% Moved from Paper I Section 8 (governance/00b, migration map); mathematical content % unchanged apart from cross-references, framing sentences, and the closing % subsection on tearing. The arithmetic motions of Section~\ref{sec:p0-aes} do not commute: addition and multiplication transport each other's rulers by the Baumslag--Solitar-type relation \eqref{eq:p0-bs-relation}, and the affine cocycle of Section~\ref{sec:p0-affine-cocycles} records the resulting order dependence in two translation coordinates. This section constructs a commutativization of that non-commutative motion, and the level at which commutation is imposed must be stated exactly. Take the history language itself: the free monoid generated by the two families of one-step histories, additive steps and multiplicative steps. Impose that generators of the two families commute. Equivalently --- and this is the form used below --- keep the total additive charge and the total logarithmic multiplicative charge of a history and forget the order in which they were accumulated; the charges satisfy the imposed commutation by construction. The resulting object is a plane with two coordinates, and we call it the \emph{accumulative commutative space}, ACS. The construction has two distinct uses, and they must not be conflated. For a positive add--scale history it records the two total charges; and it supplies a weighted integral along the charge path whose value restores the translation part that the charge endpoint alone forgets. The ACS is therefore a \emph{shadow} of the presented history: it is a commutative quotient object attached to a non-commutative one, not another presentation of the full arithmetic syntax. This is \emph{not} the abelianization of the group of affine motions, and the difference is worth recording once. In $\operatorname{Aff}^{+}(1,\mathbb R)$ one has $D_kT_pD_k^{-1}T_p^{-1}=T_{(k-1)p}$, so the commutator subgroup is the translation subgroup and the abelianization keeps the scale alone: \[ \operatorname{Aff}^{+}(1,\mathbb R)_{\mathrm{ab}} \;\cong\;\mathbb R_{>0}\;\cong\;(\mathbb R,+). \] The ACS keeps the additive charge as well, because it is attached to the presented histories \emph{before} evaluation rather than to their images in the affine group. The hierarchy --- literal histories, then charges, then evaluated motions --- is accordingly not a convenience of exposition; collapsing it changes the object. Subsection~\ref{subsec:p0-acs-forgets} states what the ACS forgets, and the distributivity example of Subsection~\ref{subsec:p0-distributivity} already shows why the charge map cannot be pushed through the operator quotient. The geometric use is the reason for the name of this chapter. Two histories with the same charge endpoint have charge paths that together bound a region of the ACS, and the exact value of their order defect is a weighted area of that region (Theorem~\ref{thm:torsion-stokes}). In Section~\ref{subsec:p0-tearing} we read that identity as a statement about the original AES: the arithmetic space carries a defect that cannot be seen in any single motion, and the geometry in which it becomes an area is the commutativized charge geometry of the presented histories. Throughout this section an elementary additive step and an elementary scaling step are denoted by \[ \mathsf A_p(x)=x+p, \qquad \mathsf M_q(x)=e^q x, \qquad p,q\in\mathbb R. \] For a history \(\gamma=(g_1,\ldots,g_n)\), the entries are performed from left to right, so that \[ \nu_x(\gamma)=g_n\circ\cdots\circ g_1(x). \] Negative values of \(p\) and \(q\) are allowed. Thus the same notation covers inverse translations and inverse positive scalings. In this section an \emph{add--scale history} means a finite word in these two families of generators. Such a word evaluates in \(\operatorname{Aff}^+(1,\mathbb R)\). The ACS construction is attached to this presented word; it is not asserted for negative multipliers or for an arbitrary affine syntax before that syntax has been written in add--scale form. \subsection{The charge path} \begin{definition}[Accumulative commutative space and charge path] \label{def:acs} The \emph{accumulative commutative space}, abbreviated ACS, is the oriented plane \[ \mathcal A=\mathbb R^2_{(A,M)}, \qquad \text{with orientation } dA\wedge dM. \] An additive step \(\mathsf A_p\) has charge increment \((p,0)\), and a scaling step \(\mathsf M_q\) has charge increment \((0,q)\). Starting at \((A_0,M_0)=(0,0)\), accumulate these increments in chronological order. The resulting oriented broken path is denoted by \(C_\gamma\), and its terminal point by \[ (A_\gamma,M_\gamma) =\left(\sum_{g_i=\mathsf A_{p_i}}p_i, \sum_{g_i=\mathsf M_{q_i}}q_i\right). \] \end{definition} The terminal charge is insensitive to order, whereas the broken path generally is not. In particular, two histories may have the same ACS endpoint while inducing different affine maps. The missing information is recovered by the following integral formula. \begin{proposition}[Direct ACS evaluation] \label{prop:acs-evaluation} For every add--scale history \(\gamma\) and every initial value \(x\), \begin{equation} \nu_x(\gamma) =e^{M_\gamma} \left(x+\int_{C_\gamma}e^{-M}\,dA\right). \label{eq:direct-acs-evaluation} \end{equation} \end{proposition} \begin{proof} Let an additive step \(\mathsf A_{p_i}\) occur when the cumulative multiplicative charge is \(M_{i-1}\). Along the corresponding horizontal edge of \(C_\gamma\), \[ \int e^{-M}\,dA=p_i e^{-M_{i-1}}. \] After that addition, the remaining multiplicative steps have total charge \(M_\gamma-M_{i-1}\). Consequently the contribution of \(p_i\) to the final value is \[ p_i e^{M_\gamma-M_{i-1}} =e^{M_\gamma}p_i e^{-M_{i-1}}. \] The initial value contributes \(e^{M_\gamma}x\), and vertical ACS edges contribute nothing to the \(dA\)-integral. Summing over the additive steps gives \eqref{eq:direct-acs-evaluation}. The argument uses oriented edge integrals, so it also covers negative increments. \end{proof} The kernel \(e^{-M}\) is therefore forced by the affine cocycle: it normalizes an additive contribution to the source frame, before the common terminal scale \(e^{M_\gamma}\) returns it to the target frame. \subsection{The history group and the descent of the charges} \label{subsec:p0-history-group} The preceding subsections attached charges to words. To say precisely what the ACS is, and in particular what it is \emph{not}, it is useful to say which words are being identified. The identifications used below are only the ones the paper has already made implicitly. \begin{convention}[The history group] \label{conv:p0-history-group} Let \[ G=(\mathbb R,+)*(\mathbb R,+) =\bigl\langle \mathsf A_p,\ \mathsf M_q\ \bigm|\ \mathsf A_p\mathsf A_{p'}=\mathsf A_{p+p'},\ \mathsf M_q\mathsf M_{q'}=\mathsf M_{q+q'},\ \mathsf A_0=\mathsf M_0=1 \bigr\rangle \] be the free product of two copies of the additive group of $\mathbb R$, written multiplicatively, with generators $\mathsf A_p$ (additive steps) and $\mathsf M_q$ (multiplicative steps). A \emph{history} is an element of $G$ together with a chosen word representing it. Two words differing only by regrouping consecutive steps of one family represent the same element of $G$ and have the same charge chain and the same evaluation, so nothing is lost by this convention; words which interleave the two families are \emph{not} identified. \end{convention} \begin{proposition}[Charge endpoint and evaluation are homomorphisms] \label{prop:p0-history-homomorphisms} The charge endpoint and the evaluation extend uniquely to homomorphisms \[ c:G\longrightarrow\mathbb R^2, \qquad \mathsf A_p\longmapsto(p,0),\quad \mathsf M_q\longmapsto(0,q), \] \[ \nu:G\longrightarrow\operatorname{Aff}^{+}(1,\mathbb R), \qquad \mathsf A_p\longmapsto T_p,\quad \mathsf M_q\longmapsto D_{e^{q}}, \] where $T_p(z)=z+p$ and $D_k(z)=kz$. Both are surjective. Moreover the charge chain $C_\gamma$ is a function of the element of $G$ and not only of the word: regrouping consecutive steps of one family replaces consecutive edges of one charge path by the single edge with the same endpoints. \end{proposition} \begin{proof} Each family is generated by elements satisfying the only relations imposed within it, so a homomorphism is determined by its values on the generators and is well defined; the images stated are themselves homomorphisms of $(\mathbb R,+)$. Surjectivity of $c$ is clear, and surjectivity of $\nu$ holds because translations and positive dilations generate $\operatorname{Aff}^{+}(1,\mathbb R)$. For the last claim, a regrouping $\mathsf A_p\mathsf A_{p'}\mapsto\mathsf A_{p+p'}$ joins two consecutive horizontal edges of $C_\gamma$ into the horizontal edge with the same endpoints and the same orientation, and the vertical case is identical. \end{proof} \begin{proposition}[The ACS endpoint is the abelianization of the history group] \label{prop:p0-acs-is-abelianization} The charge homomorphism $c$ is the abelianization of $G$: its kernel is the commutator subgroup, \[ \ker c=[G,G], \qquad\text{so}\qquad G_{\mathrm{ab}}\;\cong\;\mathbb R^2 \] and the charge plane of Definition~\ref{def:acs} is the abelianization of the history group. \end{proposition} \begin{proof} For any groups $F,H$ one has $(F*H)_{\mathrm{ab}}\cong F_{\mathrm{ab}}\oplus H_{\mathrm{ab}}$, because abelianization is left adjoint to the inclusion of abelian groups and therefore preserves coproducts. With $F=H=(\mathbb R,+)$, which is abelian, this gives $G_{\mathrm{ab}}\cong\mathbb R\oplus\mathbb R$. Since $c$ is a surjection onto the abelian group $\mathbb R^2$, it factors through $G_{\mathrm{ab}}$ as an isomorphism, so $\ker c=[G,G]$. \end{proof} Now the question the review of this paper raised: does the charge pass to the \emph{evaluated} motion? The next theorem answers it, and the answer is asymmetric. \begin{theorem}[Only the multiplicative charge descends] \label{thm:p0-descent} Fix $q\ne0$ and $p\ne0$ and put \begin{equation}\label{eq:p0-transport-element} h(p,q)=\mathsf M_q\,\mathsf A_p\,\mathsf M_{-q}\,\mathsf A_{-e^{-q}p} \;\in\; G . \end{equation} Then: \begin{enumerate} \item $h(p,q)$ is a nontrivial element of $\ker\nu$; that is, $h(p,q)$ evaluates to the identity motion while being a nonempty history; \item its charges are \[ c\bigl(h(p,q)\bigr)=\bigl(p(1-e^{-q}),\,0\bigr); \] \item the multiplicative charge $\mathsf M:G\to\mathbb R$, $\mathsf M=\mathrm{pr}_2\circ c$, factors through $\nu$: it equals $\log\circ L\circ\nu$, where $L:\operatorname{Aff}^{+}(1,\mathbb R)\to \mathbb R_{>0}$ is the linear part $(z\mapsto \Phi z+\xi)\mapsto\Phi$; \item the additive charge $\mathsf A:G\to\mathbb R$, $\mathsf A=\mathrm{pr}_1\circ c$, does \emph{not} factor through $\nu$: there is no homomorphism $\varphi:\operatorname{Aff}^{+}(1,\mathbb R)\to\mathbb R^2$ with $c=\varphi\circ\nu$. Equivalently, the additive charge is a property of the history and not of the motion it induces. \end{enumerate} \end{theorem} \begin{proof} (1) Apply the motions of $h(p,q)$ in chronological order to an arbitrary $z$: \[ z\ \xmapsto{\ \mathsf M_q\ }\ e^{q}z \ \xmapsto{\ \mathsf A_p\ }\ e^{q}z+p \ \xmapsto{\ \mathsf M_{-q}\ }\ z+e^{-q}p \ \xmapsto{\ \mathsf A_{-e^{-q}p}\ }\ z . \] Hence $\nu(h(p,q))=\mathrm{id}$. It is nontrivial in $G$ because a reduced word of a free product is trivial only if it is empty, and $q\ne0$, $p\ne0$ make \eqref{eq:p0-transport-element} reduced. (2) Add the charge increments in the same chronological order: \[ (0,q)+(p,0)+(0,-q)+(-e^{-q}p,0)=\bigl(p(1-e^{-q}),\,0\bigr). \] (3) $L$ is a homomorphism, and on generators $\log L(T_p)=0=\mathsf M(\mathsf A_p)$ and $\log L(D_{e^{q}})=q=\mathsf M(\mathsf M_q)$; since both sides are homomorphisms on $G$ agreeing on generators, they agree on $G$. (4) If $c=\varphi\circ\nu$ for some homomorphism $\varphi$ into the abelian group $\mathbb R^2$, then $c$ would vanish on $\ker\nu$. But by (1) and (2), $h(p,q)\in\ker\nu$ while $c(h(p,q))=(p(1-e^{-q}),0)\ne(0,0)$ for $p\ne0$, $q\ne0$. Contradiction. (The structural reason is Proposition~\ref{prop:p0-acs-is-abelianization} together with the computation $\operatorname{Aff}^{+}(1,\mathbb R)_{\mathrm{ab}}\cong\mathbb R_{>0}$ recorded in the opening of this section: every homomorphism from the motion group to an abelian group is one-dimensional, whereas the charges are two-dimensional. The explicit element $h(p,q)$ is what makes the failure a proof rather than a dimension count.) \end{proof} Three consequences deserve to be stated together, because they are the precise version of statements that earlier drafts left informal. \begin{enumerate} \item \emph{The ACS is the commutativization of the history language, and the sense of that phrase is now fixed.} It is the abelianization of $G$ (Proposition~\ref{prop:p0-acs-is-abelianization}), not of the motion group. The motion group's abelianization is one-dimensional and keeps the scale alone; the additive charge does not survive evaluation at all (Theorem~\ref{thm:p0-descent}). \item \emph{What the ACS retains beyond its abelianized endpoint is the path.} Two histories with the same charge endpoint may have different charge chains; the elementary rectangle $\gamma=(\mathsf A_p,\mathsf M_q)$ and $\delta=(\mathsf M_q,\mathsf A_p)$ is the smallest example, with equal endpoints $(p,q)$ and different chains. Proposition~\ref{prop:acs-evaluation} uses the chain, not the endpoint, and that is why it recovers the evaluated motion from commutative data. A charge \emph{endpoint} observable therefore cannot see the difference, exactly as Proposition~\ref{prop:p0-characters-cannot-see-tearing} states, while the endpoint together with the path can. \item \emph{The two failure directions are dual.} Histories with equal motion and different charges are the elements of $\ker\nu$ of Theorem~\ref{thm:p0-descent}; histories with equal charges and different motions are the pairs measured by the relative torsion of Definition~\ref{def:relative-torsion}. The first failure is why the charge map cannot be pushed through evaluation; the second is why the evaluation cannot be recovered from the charge endpoint. The ACS is the object in which both become computable, because it carries the endpoint (first failure visible) and the path (second failure measured by the weighted area of Theorem~\ref{thm:torsion-stokes}). \end{enumerate} \subsection{Compatibility and relative torsion} There are two useful compatibility conditions, and they should not be conflated. Histories \(\gamma\) and \(\delta\) are \emph{scale-compatible} if \[ M_\gamma=M_\delta. \] They are \emph{charge-compatible} if, in addition, \[ A_\gamma=A_\delta. \] Scale compatibility makes their affine maps have the same linear part. Charge compatibility makes their ACS paths have both endpoints in common and hence makes \(C_\gamma-C_\delta\) a closed oriented one-chain. \begin{definition}[Relative arithmetic torsion] \label{def:relative-torsion} For scale-compatible add--scale histories \(\gamma\) and \(\delta\), their \emph{relative arithmetic torsion} is \begin{equation} \tau(\gamma,\delta) :=\xi_\gamma-\xi_\delta, \label{eq:relative-torsion} \end{equation} where \(\xi_\gamma\) and \(\xi_\delta\) are the target-frame translation terms of the two affine operators. This relative translation defect is the primary torsion invariant used below. \end{definition} \begin{proposition} \label{prop:relative-torsion-independent} For every initial value \(x\), relative arithmetic torsion equals the endpoint difference \[ \tau(\gamma,\delta) =\nu_x(\gamma)-\nu_x(\delta), \] and is therefore independent of \(x\). If \(M_\gamma=M_\delta=M_*\), then \begin{equation} \tau(\gamma,\delta) =e^{M_*} \left( \int_{C_\gamma}e^{-M}\,dA -\int_{C_\delta}e^{-M}\,dA \right). \label{eq:relative-torsion-boundary-preparation} \end{equation} \end{proposition} \begin{proof} Apply Proposition~\ref{prop:acs-evaluation} to both histories. Their initial terms are respectively \(e^{M_\gamma}x\) and \(e^{M_\delta}x\), which cancel under scale compatibility. The remaining terms give \eqref{eq:relative-torsion-boundary-preparation}. \end{proof} Temporal reversal gives one possible comparison, but it is not built into the definition. The pair \((\gamma,\delta)\) may consist of any two scale-compatible add--scale histories. This extension is important because order defects occur between histories that are neither reversals nor planar mirrors of one another. \subsection{The torsion--Stokes formula} Suppose now that \(\gamma\) and \(\delta\) are charge-compatible, with common terminal charge \((A_*,M_*)\). Define the target-normalized one-form \begin{equation} \eta_*:=e^{M_*-M}\,dA \label{eq:target-normalized-acs-form} \end{equation} on the ACS. Relative torsion is then the integral of \(\eta_*\) over the closed one-chain \(C_\gamma-C_\delta\). The chosen order of the two paths fixes the sign. \begin{theorem}[Weighted torsion--Stokes theorem] \label{thm:torsion-stokes} Let \(\gamma\) and \(\delta\) be charge-compatible add--scale histories with common terminal multiplicative charge \(M_*\). Let \(\Sigma_{\gamma,\delta}\) be any oriented piecewise smooth singular two-chain in the ACS satisfying \[ \partial\Sigma_{\gamma,\delta}=C_\gamma-C_\delta. \] Then \begin{align} \tau(\gamma,\delta) &=\int_{C_\gamma-C_\delta}\eta_* \label{eq:torsion-boundary-integral}\\ &=\int_{\Sigma_{\gamma,\delta}} e^{M_*-M}\,dA\wedge dM. \label{eq:torsion-area-integral} \end{align} The value of the area integral is independent of the chosen filling chain. \end{theorem} \begin{proof} Equation \eqref{eq:relative-torsion-boundary-preparation} and charge compatibility give \[ \tau(\gamma,\delta) =\int_{C_\gamma-C_\delta}e^{M_*-M}\,dA =\int_{C_\gamma-C_\delta}\eta_*. \] With the orientation \(dA\wedge dM\), direct differentiation yields \[ d\eta_* =d(e^{M_*-M})\wedge dA =-e^{M_*-M}dM\wedge dA =e^{M_*-M}dA\wedge dM. \] Stokes' theorem for piecewise smooth chains therefore gives \[ \int_{C_\gamma-C_\delta}\eta_* =\int_{\partial\Sigma_{\gamma,\delta}}\eta_* =\int_{\Sigma_{\gamma,\delta}}d\eta_*, \] which is \eqref{eq:torsion-area-integral}. Such a filling exists because the ACS plane is contractible. Finally, if \(\Sigma\) and \(\Sigma'\) have the same boundary, applying Stokes' theorem to each shows \[ \int_\Sigma d\eta_* =\int_{\partial\Sigma}\eta_* =\int_{\partial\Sigma'}\eta_* =\int_{\Sigma'}d\eta_*. \] This also proves filling independence. The chain formulation remains valid when the broken paths self-intersect or carry signed increments; no informal notion of an enclosed region is required. \end{proof} Charge compatibility is needed for the closed-chain and area formulas, but not for Definition~\ref{def:relative-torsion}. A merely scale-compatible pair still has a well-defined translation defect, even when the endpoints of its ACS paths have different \(A\)-coordinates. Figure~\ref{fig:acs-compatible-paths-weighted-filling} depicts the chain orientation used in Theorem~\ref{thm:torsion-stokes}. It is a schematic of the theorem's data rather than an argument for filling independence. \begin{figure}[t] \centering \begin{tikzpicture}[ x=1.15cm,y=1.05cm, axis/.style={-{Latex[length=2mm]},thick,black!70}, gamma/.style={-{Latex[length=2mm]},very thick,red!65!black}, delta/.style={-{Latex[length=2mm]},very thick,blue!65!black}, every node/.style={font=\small} ] \coordinate (O) at (0,0); \coordinate (g1) at (2.2,0); \coordinate (g2) at (2.2,0.9); \coordinate (g3) at (4,0.9); \coordinate (P) at (4,2.7); \coordinate (d1) at (0,1.5); \coordinate (d2) at (1.3,1.5); \coordinate (d3) at (1.3,2.7); \path[fill=violet!11,draw=none] (O)--(g1)--(g2)--(g3)--(P)--(d3)--(d2)--(d1)--cycle; \draw[axis] (-0.25,0) -- (4.75,0) node[right] {$A$}; \draw[axis] (0,-0.25) -- (0,3.25) node[above] {$M$}; \draw[gamma] (O)--(g1); \draw[gamma] (g1)--(g2); \draw[gamma] (g2)--(g3); \draw[gamma] (g3)--(P); \draw[delta] (O)--(d1); \draw[delta] (d1)--(d2); \draw[delta] (d2)--(d3); \draw[delta] (d3)--(P); \fill (O) circle (1.7pt) node[below left] {$(0,0)$}; \fill (P) circle (1.7pt) node[above right] {$(A_*,M_*)$}; \node[red!65!black,fill=white,inner sep=1pt] at (3.3,0.72) {$C_\gamma$}; \node[blue!65!black,fill=white,inner sep=1pt] at (0.72,1.75) {$C_\delta$}; \node[align=center,violet!55!black] at (2.55,1.8) {$\Sigma_{\gamma,\delta}$\\ $e^{M_*-M}\,dA\wedge dM$}; \node[font=\scriptsize,anchor=west] at (4.35,1.72) {$\partial\Sigma=C_\gamma-C_\delta$}; \draw[-{Latex[length=1.7mm]},black!55] (4.55,0.35)--(4.95,0.35) node[right,font=\scriptsize] {$dA$}; \draw[-{Latex[length=1.7mm]},black!55] (4.55,0.35)--(4.55,0.8) node[above,font=\scriptsize] {$dM$}; \end{tikzpicture} \caption{Two charge-compatible ACS histories. Both paths run from $(0,0)$ to the common charge endpoint $(A_*,M_*)$. Traversing $C_\gamma$ forward and $C_\delta$ backward gives the positive boundary of the displayed filling for the orientation $dA\wedge dM$; the density is weighted by $e^{M_*-M}$.} \label{fig:acs-compatible-paths-weighted-filling} \end{figure} \subsection{Explicit order comparisons} \paragraph{The elementary rectangle.} Let \[ \gamma=(\mathsf A_p,\mathsf M_q), \qquad \delta=(\mathsf M_q,\mathsf A_p). \] The path \(C_\gamma-C_\delta\) is the positively oriented boundary of the rectangle \([0,p]\times[0,q]\) when \(p,q>0\). Direct evaluation and the weighted-area formula agree: \begin{align*} \tau(\gamma,\delta) &=e^q(x+p)-(e^q x+p)\\ &=p(e^q-1)\\ &=\int_0^p\int_0^q e^{q-M}\,dM\,dA. \end{align*} Thus ``add, then multiply'' minus ``multiply, then add'' is positive under the orientation fixed in Definition~\ref{def:acs}. In the flow notation used in the next chapter, \(p=\mu h\) and \(q=\lambda k\), so the defect is \[ \mu h(e^{\lambda k}-1). \] \paragraph{A four-step staircase.} For \[ \gamma=(\mathsf A_p,\mathsf M_r,\mathsf A_q,\mathsf M_s), \qquad \delta=(\mathsf M_r,\mathsf A_p,\mathsf M_s,\mathsf A_q), \] the histories have the same total charge \((p+q,r+s)\), but \(\delta\) is not the temporal reversal of \(\gamma\). Their evaluations are \[ \nu_x(\gamma)=e^{r+s}x+p e^{r+s}+q e^s, \] \[ \nu_x(\delta)=e^{r+s}x+p e^s+q. \] Consequently \begin{equation} \tau(\gamma,\delta) =p e^s(e^r-1)+q(e^s-1). \label{eq:four-step-staircase-torsion} \end{equation} For positive \(p,q,r,s\), both terms are positive weighted strips in an ACS filling. Formula \eqref{eq:four-step-staircase-torsion} also shows how a later scale changes the weight of an earlier commutation defect. \paragraph{Equal charges without reversal.} The pair \[ (\mathsf A_p,\mathsf M_r,\mathsf A_q), \qquad (\mathsf A_q,\mathsf M_r,\mathsf A_p) \] has common charge \((p+q,r)\), and its relative torsion is \[ (p-q)(e^r-1). \] It may be positive, negative, or zero. In particular, equal charge does not determine torsion, while zero torsion does not imply equality of marked histories. \subsection{What the ACS forgets} \label{subsec:p0-acs-forgets} At the formal level, the map \[ \gamma\longmapsto(A_\gamma,M_\gamma) \] is an order-forgetting charge homomorphism from concatenated marked histories to the additive group \(\mathbb R^2\). It is therefore analogous to an abelianization map. This statement concerns the formal charge language only. It does not identify the symbolic history space with the evaluated affine group, and it does not assert that either object is a free group without an additional choice of formal generators and relations. The content of Theorem~\ref{thm:torsion-stokes} is precisely what survives this loss of order: the endpoint charges specify a common frame, while the weighted filling measures the translation defect between two histories. Section~\ref{sec:p0-contact} retains the current value as a third coordinate and identifies the infinitesimal form of the same defect with contact curvature. \subsection{Tearing} \label{subsec:p0-tearing} We can now name the phenomenon that the three preceding chapters have been describing from three sides. The name is new here, the mathematics is not: it is the content of Definition~\ref{def:relative-torsion} and Theorem~\ref{thm:torsion-stokes} read geometrically. \begin{definition}[Tearing of arithmetic motion] \label{def:p0-tearing} Let $\gamma$ and $\delta$ be two add--scale histories of a regular AES that are scale-compatible in the sense of Section~\ref{sec:p0-acs-tearing}, and let $\tau(\gamma,\delta)$ be their relative torsion. The pair is said to \emph{tear} the AES by $\tau(\gamma,\delta)$: the two histories induce the same linear part, hence agree on the multiplicative content of the motion, while their additive content differs by exactly $\tau(\gamma,\delta)$. \end{definition} The identification of that scalar defect with an area requires a second hypothesis, and the two must not be merged. By Theorem~\ref{thm:torsion-stokes}, if $\gamma$ and $\delta$ are in addition \emph{charge-compatible}, then $\tau(\gamma,\delta)$ equals the integral of the weighted area form over any oriented filling of the closed chain $C_\gamma-C_\delta$. For a merely scale-compatible pair the chain $C_\gamma-C_\delta$ is not closed, because the two paths have different additive endpoints, so there is no such filling; the scalar defect \eqref{eq:relative-torsion} remains defined, but it is not yet an area. Three properties justify treating tearing as a geometric notion rather than a bookkeeping defect. \begin{enumerate} \item \emph{It exists only relative to the commutativization.} In the AES itself, $\gamma$ and $\delta$ are simply two motions; there is no region bounded by them, because the AES is where the two rulers fail to commute. The region appears only after passing to the ACS, and the weight of the filling is forced by the evaluation formula \eqref{eq:direct-acs-evaluation}. \item \emph{For charge-compatible pairs it is measured, not merely detected.} There $\tau(\gamma,\delta)$ is an exact weighted area \eqref{eq:torsion-area-integral}; for the elementary rectangle it is $\mu h(e^{\lambda k}-1)$, which is the discrete Baumslag--Solitar transport of \eqref{eq:p0-bs-relation} read as a defect. \item \emph{It is infinitesimally rigid.} Its density is the constant $\mu\lambda$ of \eqref{eq:infinitesimal-order-defect}, which is also the bracket coefficient of the arithmetic frame \eqref{eq:arithmetic-frame-bracket-on-a}. Section~\ref{sec:p0-contact} shows that this coefficient is the curvature of the specified horizontal connection, hence that tearing cannot be removed by any local recalibration of the two arithmetic directions in which both remain horizontal. \end{enumerate} The word \emph{tearing} is chosen because the two-dimensional picture is that of a surface cut along the two motions: an additive edge and a multiplicative edge that close up on the level of total charge but not on the level of motion. The standard description of a rip in a material is also a pair of paths that agree far away and separate inside; the arithmetic version of that picture is Definition~\ref{def:p0-tearing}. As with ripple geometry, the terminology is descriptive and asserts nothing beyond the displayed identities; what it adds is a single name for the phenomenon whose exact measure is the ACS area, whose infinitesimal density is $\mu\lambda$, and whose carrier in a punctured model is the subject of Section~\ref{sec:p0-holed}.