% Canonical entry point for Arithmetic Expression Geometry 0.
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\title{\textbf{Arithmetic Expression Geometry 0}\\[0.35em]
\large Arithmetic Expression Spaces: Expansion and Motion, Eikonal Equation,
Horocycle Pullbacks, Commutativization, Contact Structure, and Tearing}

\author{
Mingli Yuan\thanks{Corresponding author and sole holder of scientific
responsibility for this manuscript
(\texttt{mingli@swarma.org}).  Formalization, machine checking, computation,
proof drafting, and writing were assisted by DeepSeek Harness, an
artificial-intelligence research agent (model
\texttt{deepseek-v4-flash-vision-exp}); that assistance is credited on this
title page and in the acknowledgements, but it is not human authorship, since
the agent is not a natural person and cannot sign a copyright transfer or serve
as corresponding author.  Contribution-level provenance is recorded in
\texttt{AUTHORSHIP\_AND\_PROVENANCE.md}.}\\
\small Swarma Research\\
\small Beijing 100083, China\\[0.5em]
\small\itshape with formalization and computational assistance\\
\small\itshape from DeepSeek Harness
(model \texttt{deepseek-v4-flash-vision-exp})
}

\date{September 22, 2026\\[0.25em]
\small Draft manuscript under mathematical review.\\[0.35em]
\small
\textbf{MSC 2020:} 53D10 (primary); 53C22, 20F65;
30F45, 32G15, 11J70, 03F60}

\begin{document}
\maketitle

\begin{quote}\small
\textbf{Draft notice.}
This manuscript is a working draft in the editorial and publication sense; here
``draft'' is not a mathematical claim-status label and does not alter the status
of any theorem, proposition, computation, conjecture, or open problem.  It
replaces the earlier prelude of the same number and absorbs the geometric
chapters of Paper~I; the migration is recorded in
\texttt{governance/00b-paper-0-geometric-foundation-amendment.md}.
\end{quote}

\begin{abstract}
An arithmetic expression is a word of operations; a motion is a flow on a
manifold.  For expressions whose evaluation has a single internal spine, the two
are one structure: affine for addition and multiplication, projective once the
second operand slot is divided, contact-geometric in the order defect.  A
regular arithmetic expression space (AES) is an oriented Riemannian surface
$(M,g)$ with a smooth assignment $a\colon M\to\mathbb R$ satisfying the eikonal
equation $|\nabla a|_g^2=\mu^2+\lambda^2a^2$; the four operations then act as
motions translating and scaling $a$.  In the basic model $\mathfrak E_0$, the
upper half-plane with the left-invariant metric of
$\mathrm{Aff}^+(1,\mathbb R)$ (curvature $-\lambda^2$), addition and positive
multiplication are affine translations and dilations obeying a
Baumslag--Solitar law, and $a$ is a Laplace--Beltrami eigenfunction of
eigenvalue $2\lambda^2$.  Left-expanded words stay in the affine Borel cell
fixing $\infty$; the right-expanded language, once second-slot division is
admitted, generates all of $\mathrm{PGL}_2(\mathbb R)$ through the rank-one
Bruhat decomposition, and pulling standard horocycles back along one operator
word gives the dual reading, ripple geometry.  Commutativizing the motions
by total charge yields the accumulative commutative space, where the order
defect of two histories is an exact weighted area, the tearing.  The same defect
is the constant $\mu\lambda$ in the bracket
$[D_u,D_v]=\mu\lambda\partial_a$ of the horizontal fields of
$\alpha=da-\mu\,du-\lambda a\,dv$, a contact form whenever $\mu\lambda\ne0$, and
it is the curvature of that horizontal connection; for $\mathfrak E_0$ the
basic model is proved to be the Reeb quotient of the contact model, and
fixed-charge tearing and fixed-state Reeb drift are Legendre dual readings of
the same curvature.  The generators $\partial_a$, $a\partial_a$, and
$a\log a\,\partial_a$ span an infinite-dimensional Lie algebra, so no
finite-dimensional Lie group carries all three ranks, and no observable of the
charge endpoint detects tearing.  For every finite puncture count a conformally
compactifiable punctured AES is constructed, with an explicit four-circle zero
locus and exact reconstruction from three-ended pants blocks.  All results are
elementary, planar, or local to a three-dimensional contact model.
\end{abstract}

\noindent\textbf{Keywords:}
arithmetic expression geometry; eikonal equation on surfaces; positive affine
group; Baumslag--Solitar relation; hyperbolic upper half-plane and horocycles;
affine cocycles; contact structures and Reeb quotients; Legendre duality;
punctured Riemann surfaces and pants decompositions; infinite-dimensional Lie
algebras

\setcounter{tocdepth}{2}
\tableofcontents
\newpage

\section{Introduction}\label{sec:p0-introduction}
\input{sections/01-introduction.tex}

\section{Arithmetic Expansions and the Distributive Law}\label{sec:p0-expansion}
\input{sections/02-expansion.tex}
\input{sections/02a-evaluation-and-effectivity.tex}

\section{Arithmetic Expression Spaces and Arithmetic Motion}\label{sec:p0-aes}
\input{sections/03-aes-and-motion.tex}

\section{Ripple Geometry: The Projective Dual}\label{sec:p0-ripples}
\input{sections/04-ripple-geometry.tex}

\section{Reciprocals, Poles, Infinity, and Fixed Points}\label{sec:p0-reciprocals}
\input{sections/05-reciprocals-fixed-points.tex}

\section{Matrix and Projective Unification}\label{sec:p0-projective-unification}
\input{sections/06-projective-unification.tex}

\section{Affine Cocycles and the Relative Defect}\label{sec:p0-affine-cocycles}
\input{sections/07-affine-cocycles.tex}

\section{Commutativization: The ACS and Arithmetic Tearing}\label{sec:p0-acs-tearing}
\input{sections/08-acs-tearing.tex}

\section{The Contact Structure of Tearing}\label{sec:p0-contact}
\input{sections/09-contact-of-tearing.tex}

\section{Beyond Linear Language}\label{sec:p0-beyond-linear}
\input{sections/10-beyond-linear.tex}

\section{Holed Arithmetic Expression Spaces and the Language of Tearing}\label{sec:p0-holed}
\input{sections/11-holed-aes.tex}

\section{Finite-Type Compactification and Three-Ended Reconstruction}\label{sec:p0-compactification}
\input{sections/12-compactification.tex}

\section{Interfaces to Papers I--IV}\label{sec:p0-interface}
\input{sections/12-interface.tex}

\section*{Acknowledgements and provenance}
\addcontentsline{toc}{section}{Acknowledgements and provenance}
The research direction, the choice of objects, and the mathematical statements
of this paper are the work of Mingli Yuan, who is the author, the corresponding
author, and the sole holder of scientific responsibility for the text.

The manuscript was produced in a hybrid workflow that must be named explicitly
rather than hidden.  DeepSeek Harness, an artificial-intelligence research agent
running the model \texttt{deepseek-v4-flash-vision-exp}, assisted with proof
drafting, with the formalization and machine checking of finite statements under
\texttt{paper-0/formal}, with the exact computations under
\texttt{paper-0/scripts}, with red-team reading of the claim statuses, and with
the preparation of \LaTeX{} source.  An agent is not a natural person: it cannot
hold copyright, cannot sign a transfer, and cannot correspond, so it appears on
the title page as assistance and not as a coauthor.  Contribution-level
provenance, including the commits and model identities where they are known, is
maintained in \texttt{AUTHORSHIP\_AND\_PROVENANCE.md}; the same document records
the standing rule that machine cross-checking is not independent human review.
The mathematical statuses used throughout, and the boundary between proved
statements and structural proposals, are those registered in
\texttt{governance/05b-paper-0-status-register.md}.

\appendix
\section{Elementary Calculations}\label{app:p0-calculations}
\input{appendices/app-A-calculations.tex}

\section{Hyperbolic-Model Calculations}\label{app:p0-hyperbolic}
\input{appendices/app-B-hyperbolic-calculations.tex}

\section{Affine Cocycle Calculations}\label{app:p0-affine-cocycles}
\input{appendices/app-C-affine-cocycles.tex}

\section{ACS and Contact Calculations}\label{app:p0-acs-contact}
\input{appendices/app-D-acs-contact.tex}

\section{Circle Surgery and Branched-Sheet Models}\label{app:p0-circle-surgery}
\input{appendices/app-E-circle-surgery.tex}

\section{Cuts and the Foundations of Arithmetic Expressions}\label{app:p0-cuts-foundations}
\input{appendices/app-F-cuts-and-expression-foundations.tex}

\nocite{Beardon1983DiscreteGroups,Rosen1954ContinuedFractions}
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\bibliography{aeg-paper-0}

\end{document}