% Moved from Paper I Section 9 (governance/00b, migration map); mathematical % content unchanged apart from cross-references and framing sentences. Section~\ref{sec:p0-acs-tearing} measured tearing by a weighted area in the commutativization of the arithmetic motions. That measurement is exact at every finite scale, but it is stated in a space that has forgotten the current value. This section recovers the local picture: a three-dimensional state space that retains the value and distinguishes the two infinitesimal arithmetic directions, carrying an Ehresmann connection over the two-dimensional direction plane and, in the nondegenerate case, a contact form. Tearing then appears a third time, with the same density: as the curvature of that connection. Fix real constants \(\mu\) and \(\lambda\), and let \[ \mathcal C=\mathbb R^3_{(u,v,a)}, \qquad \pi:\mathcal C\longrightarrow\mathbb R^2_{(u,v)}. \] Here \(a\) is the current assignment value, while \(u\) and \(v\) parametrize additive and multiplicative motion. We orient \(\mathcal C\) by \[ du\wedge da\wedge dv. \] \subsection{The arithmetic contact form} Define \begin{equation} \alpha:=da-\mu\,du-\lambda a\,dv. \label{eq:arithmetic-contact-form} \end{equation} The equation \(\alpha=0\) says that a lifted base motion obeys the affine propagation law. \begin{proposition}[Contact nondegeneracy] \label{prop:contact-form} With the orientation \(du\wedge da\wedge dv\), \begin{equation} \alpha\wedge d\alpha =\mu\lambda\,du\wedge da\wedge dv. \label{eq:contact-volume-sign} \end{equation} Hence \(\alpha\) is a contact form if and only if \(\mu\lambda\ne0\). \end{proposition} \begin{proof} Since \(\mu\) and \(\lambda\) are constant, \[ d\alpha=-\lambda\,da\wedge dv. \] It follows that \begin{align*} \alpha\wedge d\alpha &=(da-\mu\,du-\lambda a\,dv) \wedge(-\lambda\,da\wedge dv)\\ &=\mu\lambda\,du\wedge da\wedge dv. \end{align*} This three-form is nowhere zero exactly when \(\mu\lambda\ne0\). \end{proof} The contact-isomorphism class is the standard local one; the arithmetic content lies in the coordinates and in the preferred horizontal frame introduced next. Even in a degenerate case, \(\ker\alpha\) remains a well-defined horizontal distribution, although it is then not contact. \subsection{Horizontal lifts and the affine flow} Set \begin{equation} D_u:=\partial_u+\mu\partial_a, \qquad D_v:=\partial_v+\lambda a\partial_a. \label{eq:horizontal-arithmetic-fields} \end{equation} Then \[ \alpha(D_u)=\alpha(D_v)=0, \] and the two fields project respectively to \(\partial_u\) and \(\partial_v\). Thus \[ \mathcal D:=\ker\alpha =\operatorname{span}\{D_u,D_v\} \] is an Ehresmann connection for \(\pi\). Its additive and multiplicative flows are \begin{align} \Phi_u^h(u,v,a)&=(u+h,v,a+\mu h), \label{eq:additive-horizontal-flow}\\ \Phi_v^k(u,v,a)&=(u,v+k,e^{\lambda k}a). \label{eq:multiplicative-horizontal-flow} \end{align} For a direction angle \(\theta\), define \[ D_\theta=\cos\theta\,D_u+\sin\theta\,D_v. \] If \((u(s),v(s),a(s))\) is tangent to \(D_\theta\), then \[ \frac{du}{ds}=\cos\theta, \qquad \frac{dv}{ds}=\sin\theta, \] and \begin{equation} \frac{da}{ds} =\mu\cos\theta+\lambda a\sin\theta. \label{eq:contact-realizes-affine-flow} \end{equation} Thus the horizontal lift reproduces the continuous affine flow rather than postulating a separate dynamics. \subsection{Horizontal curvature} \begin{theorem}[Curvature of the specified horizontal connection] \label{thm:contact-curvature} The preferred horizontal fields satisfy \begin{equation} [D_u,D_v]=\mu\lambda\partial_a. \label{eq:horizontal-curvature-bracket} \end{equation} In particular, when \(\mu\lambda\ne0\), the horizontal distribution is not integrable. Its bracket defect is vertical and has constant coefficient \(\mu\lambda\) in the arithmetic frame. \end{theorem} \begin{proof} Using \eqref{eq:horizontal-arithmetic-fields}, \begin{align*} [D_u,D_v] &=[\partial_u+\mu\partial_a, \partial_v+\lambda a\partial_a]\\ &=\mu\,\partial_a(\lambda a)\partial_a -\lambda a\,\partial_a(\mu)\partial_a\\ &=\mu\lambda\partial_a. \end{align*} All remaining coordinate brackets vanish. Since \(\partial_a\) is vertical and is not contained in \(\ker\alpha\), the bracket is the vertical failure of horizontal closure. \end{proof} The sign in \eqref{eq:horizontal-curvature-bracket} agrees with the chronological convention of the preceding chapter. Equivalently, \[ \alpha([D_u,D_v])=\mu\lambda, \qquad d\alpha(D_u,D_v)=-\mu\lambda. \] The two formulas have opposite signs by the identity \(d\alpha(X,Y)=-\alpha([X,Y])\) for horizontal fields \(X,Y\). \subsection{Open defects and closed horizontal loops} The bracket is infinitesimal. At finite scale there are two related, but different, quantities. First compare the two open lifts from \((u,v)\) to \((u+h,v+k)\): \begin{align*} \Phi_v^k\circ\Phi_u^h(u,v,a) &=(u+h,v+k,e^{\lambda k}(a+\mu h)),\\ \Phi_u^h\circ\Phi_v^k(u,v,a) &=(u+h,v+k,e^{\lambda k}a+\mu h). \end{align*} Their assignment difference, with ``add then multiply'' taken first, is \begin{equation} T_{\mathrm{open}}(h,k) =\mu h(e^{\lambda k}-1). \label{eq:exact-open-contact-defect} \end{equation} This is exactly the elementary ACS rectangle torsion with charge increments \((\mu h,\lambda k)\). Now perform the chronological four-step history \[ \mathsf A_h,\quad \mathsf M_k,\quad \mathsf A_{-h},\quad \mathsf M_{-k}, \] where \(\mathsf A_h\) denotes the flow \(\Phi_u^h\) and \(\mathsf M_k\) denotes \(\Phi_v^k\). The base point returns to \((u,v)\), and direct substitution gives \begin{align*} a&\longmapsto a+\mu h \longmapsto e^{\lambda k}(a+\mu h)\\ &\longmapsto e^{\lambda k}(a+\mu h)-\mu h \longmapsto a+\mu h(1-e^{-\lambda k}). \end{align*} The closed-loop vertical drift is therefore \begin{equation} T_{\mathrm{closed}}(h,k) =\mu h(1-e^{-\lambda k}) =e^{-\lambda k}T_{\mathrm{open}}(h,k). \label{eq:exact-closed-contact-drift} \end{equation} Thus the finite open comparison and the finite closed holonomy are not equal. Here \(T_{\mathrm{closed}}\) is the source-normalized relative translation. Converting it to the target frame gives \(e^{\lambda k}T_{\mathrm{closed}}=T_{\mathrm{open}}\) exactly. Figure~\ref{fig:contact-open-closed-commutator} keeps the two finite constructions visually separate. The square is drawn in the $(u,v)$ base; the short vertical segments represent assignment fibers, not additional base edges. \begin{figure}[t] \centering \begin{tikzpicture}[ x=1cm,y=1cm, openone/.style={-{Latex[length=2mm]},very thick,red!65!black}, opentwo/.style={-{Latex[length=2mm]},very thick,blue!65!black}, loop/.style={-{Latex[length=2mm]},very thick,green!45!black}, fiber/.style={thick,black!55}, every node/.style={font=\small} ] \node[font=\bfseries] at (-3.6,2.0) {open two-path comparison}; \coordinate (o) at (-5,-0.45); \coordinate (ou) at (-3.45,-0.45); \coordinate (ouv) at (-3.45,1.1); \coordinate (ov) at (-5,1.1); \draw[openone] (o)--node[below,font=\scriptsize] {$\mathsf A_h$} (ou); \draw[openone] (ou)--node[right,font=\scriptsize] {$\mathsf M_k$} (ouv); \draw[opentwo] (o)--node[left,font=\scriptsize] {$\mathsf M_k$} (ov); \draw[opentwo] (ov)--node[below,font=\scriptsize] {$\mathsf A_h$} (ouv); \fill (o) circle (1.6pt) node[below left] {$(u,v)$}; \fill (ouv) circle (1.6pt) node[above left,yshift=3pt,font=\scriptsize] {$(u+h,v+k)$}; \draw[densely dotted,black!45] (ouv)--(-2.72,1.1); \draw[fiber] (-2.62,0.35)--(-2.62,1.45); \fill[blue!65!black] (-2.62,0.62) circle (1.7pt) node[right,font=\scriptsize] {$a_{\mathsf M\mathsf A}$}; \fill[red!65!black] (-2.62,1.2) circle (1.7pt) node[right,font=\scriptsize] {$a_{\mathsf A\mathsf M}$}; \draw[{Latex[length=1.6mm]}-{Latex[length=1.6mm]},black!60] (-2.82,0.62)--(-2.82,1.2) node[midway,left,font=\scriptsize] {$T_{\mathrm{open}}$}; \node[font=\bfseries] at (2.6,2.0) {closed chronological lift}; \coordinate (c0) at (1.2,-0.45); \coordinate (cu) at (2.75,-0.45); \coordinate (cuv) at (2.75,1.1); \coordinate (cv) at (1.2,1.1); \draw[loop] (c0)--node[below,font=\scriptsize] {$\mathsf A_h$} (cu); \draw[loop] (cu)--node[right,font=\scriptsize] {$\mathsf M_k$} (cuv); \draw[loop] (cuv)--node[above,font=\scriptsize] {$\mathsf A_{-h}$} (cv); \draw[loop] (cv)--node[left,font=\scriptsize] {$\mathsf M_{-k}$} (c0); \fill (c0) circle (1.6pt) node[below left] {$(u,v)$}; \draw[densely dotted,black!45] (c0)--(3.48,-0.45); \draw[fiber] (3.58,-0.75)--(3.58,0.55); \fill (3.58,-0.48) circle (1.7pt) node[right,font=\scriptsize] {$a$}; \fill[green!45!black] (3.58,0.25) circle (1.7pt) node[right,font=\scriptsize] {$a+T_{\mathrm{closed}}$}; \draw[{Latex[length=1.6mm]}-{Latex[length=1.6mm]},black!60] (3.36,-0.48)--(3.36,0.25) node[midway,left,font=\scriptsize] {$T_{\mathrm{closed}}$}; \node[align=center,font=\scriptsize] at (2.45,-1.5) {$T_{\mathrm{closed}}=e^{-\lambda k}T_{\mathrm{open}}$\\ same infinitesimal density $\mu\lambda$}; \end{tikzpicture} \caption{Open defect and closed holonomy in the contact model. The two open horizontal lifts have the same base endpoint but different assignment values. The four-step closed base loop returns to its starting base point with a raw vertical drift. These finite quantities differ by the displayed scale factor; converting the closed drift to the target frame gives the open defect, while their infinitesimal densities agree as stated in the text.} \label{fig:contact-open-closed-commutator} \end{figure} \subsection{The scalar horizontal differential} For a smooth scalar field \(F\in C^\infty(\mathcal C)\), define \begin{equation} \delta_HF :=dF-(\partial_aF)\alpha =(D_uF)\,du+(D_vF)\,dv. \label{eq:scalar-horizontal-differential} \end{equation} This is the ordinary differential restricted to horizontal directions and written on the base coframe. In particular, \[ \delta_Ha=\mu\,du+\lambda a\,dv. \] For scalar fields only, define the antisymmetrized second horizontal derivative by \begin{equation} \mathcal R_H(F) :=(D_uD_vF-D_vD_uF)\,du\wedge dv. \label{eq:horizontal-scalar-curvature-definition} \end{equation} This notation deliberately avoids presupposing an extension of \(\delta_H\) to a graded complex. \begin{proposition}[Curvature on scalar fields] \label{prop:horizontal-differential-curvature} For every \(F\in C^\infty(\mathcal C)\), \begin{equation} \mathcal R_H(F) =\mu\lambda(\partial_aF)\,du\wedge dv. \label{eq:horizontal-scalar-curvature} \end{equation} In particular, \[ \mathcal R_H(a)=\mu\lambda\,du\wedge dv. \] \end{proposition} \begin{proof} By definition and Theorem~\ref{thm:contact-curvature}, \[ \mathcal R_H(F) =[D_u,D_v](F)\,du\wedge dv =\mu\lambda(\partial_aF)\,du\wedge dv. \] Taking \(F=a\) proves the final formula. \end{proof} One may use \(\delta_H^2F\) as shorthand for \(\mathcal R_H(F)\) after declaring this antisymmetrization convention. It does not mean that \(\delta_H\) has been defined on horizontal forms, nor does it assert the existence of a nilpotent differential complex. No such graded calculus is required for the foundational result. \subsection{Three levels of the same order defect} \begin{proposition}[Finite and infinitesimal comparison] \label{prop:torsion-curvature-synthesis} For the elementary rectangle with additive side \(\mu h\) and logarithmic multiplicative side \(\lambda k\), the following statements hold. \begin{enumerate} \item The open affine defect equals the ACS weighted area exactly: \[ T_{\mathrm{open}}(h,k) =\int_{[0,\mu h]\times[0,\lambda k]} e^{\lambda k-M}\,dA\wedge dM =\mu h(e^{\lambda k}-1), \] with oriented-chain interpretation when a side is negative. \item The closed chronological lift has exact drift \[ T_{\mathrm{closed}}(h,k) =e^{-\lambda k}T_{\mathrm{open}}(h,k). \] Thus \(T_{\mathrm{closed}}\) is the source-normalized relative translation; its target-frame conversion agrees with the open defect, but the raw finite values generally differ. \item Both defects have the common infinitesimal density \[ \lim_{(h,k)\to(0,0)} \frac{T_{\mathrm{open}}(h,k)}{hk} =\lim_{(h,k)\to(0,0)} \frac{T_{\mathrm{closed}}(h,k)}{hk} =\mu\lambda, \] where the quotients are taken for \(hk\ne0\). This is the coefficient of \([D_u,D_v]\) and of \(\mathcal R_H(a)\). \end{enumerate} \end{proposition} \begin{proof} The first identity is Theorem~\ref{thm:torsion-stokes} applied to the elementary rectangle, and its integral evaluates to \[ \mu h\int_0^{\lambda k}e^{\lambda k-M}\,dM =\mu h(e^{\lambda k}-1). \] The second identity is \eqref{eq:exact-closed-contact-drift}. Finally, \[ \frac{e^{\lambda k}-1}{k}\longrightarrow\lambda, \qquad \frac{1-e^{-\lambda k}}{k}\longrightarrow\lambda, \] which proves the two limits. The curvature identifications follow from Theorem~\ref{thm:contact-curvature} and Proposition~\ref{prop:horizontal-differential-curvature}. \end{proof} The synthesis has three layers, not one finite equality: exact target-frame open affine torsion is measured by ACS weighted area; source-normalized closed horizontal holonomy differs by an exact frame-conversion factor; and contact curvature is their common infinitesimal limit. The contact distribution alone supplies neither a preferred complex structure nor a full analytic function theory. The present paper uses only the horizontal connection, its scalar differential, and its curvature. \begin{remark}[What the coefficient $\mu\lambda$ is and is not] \label{rem:p0-curvature-not-contact-invariant} The number $\mu\lambda$ is the bracket coefficient of the \emph{specified} horizontal connection in the \emph{specified} arithmetic frame, and it is not a contact-isomorphism invariant: by the Darboux theorem every contact form on a three-manifold is locally equivalent to $dz-y\,dx$, and Appendix~\ref{app:p0-acs-contact} exhibits that coordinate change explicitly for $\alpha$. The arithmetic content of \eqref{eq:arithmetic-contact-form} therefore lies in the coordinates, the projection, the assignment, and the preferred horizontal frame, not in a new local contact-isomorphism class. In particular $\mu\lambda$ must not be confused with the Gaussian curvature $K=-\lambda^2$ of the basic model (Theorem~\ref{thm:basic-hyperbolic-aes}): the two numbers have different dimensions, different carriers, and different invariance groups. What \eqref{eq:horizontal-curvature-bracket} does assert is intrinsic \emph{relative to the declared structure}: the horizontal distribution is not integrable, so no choice of coordinates can make both arithmetic directions straight while keeping them horizontal. A standard account of the local theory of contact structures, used only for this comparison, is \cite{Etnyre2001IntroductoryLectures}; the elementary Darboux computation needed here is carried out in the appendix and does not depend on it. \end{remark} \begin{proposition}[The curvature form of the horizontal distribution] \label{prop:p0-curvature-form} Let $\mathcal D=\ker\alpha$ be the horizontal distribution of the arithmetic contact model, and let $\pi(u,v,a)=(u,v)$ be the charge projection, whose vertical bundle $\ker d\pi$ is spanned by $\partial_a$. For horizontal vector fields $X,Y$ define $\omega(X,Y)$ by the congruence \begin{equation}\label{eq:p0-curvature-form-congruence} [X,Y]\equiv\omega(X,Y)\,\partial_a \pmod{\mathcal D} . \end{equation} Then: \begin{enumerate} \item $\omega$ is a well-defined smooth $2$-form on $\mathcal D$: it is bilinear, alternating, and $C^\infty(\mathcal C)$-linear in each argument; \item in the arithmetic frame, $\omega(D_u,D_v)=\mu\lambda$; \item $\omega$ is unchanged when $\alpha$ is replaced by $\kappa\alpha$ for $\kappa\ne0$, and it is invariant under every diffeomorphism of $\mathcal C$ that preserves the distribution, the charge projection, and the orientation of the vertical line; \item if a horizontal metric $h$ is declared and $h$ is rescaled to $\rho^2h$, then an $h$-orthonormal positively oriented frame scales by $\rho^{-1}$ and the number $\omega(e_u,e_v)$ scales by $\rho^{-2}$, while the form $\omega$ itself is unchanged. \end{enumerate} \end{proposition} \begin{proof} (1) For $f\in C^\infty(\mathcal C)$ one has $[fX,Y]=f[X,Y]-Y(f)X$, and $Y(f)X$ is horizontal, so the class of $[X,Y]$ modulo $\mathcal D$ is unchanged; alternation is immediate. The vertical bundle is trivialized by $\partial_a$, so the congruence determines a single smooth coefficient. (2) is Theorem~\ref{thm:contact-curvature}: $[D_u,D_v]=\mu\lambda\partial_a$, whose class is already vertical, so $\omega(D_u,D_v)=\mu\lambda$. (3) Replacing $\alpha$ by $\kappa\alpha$ does not change $\ker\alpha$, and no factor of $\alpha$ appears in \eqref{eq:p0-curvature-form-congruence}; a diffeomorphism preserving the distribution maps horizontal fields to horizontal fields and preserves the trivialization of $\ker d\pi$ up to the orientation sign. (4) If $h'=\rho^2h$ then $e'_u=\rho^{-1}e_u$ and $e'_v=\rho^{-1}e_v$ are $h'$-orthonormal and positively oriented, so $\omega(e'_u,e'_v)=\rho^{-2}\omega(e_u,e_v)$. \end{proof} \begin{remark}[What the curvature scalar is invariant under] \label{rem:p0-curvature-invariance} Proposition~\ref{prop:p0-curvature-form} answers precisely the question that Remark~\ref{rem:p0-curvature-not-contact-invariant} raised. The coefficient $\mu\lambda$ is not a contact-isomorphism invariant, but neither is it a gauge artifact of coordinates. Two statements hold, and they concern different invariance groups. \begin{enumerate} \item On the contact side the canonical object is the $2$-form $\omega$ of \eqref{eq:p0-curvature-form-congruence}: it is determined by the distribution and the charge projection, and it is insensitive to the normalization of $\alpha$. The \emph{number} $\mu\lambda$ is the value of $\omega$ on the $h$-orthonormal frame for the declared horizontal metric $h=du^2+dv^2$; it is therefore invariant under every transformation preserving the triple $(\mathcal D,h,\pi)$ together with the orientation and the horizontal area normalization, and its only residual freedom is the rescaling of $h$, with the explicit law $\rho^{-2}$ of (4). A general contact transformation need not preserve $h$, and that is exactly why the number is not a contact invariant. \item On the arithmetic side the same number is a functional of the AES data alone: the canonical frame satisfies \[ da\bigl([X_u,X_v]\bigr)=X_u(\lambda a)-X_v(\mu)=\mu\lambda \] by \eqref{eq:arithmetic-frame-bracket-on-a}, an identity that mentions no chart, no projection, and no normalization of any contact form. Here the freedom is the choice of the pair $(\mu,\lambda)$ itself, which is part of the structure in Definition~\ref{def:regular-aes} rather than a choice of frame. \end{enumerate} So the correct phrase is not ``the curvature of the contact structure'' but ``the curvature scalar of the declared arithmetic frame and horizontal metric''. Rescaling the arithmetic units changes $(\mu,\lambda)$: for instance $(u,v,a)\mapsto(cu,dv,\kappa a)$ replaces $(\mu,\lambda)$ by $(\mu c/\kappa,\lambda d)$ and multiplies $\mu\lambda$ by $cd/\kappa$. Such a rescaling changes the AES data, hence the model, rather than expressing one model in another frame, which is the sense in which the theorem of Section~\ref{sec:p0-contact} concerns a specified connection. \end{remark} \subsection{The contact model as a quotient of the basic model} \label{subsec:p0-contact-quotient} Sections~\ref{sec:p0-aes} and~\ref{sec:p0-contact} describe one object from two sides: intrinsically, by the frame and metric of a regular AES, and locally, by the three coordinates $(u,v,a)$ and the contact form. The following five relations make the correspondence exact for the basic model. (The comparison was proposed in a review of this paper's first version; it is stated here with proofs, and it supersedes the earlier informal remark that the two descriptions ``describe one object''.) Define \begin{equation}\label{eq:p0-contact-quotient-map} Q:\mathcal C=\mathbb R^3_{(u,v,a)}\longrightarrow\mathbb H^2, \qquad Q(u,v,a)=(-ae^{-\lambda v},\,e^{-\lambda v})=(x,y). \end{equation} \begin{proposition}[The basic model is the Reeb quotient of the contact model] \label{prop:p0-contact-quotient} Let $\mu,\lambda>0$, and let $\alpha$, $D_u$, $D_v$, $\mathcal C$ be as in \eqref{eq:arithmetic-contact-form} and \eqref{eq:horizontal-arithmetic-fields}. Then: \begin{enumerate} \item \emph{assignment compatibility:} $a_{\mathfrak E_0}\circ Q=a$, that is, $-x/y=a$ along $Q$; \item \emph{frame descent:} $Q_*D_u=X_u$ and $Q_*D_v=X_v$, where $X_u=-\mu y\partial_x$ and $X_v=-\lambda y\partial_y$ are the canonical frame \eqref{eq:hyperbolic-arithmetic-frame}; \item \emph{horizontal length:} \[ Q^*g_{\mu,\lambda} =\frac{(da-\lambda a\,dv)^2}{\mu^2}+dv^2, \qquad\text{so}\qquad \bigl.Q^*g_{\mu,\lambda}\bigr|_{\ker\alpha}=du^2+dv^2 ; \] \item \emph{the fibers of $Q$ are the Reeb lines:} $\ker dQ=\operatorname{span}(\partial_u) =\operatorname{span}(R_\alpha)$ with $R_\alpha=-\tfrac1\mu\partial_u$, and $Q$ is surjective with global section $(x,y)\mapsto\bigl(0,-\tfrac1\lambda\log y,\,-\tfrac xy\bigr)$; \item \emph{curvature is the hyperbolic area form:} \[ d\alpha=-\mu\lambda\,Q^*\bigl(d\mathrm{vol}_{g_{\mu,\lambda}}\bigr), \qquad d\mathrm{vol}_{g_{\mu,\lambda}}=\frac{dx\wedge dy}{\mu\lambda y^2}. \] \end{enumerate} \end{proposition} \begin{proof} (1) is immediate: $-x/y=ae^{-\lambda v}/e^{-\lambda v}=a$. (2) Since $Q$ is independent of $u$ and $\partial_a Q=(-e^{-\lambda v},0)$, $\partial_v Q=(a\lambda e^{-\lambda v},-\lambda e^{-\lambda v})$, we get \[ Q_*D_u=Q_*(\partial_u+\mu\partial_a)=\mu(-e^{-\lambda v},0) =(-\mu y,0)=X_u , \] and \[ Q_*D_v=Q_*(\partial_v+\lambda a\partial_a) =(a\lambda e^{-\lambda v},-\lambda e^{-\lambda v}) +\lambda a(-e^{-\lambda v},0) =(0,-\lambda e^{-\lambda v})=X_v . \] (3) Differentiating \eqref{eq:p0-contact-quotient-map}, \[ dx=e^{-\lambda v}\bigl(-da+\lambda a\,dv\bigr)=y\bigl(-da+\lambda a\,dv\bigr), \qquad dy=-\lambda e^{-\lambda v}\,dv=-\lambda y\,dv . \] Substituting into $g_{\mu,\lambda}=(dx^2/\mu^2+dy^2/\lambda^2)/y^2$ gives the first identity. On $\ker\alpha$ one has $da=\mu\,du+\lambda a\,dv$, hence $da-\lambda a\,dv=\mu\,du$ and the restriction is $du^2+dv^2$. (4) The map $Q$ does not depend on $u$, and the pair $(\partial_v Q,\partial_a Q)$ is independent because $\det\begin{pmatrix}a\lambda e^{-\lambda v}&-\lambda e^{-\lambda v}\\ -e^{-\lambda v}&0\end{pmatrix} =\lambda e^{-2\lambda v}\ne0$. Hence $\ker dQ=\operatorname{span}(\partial_u)$. Since $\alpha(R_\alpha)=-\mu\cdot(-1/\mu)=1$, $R_\alpha$ spans the same line, and it is the Reeb field of $\alpha$. Surjectivity and the displayed section follow by solving $x=-ae^{-\lambda v}$, $y=e^{-\lambda v}$. (5) By (3), $dx\wedge dy=y(-da+\lambda a\,dv)\wedge(-\lambda y\,dv) =\lambda y^2\,da\wedge dv$. Hence \[ Q^*\bigl(d\mathrm{vol}_{g_{\mu,\lambda}}\bigr) =\frac{dx\wedge dy}{\mu\lambda y^2} =\frac{da\wedge dv}{\mu}, \] and $-\mu\lambda$ times it is $-\lambda\,da\wedge dv=d\alpha$, as computed in the proof of Proposition~\ref{prop:contact-form}. \end{proof} Three points about the proposition deserve emphasis. First, $Q$ is not the charge projection $\pi(u,v,a)=(u,v)$ of the ACS discussion: $Q$ forgets the Reeb direction $u$, whereas the ACS forgets the assignment $a$. The two projections have different fibers and different images, and neither factors through the other. Second, relation (2) says that the horizontal flow of the contact model and the arithmetic motion of the basic model are the \emph{same} curves, which is what makes the two derivations of $\frac{da}{ds}=\mu\cos\theta+\lambda a\sin\theta$ --- the intrinsic one in Theorem~\ref{thm:continuous-affine-flow} and the coordinate one in \eqref{eq:contact-realizes-affine-flow} --- statements about one object rather than two analogies. Relation (3) adds that the horizontal metric pulled back to the base coordinates is Euclidean, so the direction angle $\theta$ has the same meaning in both derivations. Third, the proposition is proved for the basic model; it must not be read as asserting that every regular AES is a Reeb quotient of a contact three-manifold, or that every AES carries such a quotient. % ====================================================================== % Insert into Paper 0, Section 9, immediately after Proposition 9.8 % ====================================================================== \subsection{Legendre duality between the charge and Reeb quotients} \label{subsec:legendre-duality} Proposition~\ref{prop:p0-contact-quotient} places the charge projection $\pi(u,v,a)=(u,v)$ of Section~8 and the Reeb quotient $Q(u,v,a)=(-ae^{-\lambda v}, e^{-\lambda v})=(x,y)$ on a common three-dimensional carrier $\mathcal{C}=\mathbb{R}^3_{(u,v,a)}$. The two maps forget complementary directions: $\pi$ forgets the assignment fiber $\partial_a$, whereas $Q$ forgets the Reeb direction $\partial_u$. We now record the exact sense in which they form a conjugate Legendre pair for horizontal lifts of arithmetic histories. In the natural charge coordinates $A=\mu u$ and $M=\lambda v$, the arithmetic contact form~\eqref{eq:arithmetic-contact-form} and its rescaled companion $e^{-M}\alpha$ satisfy the two exact decompositions \begin{align} \alpha &= da - dA - a\,dM, \label{eq:legendre-hamilton} \\ e^{-M}\alpha &= d\!\left(e^{-M}a\right) - e^{-M}dA. \label{eq:legendre-lagrange} \end{align} Since $e^{-M}>0$ everywhere on $\mathcal{C}$, both one-forms define the same horizontal distribution $\mathcal{D}=\ker\alpha$. Along any piecewise smooth horizontal curve $\tilde{\gamma}$ in $\mathcal{C}$, both~\eqref{eq:legendre-hamilton} and~\eqref{eq:legendre-lagrange} vanish identically. Notice that the quantity $\hat{a}:=e^{-M}a=-x$ in~\eqref{eq:legendre-lagrange} is simultaneously the source-normalized assignment and minus the horizontal coordinate of the basic hyperbolic model $\mathfrak{E}_0$ under $Q$. \begin{theorem}[Dual tearing and Reeb drift] \label{thm:legendre-dual-tearing} Let $\tilde{\gamma}_1$ and $\tilde{\gamma}_2$ be piecewise smooth horizontal lifts in $\mathcal{C}$ starting at a common initial state $(u_0,v_0,a_0)$, with projected charge chains $C_i=\pi\circ\tilde{\gamma}_i$ in $\mathcal{A}$ and projected state paths $\bar{\gamma}_i=Q\circ\tilde{\gamma}_i$ in $\mathbb{H}^2$. \begin{enumerate} \item \emph{(Fixed-charge comparison in $\mathcal{A}$).} Suppose $\tilde{\gamma}_1$ and $\tilde{\gamma}_2$ are charge-compatible, so that $C_1$ and $C_2$ share their terminal point $(A_*,M_*)$. Then their terminal states lie in the same $\pi$-fiber and their vertical assignment defect is \begin{equation}\label{eq:dual-value-tearing} a(\tilde{\gamma}_1(1)) - a(\tilde{\gamma}_2(1)) = e^{M_*}\int_{C_1 - C_2} e^{-M}\,dA = \int_{\Sigma_{12}} e^{M_* - M}\,dA\wedge dM, \end{equation} for any oriented piecewise smooth two-chain $\Sigma_{12}$ in $\mathcal{A}$ with $\partial\Sigma_{12}=C_1-C_2$. \item \emph{(Fixed-state comparison in $\mathbb{H}^2$).} Suppose instead that $\tilde{\gamma}_1$ and $\tilde{\gamma}_2$ are state-compatible under $Q$, so that $\bar{\gamma}_1$ and $\bar{\gamma}_2$ reach the same terminal point $p_*=(x_*,y_*)\in\mathbb{H}^2$, equivalently the same terminal pair $(M_*,a_*)$. Then their terminal states lie in the same Reeb fiber $Q^{-1}(p_*)$, and their additive charge drift $\Delta A := A(\tilde{\gamma}_1(1)) - A(\tilde{\gamma}_2(1)) = \mu\big(u(\tilde{\gamma}_1(1)) - u(\tilde{\gamma}_2(1))\big)$ satisfies \begin{equation}\label{eq:dual-reeb-drift} \Delta A = -\int_{\bar{\gamma}_1 - \bar{\gamma}_2} a\,dM = -\int_{\Omega_{12}} da\wedge dM = -\mu\lambda \int_{\Omega_{12}} d\mathrm{vol}_{g_{\mu,\lambda}}, \end{equation} for any oriented piecewise smooth two-chain $\Omega_{12}$ in $\mathbb{H}^2$ with $\partial\Omega_{12}=\bar{\gamma}_1-\bar{\gamma}_2$. \end{enumerate} \end{theorem} \begin{proof} (1) Integrating $e^{-M}\alpha=0$ in the form~\eqref{eq:legendre-lagrange} along each horizontal lift gives \[ e^{-M_*}a(\tilde{\gamma}_i(1)) - e^{-M_0}a_0 = \int_{C_i} e^{-M}\,dA, \qquad i=1,2. \] Subtracting the two identities cancels the initial term, and multiplying by $e^{M_*}$ gives the first equality in~\eqref{eq:dual-value-tearing}. The second equality follows by Stokes' theorem as in Theorem~\ref{thm:torsion-stokes}. (2) Integrating $\alpha=0$ in the form~\eqref{eq:legendre-hamilton} along each horizontal lift gives \[ A(\tilde{\gamma}_i(1)) - A_0 = \big(a(\tilde{\gamma}_i(1)) - a_0\big) - \int_{\bar{\gamma}_i} a\,dM, \qquad i=1,2. \] Under state compatibility in $\mathbb{H}^2$, both lifts terminate at the same $y_*=e^{-M_*}$ and the same $a_*=-x_*/y_*$, so the boundary term $a_*-a_0$ cancels upon subtraction. This proves the line-integral formula in~\eqref{eq:dual-reeb-drift}. Applying Stokes' theorem on $\mathbb{H}^2$ gives the integral of $-da\wedge dM = -\lambda\,da\wedge dv$. By Proposition~\ref{prop:p0-contact-quotient}(5), $-\lambda\,da\wedge dv = d\alpha = -\mu\lambda\,Q^*(d\mathrm{vol}_{g_{\mu,\lambda}})$, which yields the hyperbolic area identity. \end{proof} \begin{remark}[Unification of Theorem~8.6 and Theorem~8.9] Theorem~\ref{thm:legendre-dual-tearing} shows that the two basic non-commutativity phenomena of Section~8 are Legendre-conjugate readings of the same contact curvature: \begin{itemize} \item Holding the commutative charge endpoint $(A_*,M_*)$ fixed projects the order defect onto the assignment fiber $\ker d\pi = \mathrm{span}(\partial_a)$, recovering the weighted ACS area of Theorem~\ref{thm:torsion-stokes}. \item Holding the evaluated hyperbolic state $p_*\in\mathbb{H}^2$ fixed projects the order defect onto the Reeb fiber $\ker dQ = \mathrm{span}(\partial_u)$, measuring the failure of the additive charge to descend to the motion group. For the elementary kernel word $h(p,q)=M_q A_p M_{-q} A_{-e^{-q}p}\in\ker\nu$ of Theorem~\ref{thm:p0-descent}, the two prefixes $\gamma_1=(M_q,A_p)$ and $\gamma_2=(A_{e^{-q}p},M_q)$ connect $(0,a_0)$ to the same state $(q, e^q a_0 + p)$ in $\mathbb{H}^2$, and~\eqref{eq:dual-reeb-drift} evaluates their Reeb drift $-\int_{\bar{\gamma}_1-\bar{\gamma}_2} a\,dM = p(1-e^{-q})$ as the hyperbolic area of the enclosed strip. \end{itemize} Notice also the structural complement to Proposition~\ref{prop:p0-surface-does-not-close}: whereas a charge-compatible rectangle remains open on $\mathbb{H}^2$, a state-compatible pair closes on $\mathbb{H}^2$ and opens instead along the Reeb fiber in $\mathcal{C}$. \end{remark}