% General calculations moved from Paper I appendix B (governance/00b).
% The elementary alternating derivation and its historical motivation were
% restored on 2026-10-20. Closing conventions are imported from Paper I appendix A.
This appendix supplements Proposition~\ref{prop:affine-cocycle-formulas} with
expanded calculations.  All discrete formulas hold over an arbitrary field
$K$, and every multiplier is assumed nonzero.

\subsection{Alternating expressions: the elementary arithmetic origin}
\label{subsec:appendix-alternating-origin}

Early drafts called these objects \emph{alternating threadlike expressions}
\cite{Yuan2023AEGAlternatingExpressions}.  The elementary observation behind
them is that an addition changes a value, whereas each subsequent
multiplication transports that change to a new scale.  We recover that
calculation before using affine coordinates or a continuous model.

\begin{definition}[Alternating arithmetic expression]
\label{def:appendix-alternating-expression}
Fix $q_i\in K^\times$ and $\mu_i\in K$ for $i=1,\ldots,n$.  An
\emph{alternating expression} $\alpha$ is the chronological slot-$(1)$ word that,
in each pair, first multiplies by $q_i$ and then adds $\mu_i$.  Applied to
the initial value $\mu_0$, its successive values are
\begin{equation}
  w_0=\mu_0,\qquad w_i=q_iw_{i-1}+\mu_i.
  \label{eq:appendix-alternating-arithmetic-recursion}
\end{equation}
The two operations in a pair remain distinct marked events, even when the
pair is evaluated as one affine map.
\end{definition}

A finite word consisting of additions and nonzero scalings can be padded to
this alternating pattern by inserting $+0$ or $\times1$.  This preserves its
operator, not its literal tree or full marked history, and supplies no unique
normal form.  Slot-$(1)$ subtraction and division by a nonzero constant can
first be interpreted as addition of a negative constant and multiplication by
an inverse.  A slot-$(2)$ reciprocal $c/z$ is not covered by this argument.

\begin{proposition}[Arithmetic expansion and exact perturbation]
\label{prop:appendix-alternating-arithmetic-perturbation}
With the empty product equal to $1$, the recurrence
\eqref{eq:appendix-alternating-arithmetic-recursion} has value
\begin{equation}
  w_n=\mu_0\prod_{j=1}^n q_j
      +\sum_{i=1}^n\mu_i\prod_{j=i+1}^n q_j.
  \label{eq:appendix-alternating-arithmetic-expansion}
\end{equation}
Keep the same operations and constants, but replace the initial value by
$\widetilde\mu_0$, obtaining values $\widetilde w_i$.  Then, at every prefix,
\begin{equation}
  \widetilde w_i-w_i
    =\left(\prod_{j=1}^i q_j\right)(\widetilde\mu_0-\mu_0).
  \label{eq:appendix-alternating-arithmetic-perturbation}
\end{equation}
If $\widetilde\mu_0\ne\mu_0$, the exact difference quotient is
\begin{equation}
  \frac{\widetilde w_i-w_i}{\widetilde\mu_0-\mu_0}
    =\prod_{j=1}^i q_j.
  \label{eq:appendix-alternating-arithmetic-ratio}
\end{equation}
\end{proposition}

\begin{proof}
The first two steps, using only distributivity, give
\[
  w_1=q_1\mu_0+\mu_1,\qquad
  w_2=q_2q_1\mu_0+q_2\mu_1+\mu_2.
\]
At each next step multiplication by $q_{n+1}$ extends every displayed product,
and the final addition supplies the term $\mu_{n+1}$ with empty suffix
product.  Induction, starting at $n=0$, proves
\eqref{eq:appendix-alternating-arithmetic-expansion}.  For the perturbation,
subtract the two recurrences directly:
\[
  \widetilde w_i-w_i
   =(q_i\widetilde w_{i-1}+\mu_i)-(q_iw_{i-1}+\mu_i)
   =q_i(\widetilde w_{i-1}-w_{i-1}).
\]
Iterating from $\widetilde w_0-w_0=\widetilde\mu_0-\mu_0$ proves
\eqref{eq:appendix-alternating-arithmetic-perturbation}; division by the
nonzero initial difference proves
\eqref{eq:appendix-alternating-arithmetic-ratio}.
\end{proof}

No smallness hypothesis, derivative or limit is used.  For example, the rational
word ``multiply by $2$, add $1$, multiply by $3$, add $4$'' sends $5$ to $37$
and $6$ to $43$.  The difference $6$ is exactly $2\cdot3$ times the initial
difference $1$.  It is an exact gain formula, not a truncation of a Taylor
expansion.  With $q_i=s_i$ and $\mu_i=t_i$, the same calculation is the
elementary source of the general affine formulas below.

\subsection{Prefix recursion and closed form}

For the prefix history $(f_1,\ldots,f_k)$, write
\[
  F_k=f_k\circ\cdots\circ f_1,
  \qquad
  F_k(x)=\Phi_kx+\xi_k,
\]
with $F_0(x)=x$.  If $f_{k+1}(x)=s_{k+1}x+t_{k+1}$, then
\begin{equation}
  \Phi_{k+1}=s_{k+1}\Phi_k,
  \qquad
  \xi_{k+1}=s_{k+1}\xi_k+t_{k+1}.
  \label{eq:appendix-prefix-affine-recursion}
\end{equation}
Iterating the first recursion gives
\[
  \Phi_k=s_k\cdots s_1.
\]
Iterating the second one gives
\begin{align*}
  \xi_1&=t_1,\\
  \xi_2&=s_2t_1+t_2,\\
  \xi_3&=s_3s_2t_1+s_3t_2+t_3,\\
  \xi_4&=s_4s_3s_2t_1+s_4s_3t_2+s_4t_3+t_4.
\end{align*}
The pattern is
\[
  \xi_k=\sum_{i=1}^{k}t_i\prod_{j=i+1}^{k}s_j.
\]
Substitution in \eqref{eq:appendix-prefix-affine-recursion} proves this formula
inductively, including the final term through the empty-product convention.

The source-normalized prefix coordinate
$\widehat\xi_k=\Phi_k^{-1}\xi_k$ obeys a different recursion.  From
\eqref{eq:appendix-prefix-affine-recursion},
\begin{align}
  \widehat\xi_{k+1}
  &=\frac{s_{k+1}\xi_k+t_{k+1}}
          {s_{k+1}\Phi_k}
   =\widehat\xi_k+\frac{t_{k+1}}{\Phi_{k+1}}.
  \label{eq:appendix-normalized-prefix-recursion}
\end{align}
Starting from $\widehat\xi_0=0$ therefore gives
\[
  \widehat\xi_k
  =\sum_{i=1}^{k}\frac{t_i}{\Phi_i}
  =\sum_{i=1}^{k}\frac{t_i}{\prod_{j=1}^{i}s_j}.
\]
This second derivation makes the past normalization explicit at every prefix.

\subsection{Suffix transport and the future-weight kernel}

For $0\le i\le n$, define the scale of the strict suffix after step $i$ by
\[
  S_{i\to n}=\prod_{j=i+1}^{n}s_j,
  \qquad S_{n\to n}=1.
\]
Then the target-frame translation is simply
\begin{equation}
  \xi_n=\sum_{i=1}^{n}S_{i\to n}t_i.
  \label{eq:appendix-suffix-transport}
\end{equation}
To verify the interpretation, replace only $t_i$ by $t_i+\epsilon$.  All
states before step $i$ are unchanged.  At step $i$ the output gains
$\epsilon$, and every later affine map multiplies this change by its scale.
Thus the final endpoint changes by
\[
  \epsilon s_{i+1}\cdots s_n
  =\epsilon S_{i\to n}.
\]
The coefficient in \eqref{eq:appendix-suffix-transport} is therefore exactly
the transport of an additive event to the target frame.

\subsection{Alternating addition--scale histories}
\label{subsec:appendix-alternating-logarithmic}

Now specialize the arithmetic multipliers to $q_i=e^{m_i}>0$ over
$\mathbb R$.  This positivity concerns the multipliers; the initial value
and additive increments may have either sign.  The pairs become
\[
  x\xmapsto{\,e^{m_i}\,}e^{m_i}x
  \xmapsto{\,+\mu_i\,}e^{m_i}x+\mu_i,
  \qquad i=1,\ldots,n,
\]
applied from $i=1$ to $i=n$.  At the level of affine steps this has
$s_i=e^{m_i}$ and $t_i=\mu_i$.  Define prefix and suffix logarithmic
scales by
\[
  M_i=\sum_{j=1}^{i}m_j,
  \qquad
  M_{i+1:n}=\sum_{j=i+1}^{n}m_j.
\]
Proposition~\ref{prop:affine-cocycle-formulas} becomes
\begin{align}
  \Phi_n&=e^{M_n},
  \label{eq:appendix-alternating-scale}\\
  \xi_n&=\sum_{i=1}^{n}\mu_i e^{M_{i+1:n}},
  \label{eq:appendix-alternating-target}\\
  \widehat\xi_n&=\sum_{i=1}^{n}\mu_i e^{-M_i}.
  \label{eq:appendix-alternating-source}
\end{align}
For $n=3$, direct expansion checks the target formula:
\begin{align*}
  x
  &\longmapsto e^{m_1}x+\mu_1\\
  &\longmapsto e^{m_2+m_1}x
      +e^{m_2}\mu_1+\mu_2\\
  &\longmapsto e^{m_3+m_2+m_1}x
      +e^{m_3+m_2}\mu_1
      +e^{m_3}\mu_2+\mu_3.
\end{align*}
Dividing the translation line by
$e^{m_1+m_2+m_3}$ gives
\[
  \widehat\xi_3
  =\mu_1e^{-m_1}
   +\mu_2e^{-(m_1+m_2)}
   +\mu_3e^{-(m_1+m_2+m_3)},
\]
in agreement with \eqref{eq:appendix-alternating-source}.

The convention in this example is ``scale, then add'' within each pair.  If a
source instead groups the elementary operations as ``add, then scale'', the
translation attached to that pair is $e^{m_i}\mu_i$, not $\mu_i$.
Converting the pair to the common affine form $s_ix+t_i$ before using the
cocycle formulas prevents this convention change from being hidden.

For the original initial perturbation
$\widetilde\mu_0=e^{\eta_0}\mu_0+\epsilon_0$, the arithmetic identity gives
\[
  \widetilde w_n-w_n
   =e^{M_n}\bigl((e^{\eta_0}-1)\mu_0+\epsilon_0\bigr).
\]
The first recurrence starts at $w_0=\mu_0$, and its companion starts at
$\widetilde w_0=\widetilde\mu_0$.  Any ratio by their initial difference
requires that difference to be nonzero.

\subsection{From arithmetic weights to ACS and scale normalization}
\label{subsec:appendix-alternating-acs-scale}

For this same word, draw the ACS path of Definition~\ref{def:acs}.  The $i$th
vertical edge adds $m_i$ to $M$, and the following horizontal edge adds
$\mu_i$ to $A$ at height $M_i$.  Thus direct integration along the horizontal
edges gives
\begin{equation}
  \int_{C_\alpha}e^{-M}\,dA
    =\sum_{i=1}^n\mu_i e^{-M_i}=\widehat\xi_n,
  \qquad
  w_n=e^{M_n}(\mu_0+\widehat\xi_n).
  \label{eq:appendix-alternating-acs-bridge}
\end{equation}
This recovers Proposition~\ref{prop:acs-evaluation} for alternating words.
The initial perturbation is multiplied by the total scale $e^{M_n}$; an
additive increment $\mu_i$ is multiplied only by its remaining suffix scale
$e^{M_n-M_i}$.  These are distinct quantities forced by the same expansion.
The charge endpoint $(\sum_i\mu_i,M_n)$ forgets the heights at which the
additions occurred, whereas the weighted integral retains their effect.

Grouping consecutive pairs into blocks uses the chronological cocycle law
below: each block has an exact effective scale and translation.  Retaining
the original events distinguishes this operator summary from the full
history.  Accumulated scale and the normalization $e^{-M_n}$ supplied an
arithmetic starting point for the later renormalization discussion
\cite{Yuan2025AEGScaleRenormalization}.  The finite identities proved here
support that motivation; identifying a particular blocking and rescaling
procedure with a renormalization transformation requires further definitions.
The historical interpretation is recorded separately from those identities.

\subsection{Concatenation in target and normalized coordinates}

Let
\[
  \rho(\gamma)=g(\Phi_\gamma,\xi_\gamma),
  \qquad
  \rho(\delta)=g(\Phi_\delta,\xi_\delta),
\]
where $g(\Phi,\xi)(x)=\Phi x+\xi$.  Since $\gamma\delta$ performs $\gamma$
first,
\begin{align*}
  \rho(\gamma\delta)(x)
  &=\rho(\delta)(\rho(\gamma)(x))\\
  &=\Phi_\delta(\Phi_\gamma x+\xi_\gamma)+\xi_\delta\\
  &=\Phi_\delta\Phi_\gamma x
    +\Phi_\delta\xi_\gamma+\xi_\delta.
\end{align*}
This yields the target-coordinate pair
\[
  (\Phi_{\gamma\delta},\xi_{\gamma\delta})
  =(\Phi_\delta\Phi_\gamma,
    \Phi_\delta\xi_\gamma+\xi_\delta).
\]
For normalized coordinates,
\begin{align*}
  \widehat\xi_{\gamma\delta}
  &=\frac{\Phi_\delta\xi_\gamma+\xi_\delta}
          {\Phi_\delta\Phi_\gamma}\\
  &=\widehat\xi_\gamma
    +\Phi_\gamma^{-1}\widehat\xi_\delta.
\end{align*}
The first formula transports the earlier translation forward by the later
scale.  The second transports the later normalized translation back through
the earlier scale.

For three histories, repeated use gives
\begin{align*}
  \xi_{\gamma\delta\epsilon}
  &=\Phi_\epsilon\Phi_\delta\xi_\gamma
    +\Phi_\epsilon\xi_\delta+\xi_\epsilon,\\
  \widehat\xi_{\gamma\delta\epsilon}
  &=\widehat\xi_\gamma
    +\Phi_\gamma^{-1}\widehat\xi_\delta
    +(\Phi_\delta\Phi_\gamma)^{-1}
      \widehat\xi_\epsilon.
\end{align*}
These are the block-history versions of future weighting and past
normalization.

\subsection{Inverse histories and relative comparison}

The affine inverse is
\begin{equation}
  g(\Phi,\xi)^{-1}
  =g(\Phi^{-1},-\Phi^{-1}\xi).
  \label{eq:appendix-affine-inverse}
\end{equation}
Indeed,
\begin{align*}
  g(\Phi^{-1},-\Phi^{-1}\xi)
  \bigl(\Phi x+\xi\bigr)
  &=x+\Phi^{-1}\xi-\Phi^{-1}\xi=x.
\end{align*}
Using \eqref{eq:appendix-affine-inverse}, the relative affine defect is
\begin{align}
  \rho(\delta)^{-1}\rho(\gamma)
  &=g\left(
      \frac{\Phi_\gamma}{\Phi_\delta},
      \frac{\xi_\gamma-\xi_\delta}{\Phi_\delta}
    \right).
  \label{eq:appendix-relative-affine-matrix}
\end{align}
In matrices, the same computation is
\[
  \begin{pmatrix}
    \Phi_\delta^{-1}&-\Phi_\delta^{-1}\xi_\delta\\0&1
  \end{pmatrix}
  \begin{pmatrix}
    \Phi_\gamma&\xi_\gamma\\0&1
  \end{pmatrix}
  =
  \begin{pmatrix}
    \Phi_\delta^{-1}\Phi_\gamma&
    \Phi_\delta^{-1}(\xi_\gamma-\xi_\delta)\\0&1
  \end{pmatrix}.
\]
If $\Phi_\gamma=\Phi_\delta=\Phi$, the relative element is a pure translation
by $(\xi_\gamma-\xi_\delta)/\Phi$.  The unnormalized endpoint difference is
$\xi_\gamma-\xi_\delta$.  These two scalar quantities differ by the common
scale and should not be identified.

\subsection{The elementary order square}

Let
\[
  A_\mu=
  \begin{pmatrix}1&\mu\\0&1\end{pmatrix},
  \qquad
  S_q=
  \begin{pmatrix}q&0\\0&1\end{pmatrix}.
\]
Under the chronological convention, first adding and then scaling has matrix
\[
  S_qA_\mu
  =\begin{pmatrix}q&q\mu\\0&1\end{pmatrix},
\]
whereas first scaling and then adding has matrix
\[
  A_\mu S_q
  =\begin{pmatrix}q&\mu\\0&1\end{pmatrix}.
\]
Their endpoint difference is
\[
  (qx+q\mu)-(qx+\mu)=\mu(q-1).
\]
Taking the second history as reference gives
\begin{align*}
  (A_\mu S_q)^{-1}(S_qA_\mu)
  &=\begin{pmatrix}1&\mu(1-q^{-1})\\0&1\end{pmatrix}.
\end{align*}
Thus the relative group translation is $\mu(1-q^{-1})$, while the open
endpoint defect is $\mu(q-1)$.  The latter is the former multiplied by the
common target scale $q$.

Closing the comparison into a group commutator produces yet another ordered
quantity.  For example,
\begin{align*}
  S_qA_\mu S_q^{-1}A_\mu^{-1}
  &=\begin{pmatrix}1&(q-1)\mu\\0&1\end{pmatrix},\\
  A_\mu S_qA_\mu^{-1}S_q^{-1}
  &=\begin{pmatrix}1&(1-q)\mu\\0&1\end{pmatrix}.
\end{align*}
The signs differ because the loop orientations differ.  These closed-loop
calculations are recorded only to audit conventions; the torsion theory and
its orientation choices are introduced in Section~\ref{sec:p0-acs-tearing}.

\subsection{Exact perturbation propagation at every prefix}

Let $x$ and $\widetilde x$ be two initial values, and propagate both through
the same history.  Set
\[
  x_k=F_k(x),
  \qquad
  \widetilde x_k=F_k(\widetilde x).
\]
Taking $q_i=s_i$, $\mu_i=t_i$, $\mu_0=x$ and
$\widetilde\mu_0=\widetilde x$ in the elementary identity
\eqref{eq:appendix-alternating-arithmetic-perturbation} gives
\begin{equation}
  \widetilde x_k-x_k
  =\left(\prod_{i=1}^{k}s_i\right)(\widetilde x-x)
  =\Phi_k(\widetilde x-x).
  \label{eq:appendix-prefix-perturbation}
\end{equation}
The translations cancel at each step; only the scale cocycle propagates the
initial difference.  Over $\mathbb R$, if $s_i=e^{\lambda_i}$, then
\[
  \frac{\widetilde x_k-x_k}{\widetilde x-x}
  =e^{\lambda_1+\cdots+\lambda_k}.
\]
No limiting process is involved in this identity.
The displayed ratio requires $\widetilde x\ne x$; the difference identity
\eqref{eq:appendix-prefix-perturbation} also includes equal initial values.

\subsection{Differential conversion check}

Finally, on the positive real affine group let $\Phi=e^M$ and
$\widehat\xi=e^{-M}\xi$.  The product rule gives
\begin{align*}
  d\widehat\xi
  &=d(e^{-M}\xi)\\
  &=-e^{-M}\xi\,dM+e^{-M}d\xi\\
  &=e^{-M}(d\xi-\xi dM).
\end{align*}
Therefore
\[
  d\xi-\xi dM=e^M d\widehat\xi.
\]
This verifies that the translation component of $dg\,g^{-1}$ is the target
scale times the differential of the source-normalized coordinate.  It does
not identify $dg\,g^{-1}$ with $g^{-1}dg$, whose translation component is
$e^{-M}d\xi$.

% Imported from Paper I appendix A on 2026-09-22 (governance/00b migration map):
% affine-group conventions now used by the cocycle sections above.
\subsection{Left and right group data}

The affine group is represented by
\[
  g(\Phi,\xi)=
  \begin{pmatrix}\Phi&\xi\\0&1\end{pmatrix},
  \qquad \Phi\in K^\times.
\]
With the column action, matrix multiplication gives
\begin{equation}
  g(\Phi_2,\xi_2)g(\Phi_1,\xi_1)
  =g(\Phi_2\Phi_1,\Phi_2\xi_1+\xi_2).
  \label{eq:appendix-affine-group-law}
\end{equation}
Thus the earlier translation is rescaled by the later affine map.  Equation
\eqref{eq:appendix-affine-group-law} is the two-factor origin of the
target-frame cocycle.

Over the positive real component, write $\Phi=e^M$.  The two
Maurer--Cartan forms used in Section~\ref{sec:p0-affine-cocycles} have the
following readings.

\begin{center}
\begin{tabular}{lll}
\toprule
Form & Translation component & Frame \\
\midrule
$g^{-1}dg$ & $e^{-M}d\xi$ & body/source \\
$dg\,g^{-1}$ & $d\xi-\xi dM$ & spatial/target \\
\bottomrule
\end{tabular}
\end{center}

The words ``left'' and ``right'' here refer to invariant differential forms on
the affine group.  They do not refer to first and second operand slots.  The
operand-slot mirror is syntactic; the Maurer--Cartan distinction is
group-theoretic.

\subsection{Target and source-normalized translation coordinates}

For $x\mapsto\Phi x+\xi$, the symbol $\xi$ always denotes target-frame
translation.  The source-normalized coordinate is
\[
  \widehat\xi=\Phi^{-1}\xi.
\]
Their discrete and differential conversions are
\[
  \xi=\Phi\widehat\xi,
  \qquad
  \widehat\xi=\Phi^{-1}\xi,
\]
and, when $\Phi=e^M$,
\[
  d\xi=\xi\,dM+e^M d\widehat\xi.
\]
No formula in the main text changes from $\xi$ to $\widehat\xi$ without this
normalization being displayed.