This section introduces the geometric object of the whole series.  An arithmetic
expression space is a space together with an assignment function and a metric on
which the four arithmetic operations act as \emph{motions}: each elementary
operation moves the current arithmetic value, and the effect of that motion on
points is part of the structure.  We give the axioms, the canonical frame they
determine, the basic hyperbolic model, the four elementary motions, the point
paths they generate, and the continuous form of the same motion.

\subsection{Arithmetic motions}

Let $M$ be a smooth surface with a Riemannian metric $g$, and let
\[
  a:M\longrightarrow\mathbb R
\]
be a smooth function, called the \emph{assignment}: $a(p)$ is the arithmetic
value carried at the point $p$.  An arithmetic operation is to be realized as a
motion of $M$ whose effect on $a$ is that operation.  For instance, adding a
constant $s$ should be a map $\mathsf X_s$ with
\[
  a\circ\mathsf X_s=a+s,
\]
and scaling by a positive constant $k$ should be a map $\mathsf Y_k$ with
$a\circ\mathsf Y_k=ka$.  Two things are deliberately kept apart.

\begin{enumerate}
  \item The \emph{motions} record arithmetic propagation.  They are required to
  transport the assignment in a prescribed way, and they need not preserve the
  metric.
  \item The \emph{metric} measures the infinitesimal cost of moving.  It is part
  of the structure, and the axioms below constrain it through the assignment.
\end{enumerate}

The elementary model of Section~\ref{subsec:p0-basic-model} makes both visible at
once: its additive and multiplicative motions are affine maps of the upper
half-plane, and the metric is the invariant hyperbolic metric of the positive
affine group.

\subsection{Regular arithmetic expression spaces}

\begin{definition}[Regular arithmetic expression space]\label{def:regular-aes}
A \emph{regular arithmetic expression space}, or regular AES, is a tuple
\[
 \mathfrak E=(M,g,a;\mu,\lambda)
\]
such that $M$ is an oriented smooth surface, possibly with boundary, $g$ is a smooth Riemannian
metric, $a\in C^\infty(M,\mathbb R)$, $\mu\in\mathbb R\setminus\{0\}$,
$\lambda\in\mathbb R$, and
\begin{equation}\label{eq:regular-aes-eikonal}
 |\nabla a|_g^2=\mu^2+\lambda^2a^2
\end{equation}
everywhere on $M$.  The function $a$ is called the
\emph{assignment function}.
\end{definition}

The metric in Definition~\ref{def:regular-aes} is part of the structure.
Equation~\eqref{eq:regular-aes-eikonal} is therefore an eikonal equation on a
specified Riemannian surface, not a metric-free identity.  Its right-hand side
records the two arithmetic directions: the constant term $\mu^2$ is the
additive intensity and the assignment term $\lambda^2a^2$ the multiplicative
one.

Let $J$ denote rotation by $+\pi/2$ in the oriented tangent planes.  Since
$\mu\ne0$, the quantity
\[
 q=\sqrt{\mu^2+\lambda^2a^2}
\]
is everywhere positive.  The next proposition shows that the eikonal definition
contains exactly the oriented orthonormal directional data used in the flow
equation.

\begin{proposition}[Canonical arithmetic frame]\label{prop:canonical-arithmetic-frame}
For an oriented Riemannian surface $(M,g)$, a smooth function $a$, and
constants $\mu\ne0$ and $\lambda$, the following are equivalent.
\begin{enumerate}
\item The eikonal identity \eqref{eq:regular-aes-eikonal} holds.
\item There is a unique positively oriented orthonormal frame
      $(X_u,X_v)$ satisfying
      \begin{equation}\label{eq:arithmetic-frame-derivatives}
       X_u a=\mu,
       \qquad
       X_v a=\lambda a.
      \end{equation}
\end{enumerate}
When these conditions hold, put $E_a=\nabla a/q$.  The frame is
\begin{equation}\label{eq:explicit-arithmetic-frame}
 \boxed{
 \begin{aligned}
 X_u&=\frac{\mu}{q}E_a-\frac{\lambda a}{q}J E_a,\\
 X_v&=\frac{\lambda a}{q}E_a+\frac{\mu}{q}J E_a.
 \end{aligned}}
\end{equation}
\end{proposition}

\begin{proof}
Assume the eikonal identity.  Then $|\nabla a|=q>0$, so $E_a$ and $JE_a$ form a
smooth positively oriented orthonormal frame.  The coefficient matrix in
\eqref{eq:explicit-arithmetic-frame} is
\[
 \frac1q
 \begin{pmatrix}
  \mu&\lambda a\\
  -\lambda a&\mu
 \end{pmatrix},
\]
whose columns are orthonormal and whose determinant is one.  Hence
$(X_u,X_v)$ is positively oriented and orthonormal.  Since
$da(E_a)=q$ and $da(JE_a)=0$, the two identities in
\eqref{eq:arithmetic-frame-derivatives} follow.

For uniqueness, write any positively oriented orthonormal frame in the form
\[
 X_u=cE_a+sJE_a,
 \qquad
 X_v=-sE_a+cJE_a,
 \qquad c^2+s^2=1.
\]
The two directional identities force $qc=\mu$ and $-qs=\lambda a$.
Thus $c=\mu/q$ and $s=-\lambda a/q$, which gives
\eqref{eq:explicit-arithmetic-frame}.

Conversely, if an orthonormal frame satisfies
\eqref{eq:arithmetic-frame-derivatives}, then
\[
 |\nabla a|^2=(X_u a)^2+(X_v a)^2
 =\mu^2+\lambda^2a^2.
\]
\end{proof}

The frame is generally non-coordinate.  Applying its bracket to the assignment
function gives
\begin{equation}\label{eq:arithmetic-frame-bracket-on-a}
 [X_u,X_v]a
 =X_u(\lambda a)-X_v(\mu)
 =\mu\lambda.
\end{equation}
Thus, when $\lambda\ne0$, the fields cannot commute.  If
$(\eta^u,\eta^v)$ is their dual coframe, the valid intrinsic Pfaffian statement
is
\[
 da=\mu\,\eta^u+\lambda a\,\eta^v;
\]
the one-forms $\eta^u,\eta^v$ are not presumed exact.  Equation
\eqref{eq:arithmetic-frame-bracket-on-a} is the first appearance of a quantity
that returns in Sections~\ref{sec:p0-acs-tearing} and~\ref{sec:p0-contact}: the
additive direction changes the multiplicative datum by $\mu\lambda$ per unit
area of the arithmetic frame.

\subsection{The basic hyperbolic model}
\label{subsec:p0-basic-model}

Let
\[
 \mathbb H^2=\{(x,y)\in\mathbb R^2:y>0\}
\]
with its standard orientation, and fix $\mu,\lambda>0$.  Put
\begin{equation}\label{eq:p0-e0-data}
 g_{\mu,\lambda}
 =\frac1{y^2}
  \left(\frac{\dd x^2}{\mu^2}
       +\frac{\dd y^2}{\lambda^2}\right),
 \qquad
 a(x,y)=-\frac{x}{y},
\end{equation}
and record the assignment separately, since every later computation refers to
it:
\begin{equation}\label{eq:basic-hyperbolic-assignment}
 a(x,y)=-\frac{x}{y}.
\end{equation}
We call
\[
 \mathfrak E_0(\mu,\lambda)=(\mathbb H^2,g_{\mu,\lambda},a;\mu,\lambda)
\]
the \emph{basic hyperbolic model}.  In earlier drafts of the series and in
Paper~I this model was written $\mathfrak E_{\mathrm{hyp}}(\mu,\lambda)$; the
identification is stated once, here, and the letter $\mathfrak E_0$ is used
from now on (see also Subsection~\ref{subsec:p0-model-names}).

The metric admits the group-theoretic description that explains why addition
and multiplication are the natural motions.  Write
\[
 G=\operatorname{Aff}^{+}(1,\mathbb R)
   =\{(x,y):x\in\mathbb R,\ y>0\}
\]
with multiplication
\begin{equation}\label{eq:positive-affine-group-law}
 (x,y)(x',y')=(x+yx',yy').
\end{equation}
The element $(x,y)$ represents the affine map $t\mapsto x+yt$.  Left
translation by $(x_0,y_0)$ is
\[
 L_{(x_0,y_0)}(x,y)=(x_0+y_0x,y_0y).
\]
For fixed $\mu,\lambda>0$, consider the left-invariant vector fields
\begin{equation}\label{eq:hyperbolic-arithmetic-frame}
 X_u=-\mu y\,\partial_x,
 \qquad
 X_v=-\lambda y\,\partial_y .
\end{equation}
Both signs are chosen so that the frame is positively oriented for
$dx\wedge dy$ and has the action on the assignment stated below.

\begin{proposition}[Invariant affine metric]\label{prop:invariant-affine-metric}
The Riemannian metric for which the frame
\eqref{eq:hyperbolic-arithmetic-frame} is orthonormal is
\begin{equation}\label{eq:hyperbolic-aes-metric}
 g_{\mu,\lambda}
 =\frac1{y^2}
  \left(\frac{dx^2}{\mu^2}+\frac{dy^2}{\lambda^2}\right).
\end{equation}
It is invariant under every left translation of $G$.
\end{proposition}

\begin{proof}
The coframe dual to $(X_u,X_v)$ is
\[
 -\frac{dx}{\mu y},
 \qquad
 -\frac{dy}{\lambda y},
\]
whose sum of squares is \eqref{eq:hyperbolic-aes-metric}.  Under
$L_{(x_0,y_0)}$, one has
\[
 x\longmapsto x_0+y_0x,\qquad
 y\longmapsto y_0y,\qquad
 dx\longmapsto y_0dx,\qquad
 dy\longmapsto y_0dy.
\]
Substitution in \eqref{eq:hyperbolic-aes-metric} shows
$L_{(x_0,y_0)}^*g_{\mu,\lambda}=g_{\mu,\lambda}$.
\end{proof}

\begin{theorem}[Basic hyperbolic AES]\label{thm:basic-hyperbolic-aes}
For every $\mu,\lambda>0$, the tuple
\[
 \mathfrak E_0(\mu,\lambda)
 =(\mathbb H^2,g_{\mu,\lambda},a;\mu,\lambda)
\]
is a regular arithmetic expression space.  Its canonical arithmetic frame is
the left-invariant frame \eqref{eq:hyperbolic-arithmetic-frame}.  Moreover,
$(\mathbb H^2,g_{\mu,\lambda})$ is complete and has constant Gaussian
curvature
\[
 K=-\lambda^2.
\]
\end{theorem}

\begin{proof}
Direct differentiation gives
\[
 a_x=-\frac1y,
 \qquad
 a_y=\frac{x}{y^2}=-\frac ay.
\]
Therefore the frame \eqref{eq:hyperbolic-arithmetic-frame} satisfies
\[
 X_u a=\mu,
 \qquad
 X_v a=\lambda a.
\]
It is positively oriented and orthonormal by
Proposition~\ref{prop:invariant-affine-metric}, and the frame equivalence of
Proposition~\ref{prop:canonical-arithmetic-frame} proves the regular-AES
identity.  Equivalently, using
\[
 g_{\mu,\lambda}^{-1}
 =\mu^2y^2\,\partial_x\otimes\partial_x
  +\lambda^2y^2\,\partial_y\otimes\partial_y,
\]
one obtains directly
\[
 |\nabla a|_{g_{\mu,\lambda}}^2
 =\mu^2y^2a_x^2+\lambda^2y^2a_y^2
 =\mu^2+\lambda^2a^2.
\]

For completeness and curvature, introduce the global coordinate
\[
 X=\frac{\lambda}{\mu}x.
\]
Then
\begin{equation}\label{eq:scaled-standard-hyperbolic-metric}
 g_{\mu,\lambda}
 =\frac1{\lambda^2}\frac{dX^2+dy^2}{y^2}.
\end{equation}
The metric in parentheses is the complete standard upper-half-plane metric of
curvature $-1$.  Multiplication of a Riemannian metric by
$\lambda^{-2}$ preserves completeness and multiplies Gaussian curvature by
$\lambda^2$.  Hence $g_{\mu,\lambda}$ is complete and has curvature
$-\lambda^2$.
\end{proof}

The parameters have different geometric roles.  The change
$X=(\lambda/\mu)x$ removes $\mu$ from the metric, so $\mu$ fixes the
additive normalization of the assignment frame.  The parameter $\lambda$
sets the curvature scale.  No uniqueness claim for regular AESs follows from
this normalization.

\paragraph{Horocyclic coordinates and assignment contours.}
Set $y=e^{\lambda v}$ and $u=x$.  Then
\begin{equation}\label{eq:horocyclic-affine-coordinates}
 g_{\mu,\lambda}
 =e^{-2\lambda v}\frac{du^2}{\mu^2}+dv^2,
 \qquad
 a(u,v)=-u e^{-\lambda v}.
\end{equation}
The curves $y=\mathrm{constant}$ are horocycles based at the ideal point
$\infty$, and the vertical lines $x=\mathrm{constant}$ are their
orthogonal geodesics.  The assignment contour of value $c$ is
\[
 a^{-1}(c)=\{x=-cy,\ y>0\}.
\]
In particular, the zero locus in this model is the vertical geodesic
$\{x=0,\ y>0\}$.

\paragraph{The Laplace eigenfunction.}
We use the divergence-of-gradient sign convention
\begin{equation}\label{eq:laplacian-sign-convention}
 \Delta_g f
 =\frac1{\sqrt{\det g}}\,
  \partial_i\!\left(\sqrt{\det g}\,g^{ij}\partial_jf\right).
\end{equation}

\begin{proposition}[Laplace eigenfunction]\label{prop:laplace-eigenfunction}
The assignment function of the basic hyperbolic AES satisfies
\begin{equation}\label{eq:hyperbolic-assignment-laplacian}
 \Delta_{g_{\mu,\lambda}}a=2\lambda^2a.
\end{equation}
\end{proposition}

\begin{proof}
Since
\[
 \sqrt{\det g_{\mu,\lambda}}=\frac1{\mu\lambda y^2},
\]
the products
$\sqrt{\det g}\,g^{xx}=\mu/\lambda$ and
$\sqrt{\det g}\,g^{yy}=\lambda/\mu$ are constant.  Thus
\[
 \Delta_{g_{\mu,\lambda}}f
 =\mu^2y^2 f_{xx}+\lambda^2y^2 f_{yy}.
\]
For $a=-x/y$, one has $a_{xx}=0$ and
$a_{yy}=-2x/y^3=2a/y^2$.  Substitution gives
\eqref{eq:hyperbolic-assignment-laplacian}.
\end{proof}

\subsection{The four operations as motions}

On the basic model, define for $s\in\mathbb R$ and $k>0$
\begin{equation}\label{eq:p0-grid-moves}
  \mathsf X_s(x,y)=(x-sy,y),
  \qquad
  \mathsf Y_k(x,y)=\left(x,\frac yk\right).
\end{equation}

\begin{proposition}[Arithmetic action on the assignment]
\label{prop:p0-grid-assignment-action}
The maps in \eqref{eq:p0-grid-moves} satisfy
\[
  a\circ\mathsf X_s=a+s,
  \qquad
  a\circ\mathsf Y_k=ka.
\]
Consequently, translation and positive scaling of the current arithmetic value
can be represented by motions of points in $\mathfrak E_0$.
\end{proposition}

\begin{proof}
Direct substitution gives
\[
  a(x-sy,y)=-\frac{x-sy}{y}=a(x,y)+s,
\]
and
\[
  a\left(x,\frac yk\right)
  =-\frac{x}{y/k}=k a(x,y).
\]
\end{proof}

These transformations are assignment-compatible; they are not generally
isometries of $g_{\mu,\lambda}$.  The distinction is the one announced in
Subsection~\ref{subsec:p0-basic-model}: the motions record arithmetic
propagation, whereas the metric measures its infinitesimal geometry.

The remaining two operations are obtained from the same two families and a
reflection.

\begin{warning}[The motions are partial]
\label{warn:p0-partial-motions}
The maps below are defined only where the arithmetic value stays in the range
carried by the model, and they are not asserted to form a global group action
even there.  Concretely: the scaling motion $\mathsf Y_k$ is defined for $k>0$
and moves the assignment from $a$ to $ka$, so a word whose intermediate values
leave the range of $a$ on $M$ has no realization on $M$; a model whose
assignment misses a half-axis, such as the punctured model
$\mathfrak E_1$ of Example~\ref{ex:p0-e1-model}, cannot carry globally the
motions that would move the value into the missing half-axis; and a motion of a
punctured model may run into a puncture, as
Section~\ref{sec:p0-holed} records.  Only the basic model
$\mathfrak E_0$ carries the two affine motions globally, and only there do they
generate the full positive affine group action.  Everywhere else ``the four
operations act as motions'' means: each operation is realized by a locally
defined map wherever its assignment condition holds.
\end{warning}

\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{lll}
\toprule
operation & motion of $\mathfrak E_0$ & effect on $a$ \\
\midrule
addition of $s$ & $\mathsf X_s$ & $a\mapsto a+s$ \\
subtraction of $s$ & $\mathsf X_{-s}$ & $a\mapsto a-s$ \\
multiplication by $k>0$ & $\mathsf Y_k$ & $a\mapsto ka$ \\
division by $k>0$ & $\mathsf Y_{1/k}$ & $a\mapsto a/k$ \\
reflection (sign change) & $(x,y)\mapsto(-x,y)$ & $a\mapsto -a$ \\
\bottomrule
\end{tabular}
\end{center}

Subtraction is thus the inverse of the additive motion, and division by a
nonzero constant is the inverse of the multiplicative motion.  The reflection is
not by itself one of the four operations; it appears because the assignment of
the model is signed, and it is the elementary instance of the second-slot
subtraction $c-z$ of Section~\ref{sec:p0-expansion}.  Multiplying by a negative
constant is realized as the reflection composed with a positive scaling; the
model therefore carries every nonzero affine multiplier of a real history, while
the \emph{positive} scalings are the motions that stay in the identity component and
are the ones used in the grid, in the flow, and in the ACS.  Over a general field
the affine cocycles of Section~\ref{sec:p0-affine-cocycles} again allow every
nonzero multiplier, and the analytic statements of this paper are made over the
positive component.

For an integer $m\ge2$, the two moves satisfy the exact relation
\begin{equation}\label{eq:p0-bs-relation}
  \mathsf Y_m^{-1}\circ\mathsf X_s^m\circ\mathsf Y_m
  =\mathsf X_s.
\end{equation}
Indeed, $\mathsf X_s^m=\mathsf X_{ms}$, and conjugation by $\mathsf Y_m$
divides the additive displacement by $m$.  Equation
\eqref{eq:p0-bs-relation} is the elementary Baumslag--Solitar-type relation
visible in the grid \cite{BaumslagSolitar1962NonHopfian}; no complexity
conclusion is inferred from it.  Its arithmetic content is that a
multiplicative motion transports the additive ruler: after rescaling the
space, the additive unit that produces the same effect on $a$ is $1/m$ of the
original one.  The same transport, made infinitesimal and measured against a
commutative shadow, is the tearing discussed in
Section~\ref{sec:p0-acs-tearing}.

\paragraph{Division by the evolving value.}
The motions above are affine, and so is every word in them.  The remaining
elementary context of Section~\ref{sec:p0-expansion}, second-slot division
$c/z$, is not affine and cannot be realized by a motion of $\mathfrak E_0$ that
preserves the assignment line.  It is realized projectively: on the projective
line it is an element of $\PGL_2(K)$, and it moves the arithmetic value
$z\mapsto c/z$ on the ordinary domain $z\ne0$.  Section~\ref{sec:p0-ripples}
reads this operation not as a motion of the value but as a transport of
conditions, and Section~\ref{sec:p0-projective-unification} places both
readings in one matrix.

\subsection{Point paths}
\label{subsec:p0-point-paths}

The left-expanded comb becomes geometric once its one-hole sections are
restricted to translations and nonzero scalings.  Let a left-expanded affine
expression have chronological sections
\[
  F_i(z)=\alpha_i z+\beta_i,
  \qquad \alpha_i\ne0.
\]
For an initial arithmetic value $z_0$, define
\[
  z_i=F_i(z_{i-1}),
  \qquad i=1,\ldots,n.
\]
Choose a point $p_0\in\mathbb H^2$ with $a(p_0)=z_0$, and realize each
translation or positive scaling by the corresponding map in
\eqref{eq:p0-grid-moves}.  The successive points
\[
  p_0,p_1,\ldots,p_n
\]
form the \emph{point path} of the bounded expression in this chosen realization.
By Proposition~\ref{prop:p0-grid-assignment-action},
\[
  a(p_i)=z_i.
\]
Thus the geometry records every intermediate value, not only the endpoint.  The
word ``point path'' is a descriptive name for this orbit; it asserts no new
structure beyond the displayed construction.

\begin{example}[A four-step path]
\label{ex:p0-four-step-path}
Start from $z_0=1$ and apply
\[
  z\mapsto4z,
  \qquad z\mapsto z-1,
  \qquad z\mapsto2z,
  \qquad z\mapsto z-3.
\]
The value history is
\[
  1\longmapsto4\longmapsto3\longmapsto6\longmapsto3.
\]
In the grid this becomes two transverse scale moves and two horocyclic
translation moves.  The final value $3$ alone does not determine this path.
\end{example}

\begin{figure}[t]
\centering
\begin{tikzpicture}[
  x=0.62cm,y=0.58cm,
  axis/.style={-{Latex[length=2mm]},thick,black!70},
  hline/.style={thin,blue!45},
  vline/.style={thin,green!55!black},
  path/.style={very thick,black},
  every node/.style={font=\scriptsize}
]
  \fill[blue!2] (-8.3,0) rectangle (8.3,8.7);
  \draw[axis] (-8.5,0)--(8.7,0) node[right] {$x$};
  \draw[axis] (0,-0.1)--(0,9.0) node[above] {$y$};
  \foreach \yy in {1,2,4,8}
    {\draw[hline] (-8.2,\yy)--(8.2,\yy);}
  \foreach \xx in {-8,-6,-4,-3,-2,-1,1,2,3,4,6,8}
    {\draw[vline] (\xx,0.08)--(\xx,8.45);}
  \node[red!65!black] at (-8.0,8.35) {$1$};
  \node[red!65!black] at (-8.0,4.25) {$2$};
  \node[red!65!black] at (-8.0,2.25) {$4$};
  \node[red!65!black] at (-8.0,1.25) {$8$};
  \node[red!65!black] at (0.25,8.35) {$0$};

  \coordinate (p0) at (-8,8);
  \coordinate (p1) at (-8,2);
  \coordinate (p2) at (-6,2);
  \coordinate (p3) at (-6,1);
  \coordinate (p4) at (-3,1);
  \draw[path] (p0)--(p1)--(p2)--(p3)--(p4);
  \foreach \p/\lab/\pos in {p0/{1}/above,p1/{4}/left,p2/{3}/above,p3/{6}/left,p4/{3}/above}
    {\fill (\p) circle (2pt) node[\pos] {$\lab$};}
  \node[font=\small,anchor=west] at (-1.6,7.3)
    {$1\xrightarrow{\times4}4\xrightarrow{-1}3
      \xrightarrow{\times2}6\xrightarrow{-3}3$};
\end{tikzpicture}
\caption{A schematic point path in the basic hyperbolic arithmetic grid.  The
black polyline retains the intermediate propagation that the endpoint value
forgets.}
\label{fig:p0-path-grid}
\end{figure}

\paragraph{The affine algebra behind the path.}
Every affine prefix has a unique form
\[
  G_i(z)=A_i z+B_i,
  \qquad A_i\ne0.
\]
If $F_i(z)=\alpha_i z+\beta_i$, then
\begin{equation}\label{eq:p0-affine-prefix-recursion}
  A_i=\alpha_i A_{i-1},
  \qquad
  B_i=\alpha_i B_{i-1}+\beta_i,
  \qquad (A_0,B_0)=(1,0).
\end{equation}
The point path is therefore the evaluation of an affine matrix path
\[
  \begin{pmatrix}A_i&B_i\\0&1\end{pmatrix}
  \begin{pmatrix}z_0\\1\end{pmatrix}.
\]
The lower-left entry remains zero at every stage.  Projectively this means that
all prefixes stabilize the same point $\infty$.  Section~\ref{sec:p0-ripples}
shows what changes when right-slot division makes that entry nonzero.

\subsection{The operation-to-length assignment in the affine sector}
\label{subsec:p0-operation-to-length}

Subsection~\ref{subsec:p0-point-paths} assigns to a bounded affine expression the
affine matrix product \eqref{eq:p0-affine-prefix-recursion}, and
Proposition~\ref{prop:affine-cocycle-formulas} condenses a chronological history to
two real coordinates.  Neither statement says whether a composed arithmetic word
determines a \emph{length}, that is, a displacement of the model.  The affine sector
answers that question completely, and the answer separates the two published
coordinates of one history.  Three extensions are not available here and are listed
at the end of the subsection.

\paragraph{Length of an affine motion.}

\begin{proposition}[Translation length of a positive dilation]
\label{prop:p0-dilation-length}
Let $k>0$ and let $D_k(z)=kz$ act on $\mathbb H^2=\{z:\operatorname{Im}z>0\}$ with
the hyperbolic metric $ds^2=(dx^2+dy^2)/y^2$.  Then the imaginary axis
$i\mathbb R_{>0}$ is invariant under $D_k$, it is a complete geodesic, and the
translation length of $D_k$ along it is
\begin{equation}\label{eq:p0-dilation-length}
  \ell(D_k)=\bigl\lvert\ln k\bigr\rvert .
\end{equation}
\end{proposition}

\begin{proof}
The map $D_k$ sends $iy$ to $iky$, so $i\mathbb R_{>0}$ is invariant and the
restriction of $D_k$ to it is the translation $y\mapsto ky$.  The hyperbolic length
of the segment joining $iy$ to $iky$ is
$\bigl\lvert\int_y^{ky}dt/t\bigr\rvert=\lvert\ln k\rvert$, and it does not depend on
$y$.  The displacement is therefore $\lvert\ln k\rvert$, and it is attained on an
invariant geodesic, which is the defining property of the translation length.
\end{proof}

\begin{proposition}[The length of an affine motion is its multiplicative charge, up
to sign]
\label{prop:p0-length-multiplicative-charge}
Let $\varphi\in\operatorname{Aff}^{+}(1,\mathbb R)$ be written
$\varphi(z)=\Phi z+\xi$ with $\Phi>0$, and put $M(\varphi):=\ln\Phi$.  Then the
translation length of the action of $\varphi$ on $\mathbb H^2$ is
\begin{equation}\label{eq:p0-affine-length}
  \ell(\varphi)=\bigl\lvert M(\varphi)\bigr\rvert ,
\end{equation}
and consequently, for every history $\gamma\in G$,
\begin{equation}\label{eq:p0-history-length}
  \ell\bigl(\nu(\gamma)\bigr)=\bigl\lvert M(\gamma)\bigr\rvert ,
\end{equation}
where $\nu:G\to\operatorname{Aff}^{+}(1,\mathbb R)$ is the evaluation of
Proposition~\ref{prop:p0-history-homomorphisms} and
$M=\operatorname{pr}_2\circ c=\log\circ L\circ\nu$ is the multiplicative charge of
Theorem~\ref{thm:p0-descent}(3).
\end{proposition}

\begin{proof}
If $\Phi=1$ then $\varphi$ is the translation $z\mapsto z+\xi$; it has no fixed
point in $\mathbb H^2$, it is parabolic, and its translation length is
$0=\lvert\ln 1\rvert$.  If $\Phi\ne1$ then $\varphi$ has the fixed point
$z^{*}=\xi/(1-\Phi)$ on the boundary circle, and conjugating by the translation
$\psi(z)=z-z^{*}$, which is an isometry of $\mathbb H^2$, gives
$\psi\varphi\psi^{-1}(w)=\Phi w=D_\Phi(w)$.  The translation length is invariant
under conjugation by isometries, so
Proposition~\ref{prop:p0-dilation-length} gives $\lvert\ln\Phi\rvert$.  The second
display is the first read on histories through
Theorem~\ref{thm:p0-descent}(3).
\end{proof}

\paragraph{Two data the length does not carry.}

\begin{proposition}[The length is not a homomorphism, and it is not injective on
histories]
\label{prop:p0-length-losses}
\begin{enumerate}
\item $\ker\nu\subsetneq\ker M$.  Every nonempty pure additive history lies in
  $\ker M$ and induces a non-identity motion, so it is not in $\ker\nu$; hence the
  multiplicative charge, and with it the length, is a function of the induced
  motion, while the history-to-motion assignment is not injective.
\item By Theorem~\ref{thm:p0-descent}(4) the additive charge does not factor
  through $\nu$.  Two histories with the same induced motion, and therefore the
  same length, may differ in additive charge, and no function of the induced
  motion, the length included, recovers that difference.
\item $M$ is a homomorphism $G\to(\mathbb R,+)$, whereas $\ell=\lvert M\rvert$ is
  not.  Writing $\mathsf M_q$ for the dilation generator,
  \begin{equation}\label{eq:p0-length-not-additive}
    \ell\bigl(\mathsf M_q\mathsf M_{-q}\bigr)=0
    \;\ne\;
    2\lvert q\rvert
    =\ell(\mathsf M_q)+\ell(\mathsf M_{-q})
    \qquad(q\ne0).
  \end{equation}
  Equivalently, $\ell$ forgets the sign of $M$, that is, which of the two
  eigenvalues $e^{\pm M/2}$ of the induced affine motion is the expanding one.
\end{enumerate}
\end{proposition}

\begin{proof}
(1) For $p\ne0$ the history $\mathsf A_p$ satisfies $\nu(\mathsf A_p)=T_p\ne
\mathrm{id}$ and $M(\mathsf A_p)=0$.  (2) is the statement of
Theorem~\ref{thm:p0-descent}(4).  (3) $\mathsf M_q\mathsf M_{-q}=\mathsf M_0$ has
multiplicative charge $0$, while $\ell(\mathsf M_{\pm q})=\lvert q\rvert$ by
Proposition~\ref{prop:p0-dilation-length}.  For the eigenvalue claim: in
homogeneous coordinates $\varphi$ has the matrix
$\left(\begin{smallmatrix}\Phi&\xi\\0&1\end{smallmatrix}\right)$ with eigenvalues $\Phi$ and
$1$, so the eigenvalues of its normalised lift to $\mathrm{SL}_2(\mathbb R)$ are
$e^{\pm M/2}$.
\end{proof}

\begin{proposition}[The scale-compatible relative defect changes no length]
\label{prop:p0-relative-defect-length-zero}
Let $\gamma,\delta$ be scale-compatible affine histories with common scale $\Phi$.
Then the relative affine defect $\mathcal K_{\mathrm{aff}}(\gamma,\delta)$ of
Definition~\ref{def:relative-affine-defect} has multiplier $1$, and its translation
length is zero:
\begin{equation}\label{eq:p0-relative-defect-length-zero}
  M\bigl(\mathcal K_{\mathrm{aff}}(\gamma,\delta)\bigr)=0,
  \qquad
  \ell\bigl(\mathcal K_{\mathrm{aff}}(\gamma,\delta)\bigr)=0 .
\end{equation}
It is therefore parabolic whenever it is nontrivial, it fixes exactly one boundary
point, and it preserves every translation length.
\end{proposition}

\begin{proof}
By \eqref{eq:scale-compatible-relative-translation} the defect is the translation
$x\mapsto x+(\xi_\gamma-\xi_\delta)/\Phi$, whose homogeneous matrix
$\left(\begin{smallmatrix}1&(\xi_\gamma-\xi_\delta)/\Phi\\0&1\end{smallmatrix}\right)$
has multiplier $1$ and trace $2$.  Apply
Proposition~\ref{prop:p0-length-multiplicative-charge} in the case $\Phi=1$.
\end{proof}

\begin{proposition}[Trace classification and length of a projective motion]
\label{prop:p0-projective-trichotomy}
Let $M\in\operatorname{PSL}_2(\mathbb R)$ act on $\mathbb H^2$ and let
$\widetilde M\in\operatorname{SL}_2(\mathbb R)$ be any lift, so that
$\tau(M)^2:=\operatorname{tr}(\widetilde M)^2$ is well defined.  Then:
\begin{enumerate}
\item if $\tau^2>4$, $M$ is hyperbolic, it has two fixed points on the boundary
  circle, it preserves the geodesic joining them, and its translation length is
  \begin{equation}\label{eq:p0-projective-length}
    \ell(M)=2\operatorname{arccosh}\bigl(\lvert\tau(M)\rvert/2\bigr);
  \end{equation}
\item if $\tau^2=4$, $M$ is parabolic or the identity; a nontrivial parabolic motion
  has one fixed point on the boundary circle, preserves no geodesic, and has
  translation length $0$;
\item if $\tau^2<4$, $M$ is elliptic, it has one fixed point in $\mathbb H^2$ and
  none on the boundary circle, it is the rotation about that point by
  $2\arccos(\lvert\tau(M)\rvert/2)$, and its translation length is $0$.
\end{enumerate}
Consequently the trichotomy is decided by the sign of $\tau^2-4$, and on the affine
slice \eqref{eq:p0-projective-length} agrees with
Proposition~\ref{prop:p0-length-multiplicative-charge}.
\end{proposition}

\begin{proof}
The lift has determinant $1$, so its eigenvalues are $\lambda$ and $\lambda^{-1}$ and
$\tau=\lambda+\lambda^{-1}$.  The three cases are the three positions of the
discriminant $\tau^2-4$ of the characteristic polynomial.  If $\tau^2>4$ then
$\lambda$ is real and not $\pm1$; conjugating $M$ to the diagonal lift
$\operatorname{diag}(e^{\ell/2},e^{-\ell/2})$, which is possible in
$\operatorname{SL}_2(\mathbb R)$ for the appropriate $\ell>0$, gives
$\tau=2\cosh(\ell/2)$, and the translation length is $\ell$ by
Proposition~\ref{prop:p0-dilation-length} together with the invariance of the length
under conjugation by isometries.  If $\tau^2=4$ then $\lambda=\pm1$ and the lift is
unipotent, $\widetilde M=\pm(I+N)$ with $N^2=0$ and $N\neq0$ unless $M$ is the
identity; such an element fixes exactly one boundary point.  If $\tau^2<4$ then
$\lambda=e^{\pm i\theta}$ with $\theta\notin\{0,\pi\}$, the lift is conjugate in
$\operatorname{SL}_2(\mathbb R)$ to a rotation by $2\theta$, and
$\tau=2\cos\theta$.  For the last claim, an affine motion
$\varphi(z)=\Phi z+\xi$ with $\Phi>0$ has the lift
$\Phi^{-1/2}\left(\begin{smallmatrix}\Phi&\xi\\0&1\end{smallmatrix}\right)$, whose
trace is $\Phi^{1/2}+\Phi^{-1/2}=2\cosh(\tfrac12\ln\Phi)$; so
\eqref{eq:p0-projective-length} returns $\lvert\ln\Phi\rvert$, which is
Proposition~\ref{prop:p0-length-multiplicative-charge}.
\end{proof}

\begin{proposition}[The two layers of the affine conjugacy classification]
\label{prop:p0-affine-two-layers}
Work in $\operatorname{Aff}^{+}(1,\mathbb R)$, write $\varphi(z)=\Phi z+\xi$ with
$\Phi>0$, and write $T_c(z)=z+c$ for the translation by $c$.
\begin{enumerate}
\item For every $\Phi>0$ one has $D_\Phi T_c D_\Phi^{-1}=T_{\Phi c}$.  Hence all
  nontrivial translations of a fixed sign are mutually conjugate, the stabiliser in
  $\operatorname{Aff}^{+}(1,\mathbb R)$ of $T_c$ is the translation subgroup, and
  the conjugacy class $\{T_{\Phi c}:\Phi>0\}$ is a torsor under the dilations.
\item A dilation $D_\Phi$ is conjugate to $D_{\Phi'}$ if and only if
  $\Phi=\Phi'$; its conjugacy class is therefore determined by the multiplier, so
  the length $\lvert\ln\Phi\rvert$ is a function of the class.
\end{enumerate}
Consequently the conjugacy class carries one continuous invariant in the dilation
layer and none in the translation layer, where only the sign survives.
\end{proposition}

\begin{proof}
(1) $D_\Phi\circ T_c\circ D_\Phi^{-1}(z)=D_\Phi(z/\Phi+c)=z+\Phi c$.  Conjugation by
a translation fixes $T_c$, so the stabiliser is the translation subgroup, and
$\operatorname{Aff}^{+}(1,\mathbb R)$ modulo that subgroup is
$\mathbb R_{>0}$; the orbit is therefore a torsor under the dilations.
(2) dilations commute, so $D_\Phi D_{\Phi'}D_\Phi^{-1}=D_{\Phi'}$ and conjugacy
forces $\Phi=\Phi'$.  The length is
$\lvert\ln\Phi\rvert$ by Proposition~\ref{prop:p0-length-multiplicative-charge}.
\end{proof}

\begin{remark}[Status: structural proposal --- the two coordinates of one history]
\label{rem:p0-length-two-coordinates}
Proposition~\ref{prop:affine-cocycle-formulas} gives one chronological history two
real coordinates, the accumulated scale $\Phi_n$ of
\eqref{eq:accumulated-affine-scale} and the target affine cocycle $\xi_n$ of
\eqref{eq:target-affine-cocycle}.  By
Proposition~\ref{prop:p0-length-multiplicative-charge} the quantity
$\log\lvert\Phi_n\rvert$ \emph{is} the translation length of the induced motion, and
by Proposition~\ref{prop:p0-length-losses}(2) the additive data that $\xi_n$ records
is not a function of the induced motion at all.  We propose reading
$(\log\Phi_n,\xi_n)$ as one assignment with two sides: a side that is a function of
the induced motion, and a side that is a function of the history only.  This is a
proposal about how to organise statements already proved above; it adds no theorem,
it does not identify $\xi_n$ with any geometric coordinate, and it makes no claim
about the Fenchel--Nielsen coordinates of
Section~\ref{sec:p0-compactification}.  One negative point can be recorded without
leaving the affine sector: neither $\xi_n$ nor the relative translation
$(\xi_\gamma-\xi_\delta)/\Phi$ can be a Fenchel--Nielsen twist, because by
Proposition~\ref{prop:p0-relative-defect-length-zero} both are parabolic of length
zero, whereas the holonomy of a closed geodesic of length $\ell>0$ is hyperbolic of
trace $2\cosh(\ell/2)>2$; a twist changes no length, and no rescaling of a
coefficient changes a conjugacy type.  What does transfer is the compatibility
condition: ``scale-compatible'' in
Definition~\ref{def:relative-affine-defect} is exactly the requirement that a glued
curve carry one agreed length.  The length-zero verdict should be read with
Proposition~\ref{prop:p0-affine-two-layers}: a translation layer whose class carries
no continuous invariant is not an empty layer but the layer in which the only datum
is the choice of conjugate, i.e. a torsor.  So the relative affine defect is not a
failed length; it is the stratum on which a twist, if this geometry carries one,
would have to live.  Whether that torsor coincides with a resolution torsor of the kind used in a
through-crossing theory is not decided here, and the series architecture gives
Paper~0 no route to say so.
\end{remark}

\paragraph{What this subsection does not do.}
\begin{enumerate}
\item It does not pass from $\operatorname{Aff}^{+}(1,\mathbb R)$ to
  $\operatorname{PGL}_2(K)$ for a general field $K$.  Over $\mathbb R$ the passage is
  now made: Proposition~\ref{prop:p0-projective-trichotomy} classifies a projective
  motion by the sign of $\tau^2-4$ and gives its translation length, so second-slot
  division, whose affine/projective break is
  Corollary~\ref{cor:p0-affine-projective-break}, is covered in the real case.  What
  remains outside is a general $K$: for a field with no order the sign of
  $\tau^2-4$ is not available, and the classification must be stated through the
  discriminant's multiplicative character together with the number of rational fixed
  points.  Paper~0 does not develop that form, and it asserts no length outside
  $\mathbb R$.
\item It does not pass from left- and right-expanded combs to arbitrary binary
  expression trees.  The source of the assignment is the comb grammar of
  Definition~\ref{def:p0-left-right-combs}, with the unique internal order of
  Proposition~\ref{prop:p0-combs-unique-order}.  The sequential-tree classification
  is a Paper~I result, cited here prospectively and not restated.
\item It does not settle the category in which the assignment lives.  Whether the
  source should be a free category of partial contexts, an action groupoid, a path
  groupoid, or a rewriting $2$-category is OQ-047, deferred to Paper~IV.  The
  affine-sector instance stated here is compatible with each of those choices and
  selects none of them.
\end{enumerate}

\subsection{Continuous arithmetic motion}
\label{subsec:p0-continuous-motion}

The discrete motions of Subsection~\ref{subsec:p0-point-paths} have an
infinitesimal counterpart.  Translation and dilation generate the Lie algebra
of the positive affine group, and their linear combinations give the continuous
propagation law.  The formulation below is intrinsic: the symbols $u$ and $v$
label arithmetic directions, but need not be coordinates.

\paragraph{The affine Lie algebra and its flow.}
On the assignment line with coordinate $z$, let
\[
 E=\partial_z,
 \qquad
 H=z\partial_z .
\]
These are respectively the infinitesimal generators of translation and positive
dilation, and they satisfy $[E,H]=E$.  Fix constants
$\mu,\lambda\in\mathbb R$, with $\mu\ne0$.  The generator selected by an angle
$\theta$ is
\begin{equation}\label{eq:angular-affine-generator}
 \Omega_\theta
 =\mu\cos\theta\,E+\lambda\sin\theta\,H .
\end{equation}
Consequently, an integral curve $z=z(s)$ satisfies
\begin{equation}\label{eq:continuous-affine-flow}
 \frac{dz}{ds}
 =\mu\cos\theta+\lambda z\sin\theta .
\end{equation}
This Lie-algebra calculation is a derivation of the continuous form of the
arithmetic motion, and it fixes the convention: the translation coefficient is
$\mu\cos\theta$, while the dilation coefficient is $\lambda\sin\theta$.

When $\theta$ is constant, set $b=\lambda\sin\theta$.  The solution through
$z(0)=z_0$ is
\begin{equation}\label{eq:directional-affine-solution}
 z(s)=
 \begin{cases}
  e^{bs}z_0+\displaystyle\mu\cos\theta\,\frac{e^{bs}-1}{b},
     & b\ne0,\\[7pt]
  z_0+\mu s\cos\theta, & b=0.
 \end{cases}
\end{equation}
In particular, the positive arithmetic axes give
\[
 \theta=0:\quad z(s)=z_0+\mu s,
 \qquad
 \theta=\frac{\pi}{2}:\quad z(s)=e^{\lambda s}z_0.
\]
The opposite axes give subtraction and inverse dilation.

\paragraph{The Pfaffian propagation constraint.}
There is a useful three-dimensional local state space with coordinates
$(u,v,z)$.  On it define
\begin{equation}\label{eq:affine-pfaffian-form}
 \alpha=dz-\mu\,du-\lambda z\,dv
\end{equation}
and the two horizontal vector fields
\begin{equation}\label{eq:affine-horizontal-fields}
 D_u=\partial_u+\mu\partial_z,
 \qquad
 D_v=\partial_v+\lambda z\partial_z.
\end{equation}
A curve tangent to
$\cos\theta\,D_u+\sin\theta\,D_v$ satisfies
\eqref{eq:continuous-affine-flow}.  Thus the frequently used notation
\[
 dz=\mu\,du+\lambda z\,dv
\]
means that $\alpha$ vanishes on admissible tangent vectors.  It is not an
identity asserting that $z$ is a function of two independent coordinates whose
ordinary partial derivatives are simultaneously $\mu$ and $\lambda z$.
Indeed,
\begin{equation}\label{eq:horizontal-noncommutation}
 [D_u,D_v]=\mu\lambda\,\partial_z.
\end{equation}
When $\mu\lambda\ne0$, this bracket already rules out such an integrable
coordinate interpretation.  The same bracket is computed intrinsically in
\eqref{eq:arithmetic-frame-bracket-on-a}, and
Proposition~\ref{prop:p0-contact-quotient} proves that the local model here and
the frame there are two descriptions of one object for the basic model: the
horizontal fields descend to the canonical frame and the curvature form is the
hyperbolic area form.  Its geometric reading as a contact structure is
taken up in Section~\ref{sec:p0-contact}.

\begin{theorem}[Continuous affine flow]\label{thm:continuous-affine-flow}
Let $\mathfrak E=(M,g,a;\mu,\lambda)$ be a regular AES with its canonical
arithmetic frame.  If a unit-speed curve $\gamma$ has tangent
\[
 \dot\gamma
 =\cos\theta\,X_u+\sin\theta\,X_v,
\]
where $\theta$ may vary along the curve, then
\begin{equation}\label{eq:intrinsic-continuous-affine-flow}
 \frac{d}{ds}(a\circ\gamma)
 =\mu\cos\theta+\lambda(a\circ\gamma)\sin\theta.
\end{equation}
For constant $\theta$, its solutions are given by \eqref{eq:directional-affine-solution}.
\end{theorem}

\begin{proof}
By the chain rule and \eqref{eq:arithmetic-frame-derivatives},
\[
 \frac{d}{ds}(a\circ\gamma)
 =da(\dot\gamma)
 =\cos\theta\,X_u a+\sin\theta\,X_v a
 =\mu\cos\theta+\lambda a\sin\theta.
\]
For constant $\theta$, this is the linear equation already integrated in
\eqref{eq:directional-affine-solution}.
\end{proof}

\paragraph{Rectification.}
For $\lambda\ne0$, define
\begin{equation}\label{eq:rectifying-assignment}
 r=\operatorname{arcsinh}\!\left(\frac{\lambda a}{\mu}\right).
\end{equation}
The chain rule and \eqref{eq:regular-aes-eikonal} give
\begin{equation}\label{eq:rectified-eikonal}
 |\nabla r|=|\lambda|.
\end{equation}
Thus $r$ has constant gradient magnitude.  When $\lambda=0$, the original
equation already reduces to $|\nabla a|=|\mu|$, and $a/\mu$ is a
unit-gradient rectifying function.  Rectification of the assignment is
possible; rectification of \emph{both} arithmetic directions at once is not, by
\eqref{eq:horizontal-noncommutation}.  This asymmetry is the discrete root of
the tearing language of Section~\ref{sec:p0-holed}.

The same affine convention fixes the sign of the elementary local order defect.
For small increments $h$ and $k$, let
\[
 A_h(z)=z+\mu h,
 \qquad
 M_k(z)=e^{\lambda k}z.
\]
Then
\begin{align}
 (M_k\circ A_h)(z)-(A_h\circ M_k)(z)
 &=\mu h\bigl(e^{\lambda k}-1\bigr)\notag\\
 &=\mu\lambda\,hk+O\!\left(\lVert(h,k)\rVert^3\right).
 \label{eq:infinitesimal-order-defect}
\end{align}
This is an infinitesimal two-order comparison, not an exact area formula for an
arbitrary finite region.  The exact finite statement, and its interpretation as
a defect measured in the commutativized space, occupy
Section~\ref{sec:p0-acs-tearing}.

\paragraph{Affine motion inside projective motion.}
An infinitesimal projective transformation of the line has the Riccati form
\[
 \dot z=\beta+\alpha z+\kappa z^2.
\]
The theory developed in this section is the affine slice $\kappa=0$, with
$\beta=\mu\cos\theta$ and $\alpha=\lambda\sin\theta$.  Results proved for
regular affine AESs do not automatically extend to the quadratic projective
flow.

\subsection{Model names: $E_0$, $E_1$, and later examples}
\label{subsec:p0-model-names}

The historical drafts of this programme used the labels $E_0$ and $E_1$
inconsistently; in some notes the hyperbolic half-plane is called $E_0$, in
others $E_1$.  Because later chapters introduce spaces that are \emph{not} smooth,
we fix the convention now, once, and record it as a convention rather than a
theorem.

\begin{convention}[Model labels]
\label{conv:p0-model-labels}
$\mathfrak E_0(\mu,\lambda)$ denotes the basic regular hyperbolic model of
Subsection~\ref{subsec:p0-basic-model}.  For $k\ge1$, the label
$\mathfrak E_k$ is used descriptively for an arithmetic expression space whose
regular locus is a surface with $k$ declared punctures: points removed from the
regular structure and carrying no regular AES data.  No classification by $k$
is asserted; the label records the number of declared punctures and nothing
else.  The reserved historical labels $E_k$ and $E_{\log}$ of earlier drafts
are not model classes, and this paper defines neither of them.
\end{convention}

Two examples make the convention concrete; both are developed later, and the
first one is the only example in this paper whose punctured model is verified
in detail.

\begin{example}[The basic model, $k=0$]
\label{ex:p0-e0-model}
$\mathfrak E_0(\mu,\lambda)$ is the upper half-plane with the metric
\eqref{eq:p0-e0-data} and the assignment $a=-x/y$.  It has no puncture, its
assignment vanishes along one complete geodesic, and by
Theorem~\ref{thm:basic-hyperbolic-aes} it is complete with constant curvature.
This is the model in which the arithmetic motions of
Subsection~\ref{subsec:p0-point-paths} are read as point paths.
\end{example}

\begin{example}[A one-puncture model, $k=1$]
\label{ex:p0-e1-model}
Let $\mathbb D=\{\rho<1\}$ be the unit disc with the complete Poincar\'e
metric $g_{\mathbb D}=4\,(d\xi^2+d\eta^2)/\lambda^2(1-\rho^2)^2$ and the radial
assignment
\[
 a_{\mathbb D}(\xi,\eta)=
 \begin{cases}
  \dfrac{2\mu}{\lambda}\,\dfrac{\rho}{1-\rho^2},&\rho\ne0,\\[6pt]
  0,&\rho=0 .
 \end{cases}
\]
On the punctured disc $\mathbb D\setminus\{0\}$ this is a regular AES with
$|\nabla a_{\mathbb D}|^2=\mu^2+\lambda^2a_{\mathbb D}^2$, while at the centre
the assignment fails to be $C^1$ although the metric extends smoothly.  The
regular locus is therefore $\mathbb D\setminus\{0\}$ and the model carries
exactly one puncture.  Paper~I verifies the details of this model as its
isolated-zero example; the global invariant $\lambda a_{\mathbb D}
=\mu\sinh(\lambda s)$ along hyperbolic distance $s$ from the centre shows that
the assignment, not the metric, is what fails at the puncture.
\end{example}

Example~\ref{ex:p0-e1-model} is the first holed arithmetic expression space, and
Section~\ref{sec:p0-holed} returns to it.  Its puncture is not a hole in the
metric; it is a point at which the regular structure stops being defined, and
it is where the arithmetic motions of the punctured part can no longer be
extended.