This paper was enlarged to become the geometric foundation of the series. The division of labour between Paper~0 and Paper~I is fixed by the authoritative amendment \path{governance/00b-paper-0-geometric-foundation-amendment.md}, and this section records the resulting interface without restating either paper's theorems. \subsection{What Paper 0 establishes} The proved spine of this paper is \[ \begin{gathered} \text{left and right comb grammars} \longrightarrow \text{slot-$(1)$/slot-$(2)$ one-hole maps},\\ \text{distributive expansion} \longrightarrow \text{same operator, different charges},\\ \text{arithmetic motions on an AES} \longrightarrow \text{point path in } \mathfrak E_0,\\ \text{projective operator} \longrightarrow \text{pole, fixed points, ripple pencil},\\ \text{affine cocycles} \longrightarrow \text{relative defect} \longrightarrow \text{ACS weighted area},\\ \text{contact form} \longrightarrow \text{non-integrable horizontal distribution} \longrightarrow \text{curvature } \mu\lambda,\\ \Aff(1,K)=\Stab(\infty) \subset \PGL_2(K). \end{gathered} \] The paper also supplies four laboratories: the basic hyperbolic model $\mathfrak E_0$, the one-puncture model $\mathfrak E_1$, the pulled-back horocycle pencil, and the three-dimensional contact model. The finite geometric-series and continued-fraction examples are exact at every truncation; their infinite values are asserted only under the stated formal or analytic convergence hypotheses. Two statements in the paper are not theorems about AES and are labelled as proposals where they occur: the descriptive name \emph{ripple geometry} for the dual reading of Section~\ref{sec:p0-ripples}, and the language of \emph{tearing} of Section~\ref{sec:p0-acs-tearing} together with the punctured reading of Section~\ref{sec:p0-holed}. Both rest on proved computations --- the line-to-circle theorem, the weighted torsion--Stokes theorem, and the contact curvature --- and neither claims a new isomorphism class of geometry. \subsection{What remains in Paper I} Paper~I begins one level earlier and keeps the history layer. It retains responsibility for: \begin{enumerate} \item the intrinsic dependency-poset classification of all sequential binary trees, including mixed slot words; \item marked planar sequential trees and bounded/free marked spinal histories, with operand-slot chirality, mirror, temporal reversal, and path inverse kept distinct; \item projective evaluation of chronological marked histories, including ordinary-domain data and the affine sector; \item the $q=4$ Hecke sublanguage and its history-level relations; \item the regular-zero theorems, the complete splitting theorem, the foundational definition of a singular arithmetic expression space, the verified isolated-zero model, and the smooth parameter-family zero-set lemma; \item the interfaces to Papers~II, III, and~IV. \end{enumerate} Paper~I states the definition of a regular AES and its canonical frame as a concise imported interface, because its zero theorems are stated in those terms; everything else that moved here is referenced rather than repeated. The earlier version of that interface, which ran in the opposite direction, is superseded. Two chapters of this paper are deliberately \emph{not} offered as stable interfaces to the later papers. Section~\ref{sec:p0-beyond-linear} states two proved boundaries --- the boundary of finite-dimensional Lie-group closure and the boundary of observables of the charge endpoint --- together with a vocabulary imported from a companion programme; the boundaries are theorems, the vocabulary is a research stance, and the later papers depend on neither. Similarly, the \emph{witness register} of Section~\ref{subsec:p0-holed-witnesses} and the two registered problems of Subsection~\ref{subsec:p0-holed-open} are research reports. The new Section~\ref{sec:p0-compactification}, in contrast, provides the stable finite-type construction and collar interface. Section~\ref{sec:p0-holed} is not without proved content: the definition of a punctured space, the non-vanishing of the assignment gradient in a regular AES, and the explicit punctured model realizing the four-circle configuration (Theorem~\ref{thm:p0-four-circle-model}) are theorems, and the last of them may be imported wherever a concrete punctured instance is needed in place of the disc model of Paper~I. \subsection{The dependency architecture} The series dependency is \[ \boxed{ \text{Paper 0} \longrightarrow \text{Paper I} \longrightarrow \{\text{Papers II, III, IV}\}. } \] The first arrow exports the following interfaces: \begin{center} \renewcommand{\arraystretch}{1.3} \begin{tabular}{ll} \toprule Paper 0 object & use in later papers \\ \midrule comb and slot conventions & operand-slot chirality on sequential spines \\ distributive expansion & comparison of expansions with equal operator \\ regular AES and canonical frame & imported definition for zero theory \\ arithmetic motions and $\mathfrak E_0$ & model for analysis and for the flow \\ one-puncture model $\mathfrak E_1$ & first punctured instance \\ point path & bounded marked-history evaluation in the affine sector \\ ripple pencil and pole & motivation for bilateral projective continuation \\ affine cocycles & discrete order defect \\ history group and charge descent & what the ACS is, and what the motions keep \\ ACS and relative torsion & global torsion and its measures \\ contact form and horizontal curvature & horizontal calculus and function theory \\ the Reeb-quotient identification & the two models are one object \\ punctured models and their zero loci & verified finite-puncture instances \\ finite-type carrier and collar contracts & geometric input for Paper IV ports\\ tearing and the hole witnesses & research report; not yet an interface \\ \bottomrule \end{tabular} \end{center} Papers~II and~III import the regular-AES definition, the canonical frame, the basic hyperbolic model, the contact model, and the elementary bilateral operators from this paper; they do not redefine them. Paper~IV imports the history layer from Paper~I and the affine torsion, finite-type carriers and collar-reconstruction theorem from this paper. Its finite program substitution and numerical evaluation do not establish that every gate is an AES motion. \subsection{Open extensions} Several genuine problems remain, and they are recorded here rather than advertised as results. \begin{enumerate} \item Construct a projective analogue of the affine arithmetic expression space in which the pole and the ripple pencil are intrinsic rather than chosen through one upper-half-plane chart, and decide which path or ripple data descend through relations among history words. \item Decide whether the contact structure of Section~\ref{sec:p0-contact} admits a canonical compatible structure beyond the horizontal distribution, and if so which. Neither a complex structure nor a horizontal metric is determined by the contact form alone. \item Determine whether the choice of punctures in Section~\ref{sec:p0-holed} can be made canonical, and whether a punctured model carries a ripple structure that survives at the puncture; the three registered programmes of Subsection~\ref{subsec:p0-holed-open} are the concrete routes. \item Extend the one-hole analysis to multi-wire expression networks while retaining ordinary-domain, pole, and fixed-point information, and develop a controlled theory of infinite arithmetic expressions that separates formal, analytic, projective, and computational convergence. \item Determine whether the non-linear algebraic structures proposed in Section~\ref{sec:p0-beyond-linear} can be defined intrinsically for AES, so that the two proved obstructions of that section are replaced by a positive description rather than a refusal. \end{enumerate} None of these problems is solved merely by the matrix representation or by the existence of the contact form. The stable conclusion of the present paper is more elementary: the four arithmetic operations act as motions of one space, their order defect is measured in the commutativization of that space, and the contact structure of the same defect is what prevents the two motions from being calibrated independently.