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%  Uniform Permanence in Weakly Reversible Mass-Action Systems
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\title{Uniform Permanence in\\Weakly Reversible Mass-Action Systems}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
A chemical reaction network records which mixtures of molecules turn
into which. Under mass-action kinetics the rate of a reaction is
proportional to the concentrations of its reactants, so the concentrations
obey a polynomial differential equation. A network is \emph{weakly
reversible} when every reaction can be undone by a chain of reactions.
We prove that every finite weakly reversible network with fixed positive
rate constants is \emph{permanent} on each positive stoichiometric class:
every solution is global and positive, and after a transient that may
depend on the initial point the whole trajectory is trapped in one compact
convex set that depends only on the network, the rates, and the class.
It follows that there are constants $0<\eps<\eps^{-1}$ bounding every
concentration from below and above for all large time, uniformly over all
initial points in the class, even when that class is unbounded. The
central tool, proved in full, is a lemma saying that a minimum of finitely
many affine functions can be arranged to reveal the exponent vector of any
point at which it is active, to any prescribed tolerance.
\end{abstract}

\tableofcontents

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

\subsection{Reaction networks and mass-action kinetics}

Fix $d$ chemical species. A \emph{complex} is a vector
$y\in\Z_{\ge0}^d$ whose $i$-th entry counts molecules of species $i$ in a
mixture. A \emph{reaction network} consists of a finite set
$C\subset\Z_{\ge0}^d$ of complexes and a finite set $R\subset C\times C$
of \emph{reactions}; we write $y\to y'$ for $(y,y')\in R$. For example,
$A+B\to C$ is the reaction from $e_A+e_B$ to $e_C$.

Assign to each reaction a positive \emph{rate constant} $k_{y\to y'}$.
Mass-action kinetics postulates that $y\to y'$ runs at speed
$k_{y\to y'}x^y$, where $x_i\ge0$ is the concentration of species $i$ and
$x^y=\prod_i x_i^{y_i}$. Since that reaction changes the state by
$y'-y$, the concentrations satisfy
\begin{equation}\label{eq:ma}
  \dot x=f(x):=\sum_{y\to y'\in R} k_{y\to y'}\,x^y\,(y'-y).
\end{equation}
This is the mass-action ordinary differential equation; we always use
fixed positive rates. Its right-hand side is a combination of the
difference vectors $y'-y$, so $f(x)$ always lies in the
\emph{stoichiometric subspace}
\[
  S:=\operatorname{span}_{\R}\{y'-y:y\to y'\in R\}.
\]
Hence every solution starting at $c$ stays in $c+S$. Because concentrations
are positive, the relevant state space is the \emph{positive stoichiometric
class}
\[
  P:=(c+S)\cap\R_{>0}^d,
\]
the part of the conservation surface on which every species is present.
The class $P$ may be bounded or unbounded; the unbounded case is where the
statement below is delicate.

\subsection{An example}

Take $d=2$, $C=\{e_1,e_2\}$, and $R=\{e_1\to e_2,\ e_2\to e_1\}$ with rates
$k_1,k_2>0$. Then
\[
  \dot x_1=-k_1x_1+k_2x_2,\qquad \dot x_2=k_1x_1-k_2x_2,
\]
and $S=\operatorname{span}\{e_2-e_1\}$, so each class is a segment
$x_1+x_2=c_0$ with $c_0>0$. Every solution on the class converges to the
interior equilibrium
$\bigl(k_2c_0/(k_1+k_2),\ k_1c_0/(k_1+k_2)\bigr)$, so the class is permanent
and $K_P$ can be that single point. This is the easy, bounded,
single-equilibrium case. The theorem also covers unbounded classes and
networks whose long-time behavior is far richer than convergence to one
point.

\subsection{Persistence and permanence}

Two failures are possible: a concentration can approach $0$, or it can grow
without bound. A global positive solution is \emph{persistent} if
$\liminf_{t\to\infty}x_i(t)>0$ for every $i$; these lower bounds may depend
on the trajectory. We call the system \emph{permanent on the class $P$} if
every solution starting in $P$ is global and positive, and if there is a
single compact set $K\subset P$ that eventually contains every such
solution. The entry time into $K$ may depend on the initial point; $K$ may
not. Permanence is thus stronger than persistence: it demands uniform
lower and upper bounds and a single absorbing set for the whole class.

\subsection{Weak reversibility and the main result}

A network is \emph{weakly reversible} if for every reaction $y\to y'$
there is a directed chain of reactions leading from $y'$ back to $y$.
Informally, no reaction is a one-way dead end. This purely combinatorial
condition turns out to control the long-time behavior of \eqref{eq:ma}.

\begin{theorem}\label{thm:main}
Let $C\subset\Z_{\ge0}^d$ be finite and $R\subset C\times C$ weakly
reversible, and fix positive rates in \eqref{eq:ma}. For every positive
stoichiometric class $P$ there is a nonempty compact convex set
$K_P\subset P$ with the following property. Every $x^0\in P$ gives a unique
global positive solution of \eqref{eq:ma}, and there is a finite
$T(x^0)\ge0$ with $x(t)\in K_P$ for all $t\ge T(x^0)$. The set $K_P$ is
forward invariant. Hence there is $\eps_P\in(0,1)$, depending only on the
network, the rates, and $P$, such that every such solution satisfies
\begin{equation}\label{eq:bounds}
  \eps_P\le x_i(t)\le\eps_P^{-1}
  \qquad(1\le i\le d,\ t\ge T(x^0)).
\end{equation}
\end{theorem}

The uniformity in \eqref{eq:bounds} holds within a fixed class and fixed
rates. The transient time may depend on $x^0$, but the bounds do not: they
are decided before $x^0$ is chosen.

\subsection{Background}

The structural theory starts with Horn and Jackson and with Feinberg
\cite{HornJackson1972,Feinberg1972}. A system is \emph{complex balanced} if
at some positive state the total incoming and outgoing flux agree at every
complex; then each positive class contains a unique positive equilibrium
and carries an entropy Lyapunov function. Complex balance constrains the
rates, while weak reversibility is purely combinatorial, and the gap
between them is what makes weak reversibility hard. Feinberg
\cite{Feinberg1987} raised the question of excluding boundary limits.
Partial answers followed: Anderson \cite{Anderson2011} proved convergence
for complex-balanced systems with a single linkage class; Craciun, Nazarov,
and Pantea \cite{CraciunNazarovPantea2013} proved permanence for two-species
weakly reversible (more generally endotactic) systems; Pantea
\cite{Pantea2012} proved persistence of bounded trajectories when
$\dim S=2$; Gopalkrishnan, Miller, and Shiu
\cite{GopalkrishnanMillerShiu2014} proved permanence under strong
endotacticity, which covers every weakly reversible network with one
linkage class; and Boros and Hofbauer \cite{BorosHofbauer2020} gave another
proof for one linkage class, producing a compact forward-invariant
absorbing set. Craciun's toric differential inclusions \cite{Craciun2026}
prove persistence for bounded positive trajectories with rates bounded
above and below. The immediate input here is the affine lemma and trapping
construction of \cite{OpenAI2026}, which produce a compact convex invariant
set depending on the chosen initial point; the present argument removes
that dependence.

\subsection{Proof strategy}

Fix a representative $c$ of a class. For $h\in(0,1)$ and an exponent vector
$p\in\R^d$, write $X_h(p)=(h^{p_1},\dots,h^{p_d})$. Every point of the box
$B_h=[h,h^{-1}]^d$ is $X_h(p)$ for $p=\log x/\log h\in[-1,1]^d$. As
$h\to0$, monomials are ordered by exponents, because $h^{p\cdot v}\to0$
whenever $p\cdot v>0$. This turns monomial algebra into the geometry of
exponent vectors.

The proof builds a concave piecewise-linear function
$F_h(x)=\min_\lambda (r_\lambda\cdot x+b_\lambda(h))$, a minimum of finitely
many affine functions whose slopes $r_\lambda\in[-1,1]^d$ are fixed as
$h\to0$. The affine lemma (Section~\ref{sec:affine}) guarantees that at any
point where the label with slope $r$ is active, the true exponent $p$ is
within a chosen tolerance of $r$. Choosing the tolerance small relative to
the directions $v$ that matter (coordinate vectors and differences of
complexes), an active slope reproduces every sign $p\cdot v$ we need.

At $c$ the active labels have zero slope, so $F_h(c)=M_h$ is the global
maximum and the plateau $K_h=\{x:F_h(x)=M_h\}$ is compact, convex, contains
$c$, and lies inside $\interior B_{\sqrt h}$.

Along a trajectory in $P$, the function $F_h$ does not decrease, and outside
$K_h$ it strictly increases at a uniform rate $\delta_h>0$. The increase
comes from a flux estimate: order the complexes by the level $r\cdot y$, cut
between consecutive levels, and use weak reversibility to show that the net
upward flux across each cut is nonnegative, and positive when a reaction
crosses it. Hence $r\cdot f(x)\ge0$, strictly positive when
$r\notin S^\perp$. At points of $P\setminus K_h$ every active slope lies
outside $S^\perp$, so the increase is strict. A trajectory that stays in
$B_h$ must therefore reach $K_h$ in finite time and remain there.

Finally, $K_h\subset\interior B_{\sqrt h}$ lets us change scale. Repeated
square roots, capped at a scale $h_P$ fixed by the class, produce the single
absorbing set $K_P=K_{h_P}\cap(c+S)$.

Section~\ref{sec:affine} proves the affine lemma,
Section~\ref{sec:trapping} builds the plateaus and the strict increase, and
Section~\ref{sec:perm} assembles the common absorbing set.

\section{Affine minima with arbitrary tolerance}\label{sec:affine}

We use a finite minimum of affine functions on the positive orthant. Its key
feature is that an active slope approximates the exponent vector of the
point where it is active, to a tolerance chosen in advance. The tolerance
may be a wild, even discontinuous, function of the slope.

Fix $m\ge1$, $u\in\R^m$, and $0<h<1$, and put
$X_h(u)=(h^{u_1},\dots,h^{u_m})$. A \emph{label}
$\lambda=(r_\lambda,b_\lambda)$ consists of a slope $r_\lambda\in\R^m$,
independent of $h$, and an offset $b_\lambda(h)$ defined for all
sufficiently small $h>0$. For a finite nonempty family $\Lambda$ of labels,
set
\[
  F_h(x)=\min_{\lambda\in\Lambda}\bigl(r_\lambda\cdot x+b_\lambda(h)\bigr).
\]
A label is \emph{active} at $x$ if its affine value equals $F_h(x)$; every
minimizing label is active in case of a tie. Offsets in a finite family are
restricted to a common interval of small $h$.

\begin{lemma}[Affine approximation]\label{lem:affine}
Let $m\ge1$, let $a<b$ be real, and let $E:[a,b]^m\to(0,\infty)$ be any
function. There is a finite nonempty family $\Lambda$ of labels with
$r_\lambda\in[a,b]^m$ and $b_\lambda(h)=o(h^a)$ $(h\to0+)$ such that the
following holds. If $h_\nu\to0+$, if $p^\nu\in\R^m$ converges to some
$p\in[a,b]^m$, and if a fixed $\lambda\in\Lambda$ is active at
$X_{h_\nu}(p^\nu)$ for every $\nu$, then
\[
  \norm{p-r_\lambda}_\infty<E(r_\lambda).
\]
The slopes and the family do not depend on $h$. No regularity of $E$ is
assumed, and the points $p^\nu$ need not lie in $[a,b]^m$.
\end{lemma}

\begin{proof}
We induct on $m$. Write $\mu(u)=\min_i u_i$. The largest coordinate of
$X_h(u)$ is $h^{\mu(u)}$. The proof first builds a family that works near a
prescribed value $t$ of the minimum exponent, then patches finitely many
such families together using decreasing offsets $-h^{q_j}$ that force the
minimum exponent of any active label into the correct range.

Throughout, asymptotic comparisons use fixed gaps between limiting
exponents: if $p_i>q$ then eventually $p_i^\nu\ge q+\delta$ for some
$\delta>0$, so $h_\nu^{p_i^\nu}=o(h_\nu^q)$; if $p_i<q$ then
$h_\nu^{p_i^\nu}/h_\nu^q\to\infty$. We only compare across strict gaps.

\medskip\noindent\emph{Step 1: a family for a fixed minimum.}
Fix $t\in[a,b]$ and build a finite family $\Lambda_t$ containing $(t\one,0)$
with $\mu(r)=t$ for every slope, working whenever $\mu(p)=t$. If $m=1$ or
$t=b$, take $\Lambda_t=\{(t\one,0)\}$: a point $p\in[a,b]^m$ with
$\mu(p)=t$ then equals $t\one$, giving zero error. Otherwise $m>1$ and
$t<b$. Call a label \emph{of type $k$} if exactly $k$ entries of its slope
exceed $t$. Build the types $k=1,\dots,m-1$ in order, after choosing
$\beta_k$ with
\[
  t<\beta_k<b,\qquad
  \beta_k<\beta_{k-1}\ (k>1),\qquad
  \beta_k-t<E(r)\ \text{for every slope $r$ of type $k-1$}.
\]
These finitely many positive conditions are satisfiable, and
$\beta_{k-1}>\beta_k>t$.

For each $k$-element set $U\subset\{1,\dots,m\}$ and $s\in[\beta_k,b]^U$,
let $r(s)$ agree with $s$ on $U$ and equal $t$ off $U$. Apply the induction
hypothesis in dimension $k$ to the cube $[\beta_k,b]^U$ with tolerance
$s\mapsto E(r(s))$. For every resulting label $(s,d)$, add to $\Lambda_t$
the type-$k$ label
\begin{equation}\label{eq:local-labels}
  r=r(s),\qquad c(h)=-h^{\beta_k}+d(h),\qquad d(h)=o(h^{\beta_k}).
\end{equation}
All slopes of $\Lambda_t$ have minimum exponent $t$, and all offsets are
$o(h^t)$. The negative offset favors raising slope entries on the
coordinates where $X_h(p)$ is small, and the decreasing $\beta_k$ make that
preference stronger with each type.

To verify the guarantee, let $p^\nu\to p\in[a,b]^m$ with $\mu(p)=t$ and
suppose a type-$k$ label $(r,c)$ stays active within $\Lambda_t$ along
$h_\nu\to0+$. Let $U$ be its raised set. For $k>0$, comparing with
$(t\one,0)$ gives
\begin{equation}\label{eq:raise-ineq}
  \sum_{i\in U}(r_i-t)h^{p_i^\nu}\le h^{\beta_k}-d(h)=(1+o(1))h^{\beta_k}.
\end{equation}
Each coefficient $r_i-t\ge\beta_k-t>0$. If some $p_i<\beta_k$, the left
side divided by $h^{\beta_k}$ tends to $+\infty$, contradicting the bounded
right side. Hence
\begin{equation}\label{eq:raised}
  p_i\ge\beta_k\qquad(i\in U).
\end{equation}
On the same $U$, all labels share their behaviour off $U$ and the term
$-h^{\beta_k}$, so $(s,d)$ is active in the family furnished by induction
for the sub-vector $p_U^\nu$. Its limit $p_U$ lies in $[\beta_k,b]^U$ by
\eqref{eq:raised} and $p_i\le b$; the induction hypothesis, which allows
sub-vectors approaching from outside the cube, gives
\begin{equation}\label{eq:onU}
  \max_{i\in U}\abs{p_i-r_i}<E(r).
\end{equation}

Coordinates off $U$ remain (this includes type $0$, where $U=\varnothing$).
We claim that if $k<m-1$ then
\begin{equation}\label{eq:offU}
  p_i\le\beta_{k+1}\qquad(i\notin U).
\end{equation}
Otherwise pick $j\notin U$ with $p_j>\beta_{k+1}$ and a type-$(k+1)$ label
raised on $W=U\cup\{j\}$. Every exponent on $W$ exceeds $\beta_{k+1}$: for
$i\in U$, $p_i\ge\beta_k>\beta_{k+1}$, and $p_j>\beta_{k+1}$. So the
competitor's value minus that of $(t\one,0)$ equals
$-h^{\beta_{k+1}}+o(h^{\beta_{k+1}})$, because each monomial on $W$ is
$o(h^{\beta_{k+1}})$ and the offset is $o(h^{\beta_{k+1}})$. For the active
label the same difference is $0$ if $k=0$, and is at least
$-h^{\beta_k}+o(h^{\beta_k})$ (each term being $\ge0$) if $k>0$. Since
$h^{\beta_k}=o(h^{\beta_{k+1}})$, the competitor is strictly smaller for
small $h$, contradicting activity. This proves \eqref{eq:offU}. Hence off
$U$,
\[
  \abs{p_i-r_i}\le\beta_{k+1}-t<E(r),
\]
because the choice of $\beta_{k+1}$ gives $\beta_{k+1}-t<E(r)$ for every
slope $r$ of type $k$. If $k=m-1$, exactly one coordinate lies off $U$; by
$\mu(p)=t$ and $p_i\ge\beta_k>t$ on $U$, that coordinate equals $t$, where
$r$ also equals $t$, so its error is $0<E(r)$. This proves the required
strict bound for every active label at the fixed minimum.

\medskip\noindent\emph{Step 2: extension to nearby minima.}
For a fixed label $\lambda=(r,c)\in\Lambda_t$, let $A_\lambda\subset[a,b]^m$
be the set of limits $p\in[a,b]^m$ of sequences $p^\nu$ with $h_\nu\to0+$
such that $\lambda$ is active within $\Lambda_t$ at $X_{h_\nu}(p^\nu)$.
$A_\lambda$ is closed: if $z_n\in A_\lambda$ and $z_n\to z$, choose
$0<h_n<1/n$ and $u_n$ with $\norm{u_n-z_n}_\infty<1/n$ and $\lambda$ active
at $X_{h_n}(u_n)$; then $u_n\to z$, so $z\in A_\lambda$. This argument uses
actual active points and needs no continuity of the offsets in $h$.

The set of bad limits
\[
  B_t=\bigcup_{\lambda=(r,c)\in\Lambda_t}
  \{p\in A_\lambda:\norm{p-r}_\infty\ge E(r)\}
\]
is compact, being a finite union of closed subsets of the compact $[a,b]^m$.
Step 1 shows $B_t$ contains no point with $\mu(p)=t$. Since $\mu$ is
continuous and $B_t$ is compact, there is $\rho_t>0$ with $B_t$ disjoint
from $\{\mu\in I_t\}$, $I_t:=(t-\rho_t,t+\rho_t)$: if $B_t$ is nonempty take
$\rho_t<\min_{p\in B_t}\abs{\mu(p)-t}$, and if $B_t$ is empty any
$\rho_t>0$ works. Thus $\Lambda_t$ has the limiting activity property
whenever $\mu(p)\in I_t$. Shrink $\rho_t$ if necessary so that every local
offset obeys
\begin{equation}\label{eq:local-decay}
  c(h)=o(h^{t+\rho_t}).
\end{equation}
For a nonsingleton family choose $t+\rho_t<\min_k\beta_k$ and use
\eqref{eq:local-labels}; for a singleton the offset is zero.

\medskip\noindent\emph{Step 3: combining the families.}
Take a finite subcover of $[a,b]$ by the intervals $I_t$, discard contained
intervals (keeping one of any coinciding pair), and order the surviving
centers as $t_1<\cdots<t_N$. Write $I_{t_j}=(L_j,R_j)$. Noncontainment
forces
\[
  L_1<\cdots<L_N,\qquad R_1<\cdots<R_N.
\]
Consecutive intervals overlap, since if $R_j\le L_{j+1}$ a point of
$[R_j,L_{j+1}]\subset(t_j,t_{j+1})\subset[a,b]$ would be uncovered. The
first interval contains $a$ and the last contains $b$. For $1\le j<N$
choose a cut
\[
  q_j\in\bigl(\max\{t_j,L_{j+1}\},\ \min\{t_{j+1},R_j\}\bigr),
\]
which is nonempty because consecutive intervals overlap and each contains
its own center. With $q_0=a$ and $q_N=b$,
\begin{equation}\label{eq:affine-cuts}
  t_j<q_j<t_{j+1}\ (1\le j<N),\qquad
  [q_{j-1},q_j]\subset I_{t_j}\ (1\le j\le N).
\end{equation}
When $N=1$ the sole interval contains $[a,b]$ and there are no interior
cuts; see Figure~\ref{fig:intervals}.

\begin{figure}[tb]
\centering
\begin{tikzpicture}[x=0.9cm,y=0.8cm,font=\small]
  \draw[->,thin] (-0.4,0) -- (11.0,0) node[right] {$\mu(p)$};
  \foreach \x/\lb in {0.6/{a=q_0},2/{t_1},3.5/{q_1},5/{t_2},6.5/{q_2},8/{t_3},9.5/{b=q_3}} {
    \draw (\x,0.08) -- (\x,-0.08);
    \node[below] at (\x,-0.12) {$\lb$};
  }
  \foreach \x in {0.6,3.5,6.5,9.5} {\draw[gray,densely dashed,thin] (\x,0.2) -- (\x,2.75);}
  \draw[thick] (-0.1,2.5) -- (4.1,2.5);
  \draw[fill=white,thick] (-0.1,2.5) circle (0.06);
  \draw[fill=white,thick] (4.1,2.5) circle (0.06);
  \draw[line width=2pt] (0.6,2.5) -- (3.5,2.5);
  \fill (0.6,2.5) circle (0.07);
  \fill (3.5,2.5) circle (0.07);
  \node[right] at (4.25,2.5) {$I_{t_1}$};
  \draw[thick] (2.9,1.7) -- (7.1,1.7);
  \draw[fill=white,thick] (2.9,1.7) circle (0.06);
  \draw[fill=white,thick] (7.1,1.7) circle (0.06);
  \draw[line width=2pt] (3.5,1.7) -- (6.5,1.7);
  \fill (3.5,1.7) circle (0.07);
  \fill (6.5,1.7) circle (0.07);
  \node[right] at (7.25,1.7) {$I_{t_2}$};
  \draw[thick] (5.9,0.9) -- (10.1,0.9);
  \draw[fill=white,thick] (5.9,0.9) circle (0.06);
  \draw[fill=white,thick] (10.1,0.9) circle (0.06);
  \draw[line width=2pt] (6.5,0.9) -- (9.5,0.9);
  \fill (6.5,0.9) circle (0.07);
  \fill (9.5,0.9) circle (0.07);
  \node[right] at (10.25,0.9) {$I_{t_3}$};
\end{tikzpicture}
\caption{Combining three local families. Each thick closed interval
$[q_{j-1},q_j]$ lies inside the open neighborhood $I_{t_j}$ where its
family is valid. The shifts \eqref{eq:affine-shifts} force the limiting
minimum exponent of an active label into its closed interval. At a cut
either neighbouring family remains valid.}
\label{fig:intervals}
\end{figure}

Combine the selected families, replacing every local offset $c(h)$ in
$\Lambda_{t_j}$ by
\begin{equation}\label{eq:affine-shifts}
  \widetilde c(h)=c(h)-\sum_{\ell=1}^{j-1}h^{q_\ell}.
\end{equation}
These offsets are $o(h^a)$: \eqref{eq:local-decay} gives
$t_j+\rho_{t_j}>a$, and each shift exponent satisfies $q_\ell>t_\ell\ge a$.
A common shift preserves activity within a family; between consecutive
families the new term $-h^{q_j}$ pins the limiting minimum exponent to the
correct side of the cut.

Suppose a fixed label from $\Lambda_{t_j}$ is active in the combined family
along $h_\nu\to0+$ and $p^\nu\to p\in[a,b]^m$; write $c(h)$ for its local
offset and $h=h_\nu$. We claim $q_{j-1}\le\mu(p)\le q_j$. If $j<N$ and
$\mu(p)>q_j$, each coordinate $h^{p_i^\nu}$ is $o(h^{q_j})$ and
$c(h)=o(h^{q_j})$ since $q_j<R_j=t_j+\rho_{t_j}$. Subtracting the value of
the shifted label $(t_{j+1}\one,0)$ gives
\[
  (r-t_{j+1}\one)\cdot X_h(p^\nu)+c(h)+h^{q_j}=h^{q_j}+o(h^{q_j})>0,
\]
a contradiction. If $j>1$ and $\mu(p)<q_{j-1}$, compare instead with the
shifted $(t_{j-1}\one,0)$; the difference is
\[
  (r-t_{j-1}\one)\cdot X_h(p^\nu)+c(h)-h^{q_{j-1}}.
\]
Every coefficient $r_i-t_{j-1}$ is at least $t_j-t_{j-1}>0$; some coordinate
has $p_i<q_{j-1}$, so its monomial divided by $h^{q_{j-1}}$ tends to
$+\infty$; and $c(h)=o(h^{q_{j-1}})$ since $q_{j-1}<t_j<t_j+\rho_{t_j}$. The
whole expression divided by $h^{q_{j-1}}$ tends to $+\infty$, again
contradicting activity. For $j=1$ or $j=N$ the outer bounds follow from
$\mu(p)\in[a,b]$.

By \eqref{eq:affine-cuts} the claimed bounds place $\mu(p)$ in $I_{t_j}$,
including when it equals a cut. The label is active within its original
local family, so the guarantee on $I_{t_j}$ gives
$\norm{p-r}_\infty<E(r)$. This completes the induction and the lemma.
\end{proof}

The sequential formulation yields the uniform statement used below. The
order of choices matters: the family, hence its finite slope set, is fixed
before the upper bound on $h$ is chosen.

\begin{corollary}[Uniform activity bound]\label{cor:activity}
Let $m\ge1$, $a<b$, and $E:[a,b]^m\to(0,\infty)$ be as in
Lemma~\ref{lem:affine}, and fix a family $\Lambda$ supplied by that lemma.
There is $h_0\in(0,1)$ such that for every $0<h<h_0$ and every
$p\in[a,b]^m$, each label $\lambda\in\Lambda$ active at $X_h(p)$ satisfies
$\norm{p-r_\lambda}_\infty<E(r_\lambda)$.
\end{corollary}

\begin{proof}
Otherwise there are $h_\nu\to0+$, points $p^\nu\in[a,b]^m$, and active
labels $\lambda_\nu$ with
$\norm{p^\nu-r_{\lambda_\nu}}_\infty\ge E(r_{\lambda_\nu})$. By finiteness
and compactness, pass to a subsequence on which $\lambda_\nu=\lambda$ is
fixed and $p^\nu\to p\in[a,b]^m$. The inequality passes to the limit and
contradicts Lemma~\ref{lem:affine}. No continuity of $E$ is needed, because
its argument $r_\lambda$ is fixed on the subsequence.
\end{proof}