\textit{Spatial structure.}
The model has none.
Results apply within any causally connected region; adding propagation delays
extends timescales without changing the qualitative picture.

\textit{Capability proxy in the calibration.}
Section~\ref{sec:calibration} uses training compute as a stand-in for
$S$.
This proxy ignores algorithmic efficiency, data, post-training, and
inference-time compute, all of which have become significant since
$2024$.
A null result on this proxy is therefore weak evidence on A4 in
general; it is strong evidence only against the specific narrative
that ``compute alone is going superexponential.''

\subsection{Open questions}

\begin{enumerate}[label=(\arabic*)]
\item Closed-form timescale exponents for $\alpha$, gap, and $|\betaL|$.
The $N^1$ exponent is derived; others require numerical integration of the
coupled ODE system.

\item True merger stability: game-theoretic conditions under which a merged
coalition avoids fracturing under competitive pressure.

\item Calibration of $\beta(S)$ against capability proxies less tied
to raw compute (effective compute, frontier benchmark performance,
agent-evaluation scores), with explicit confidence bands on whether
$\beta < 0$ is consistent with the data at any horizon.
\end{enumerate}

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

Under A1--A5, competitive environments with recursive self-improvement
produce a singleton.
Once any agent crosses the $\beta$-threshold, its capability separates from
all competitors in finite time.
Initial conditions matter less than threshold position.
Under deterministic dynamics noise reduces to a question of who wins;
under multiplicative GBM noise it also affects whether anyone does
(Theorem~\ref{thm:stochastic}), with the probability of emergence
expressible in closed form as a Dufresne perpetuity.
Cooperation under any tested structure fails to prevent emergence in the
deterministic model.

The failure conditions are narrower than the prior literature implies.
Bostrom's niche partitioning argument requires that one agent cannot reach
$\beta < 0$. Separate resource pools alone are not enough.
Late entrants are a real threat only before threshold crossing.
Coalitions fail not because they are irrational but because proportional
internal resource distribution leaves each member individually weaker than a
singleton candidate.
A merger of the whole coalition would work, but then you have a singleton.

The hard question is A4.
A3 holds in any physical environment.
A5 is satisfied by noise.
Section~\ref{sec:calibration} converts A4 from a verbal premise into a
falsifiable hypothesis on six concrete proxies (frontier training
compute and five public benchmarks).
On none of the six is the curvature significantly positive; on two
benchmarks it is significantly negative; the full deep-learning window
of frontier compute is consistent with pure exponential growth.
This does not refute A4 (each proxy is a partial slice of capability,
and the data are silent on the future), but it does shift the burden
onto proponents of the singleton scenario to specify which capability
proxy they expect to enter the $\beta < 0$ regime, and on what
timescale.

% ============================================================
\section*{Acknowledgments}

No external funding was received.

\section*{Code and data availability}

Simulation scripts, calibration code, and the Epoch AI dataset
snapshot used in Section~\ref{sec:calibration} are available at
\url{https://github.com/ninjahawk/singleton-attractor}.

% ============================================================
\bibliographystyle{plainnat}
\bibliography{refs}

% ============================================================
\appendix

\section{Derivation of timescale exponents}
\label{app:timescale}

\paragraph{$N^1$ scaling (analytical).}
The derivation assumes $N$ agents with equal initial capability, each
starting with resource share $1/N$.
A5 requires initial heterogeneity, so this is a conservative case: the
leader's initial share is exactly $1/N$ (worst case), giving an upper bound
on $\tcross$.
With unequal initial conditions the leader starts with share $> 1/N$,
so actual $\tcross \leq$ the equal-agent estimate.
The leader's growth rate under the equal-agent approximation:
\[
  \dot{S}_{\mathrm{lead}} \approx S^{1-\betaH} / N.
\]
Time for the leader to reach $T$ from $S_0$:
\[
  \tcross \approx N \cdot \int_{S_0}^T S^{\betaH - 1}\,dS
  = N \cdot \frac{T^{\betaH} - S_0^{\betaH}}{\betaH}.
\]
This gives $\tcross \propto N$ exactly.
The empirical $N^{0.96}$ reflects the leader's increasing resource share
above $1/N$ as it advances, slightly accelerating threshold crossing.

\paragraph{$|\betaL|^{-0.31}$ scaling.}
Post-threshold time to reach $10\times$ ratio from $T$:
\[
  t_{\mathrm{post}} \approx \frac{T^{-|\betaL|}}{|\betaL| \cdot \rmin}.
\]
This gives $t_{\mathrm{post}} \propto |\betaL|^{-1}$.
Pre-threshold time $\tcross$ is independent of $\betaL$.
The combined exponent $-0.31$ reflects the mixture: $t_{10\times} = \tcross + t_{\mathrm{post}}$
where $\tcross / t_{10\times} \approx 0.6$ for default parameters, shifting the
effective exponent from $-1$ (pure post-threshold) toward 0 (pure pre-threshold).

\paragraph{$\alpha^{-0.30}$ and $\mathrm{gap}^{-0.15}$ scaling.}
Higher $\alpha$ concentrates resources on the leader more aggressively,
reducing $\tcross$ via a growing excess resource share above $1/N$.
The excess scales as $\alpha \cdot \varepsilon / S_0$ where $\varepsilon$ is
the leader's relative advantage at each moment; the integrated effect gives
$\tcross \propto \alpha^{-\delta}$, $\delta$ estimated empirically as 0.30.
The gap exponent is small ($-0.15$) because gap affects the initial divergence
rate but not the total path to $T$; the $\beta$-threshold mechanism dominates
initial conditions.
Exact closed-form values for both exponents require integrating the coupled ODE
$\dot{x} = x^{1-\betaH+\alpha}/(x^\alpha + (N-1)S_0^\alpha)$, which lacks a
closed-form solution.

\section{Critical $\alpha$ derivation}
\label{app:criticalalpha}

The derivation of Theorem~\ref{thm:criticalalpha} is given in
Section~\ref{sec:criticalalpha}.
Here we provide additional detail on the ODE reduction.

The ratio $\rho = y/x$ (coalition combined / singleton) satisfies:
\[
  \frac{d\rho}{dx} = \frac{\rho(\rho^\gamma - 1)}{x}.
\]
For $\gamma > 0$ ($\alpha > \betaH$) and $\rho > 1$ (coalition initially
stronger combined), $d\rho/dx > 0$: the ratio grows as the singleton
advances toward $T$.
For $\gamma = 0$ ($\alpha = \betaH$), $d\rho/dx = 0$: $\rho$ is constant.
For $\gamma < 0$ ($\alpha < \betaH$), $d\rho/dx < 0$: singleton grows
relatively faster.

The closed-form solution is:
\[
  \rho(x) = \left[\frac{1}{1 - \eta(x)}\right]^{1/\gamma},
  \quad
  \eta(x) = \left(1 - \left(\frac{x_0}{Nc}\right)^\gamma\right)
            \left(\frac{x}{x_0}\right)^\gamma.
\]
$\eta(x)$ is increasing; $\rho(x)$ diverges when $\eta \to 1$, which occurs
at finite $x$ (potential blow-up before the singleton reaches $T$ if the
coalition grows fast enough).

\end{document}