The picture developed so far is deliberately written in the language of
surfaces, metrics, vector fields, and contact forms.  That language is
convenient, and it is also, in a precise sense, too small.  This section records
two elementary obstructions that explain why we do not take linear language
(manifolds, tangent spaces, Lie groups, and character theory built on modules)
as the foundation of AEG, and it records the alternative that a companion
programme, process geometry, has begun to develop.  The obstructions are
theorems; the alternative is a structural proposal, and the two are kept apart
throughout.

\subsection{Two obstructions to linear language}

The first obstruction concerns the Lie-algebraic reading of the arithmetic
tower.  Addition and multiplication are the first two ranks of that tower: an
additive step moves the assignment by a constant, a multiplicative step scales
it.  On the positive assignment line $a>0$ these two kinds of motion are
generated by
\[
  A=\partial_a,
  \qquad
  M=a\partial_a .
\]
Together with the third rank, multiplication viewed as an endomorphism of
addition, they generate more than a three-dimensional Lie algebra.

\begin{proposition}[The arithmetic generators do not close]
\label{prop:p0-arithmetic-generators}
Let
\[
  A=\partial_a,
  \qquad
  M=a\partial_a,
  \qquad
  P=a\log a\,\partial_a
\]
be defined for $a>0$.  Then
\[
  [A,M]=A,
  \qquad
  [M,P]=M,
  \qquad
  [A,P]=(1+\log a)\partial_a .
\]
The last field does not lie in the real span of $A,M,P$, and the Lie algebra
generated by $A,M,P$ is infinite-dimensional.
\end{proposition}

\begin{proof}
The three brackets are direct: for any smooth $f$,
\[
  [A,M]f=\partial_a\!\left(a f'\right)-a\partial_a f'=f' ,
\]
\[
  [M,P]f=a\partial_a\!\left(a\log a\,f'\right)
        -a\log a\,\partial_a\!\left(a f'\right)
        =a f' ,
\]
\[
  [A,P]f=\partial_a\!\left(a\log a\,f'\right)
        -a\log a\,\partial_a f'
        =\left(1+\log a\right)f' .
\]

For the last claim, suppose first that $(1+\log a)\partial_a=\alpha A+\beta M+\gamma P$
for constants $\alpha,\beta,\gamma$.  Equivalently
$1+\log a=\alpha+\beta a+\gamma a\log a$ for all $a>0$.  Letting
$a\to0^{+}$ makes the left-hand side tend to $-\infty$ while the right-hand
side tends to $\alpha$, a contradiction.  Hence the generated Lie algebra is
strictly larger than $\operatorname{span}\{A,M,P\}$.

More is true, and the argument is shorter than the one just given: it suffices
to iterate $\operatorname{ad}_A$ on $P$.  The derived algebra is closed under
$\operatorname{ad}_A$, and $P=a\log a\,\partial_a$ belongs to it, so it contains
\[
  \operatorname{ad}_A^{\,n}(P)
  =\frac{d^n}{da^n}\bigl(a\log a\bigr)\,\partial_a
  =(-1)^n(n-2)!\,a^{1-n}\,\partial_a,
  \qquad n\ge2 ,
\]
where the derivative is computed for $a>0$.  Hence the derived algebra contains
$a^{-1}\partial_a,\ a^{-2}\partial_a,\ a^{-3}\partial_a,\ldots$, and these
fields are real-linearly independent on every nonempty open subinterval of
$a>0$: multiplying a finite linear relation by a sufficiently high power of $a$
turns it into the vanishing of a polynomial.  Therefore the Lie algebra
generated by $A$ and $P$ alone is infinite-dimensional, and so is the algebra
generated by $A,M,P$; the dilation $M$ is not needed for this conclusion.
\end{proof}

The three generators have the following arithmetic readings, which fix the
calibration of this section.  $A$ generates translation of the assignment
($a\mapsto a+s$).  $M$ generates multiplication by a positive constant
($a\mapsto e^{s}a$), so its parameter is the logarithm of the multiplier, which
is why the multiplicative data enter the ACS as a logarithmic charge.  $P$
generates the one-parameter group of power maps: its flow solves
$\dot a=a\log a$, hence
\[
  a(t)=a(0)^{\,e^{t}} .
\]
Thus $P$ moves the \emph{exponent} of the assignment, and it is the infinitesimal
form of the third rank of the arithmetic tower, in which multiplication is
iterated.  We use ``third rank'' here as a name for this calibration only; no
classification of arithmetic ranks is attempted in this paper.

The consequence for this paper is narrow but decisive.  A finite-dimensional Lie
group acting locally near the assignment line has a finite-dimensional Lie
algebra of fundamental vector fields.  If the rank-one, rank-two, and rank-three
motions above were all fundamental vector fields of one such action, then their
Lie closure — infinite-dimensional by Proposition~\ref{prop:p0-arithmetic-generators}
— would have to be finite-dimensional.  Hence no finite-dimensional Lie group
action realizes all three motions simultaneously.

Two weaker statements must not be confused with that one, and the distinction
matters for everything that follows.  First, a finite-dimensional \emph{manifold} can
carry infinitely many vector fields: the three fields above are defined on the
one-dimensional half-line $a>0$, and every smooth manifold has an
infinite-dimensional space of vector fields.  What is excluded is a
finite-dimensional Lie \emph{algebra of fundamental fields}, not a finite-dimensional
carrier.  Second, linear language is not thereby refuted.  The affine sector has
the faithful two-dimensional matrix representation
$\bigl(\begin{smallmatrix}\Phi&\xi\\0&1\end{smallmatrix}\bigr)$ used throughout
Section~\ref{sec:p0-affine-cocycles}, and it does detect order defects; the
non-abelian representation theory of that group is not touched by
Proposition~\ref{prop:p0-arithmetic-generators}.  The boundary established here
is the boundary of finite-dimensional Lie-group closure, and it says only that
the arithmetic tower cannot be \emph{linearized into a finite-dimensional Lie
algebra without loss}.

The second obstruction concerns what an observable of the \emph{commutativized} data
can see.  It is the obstruction that makes the tearing of
Section~\ref{sec:p0-acs-tearing} irreducible in that class of descriptions.

\begin{proposition}[Observables of the charge endpoint cannot see tearing]
\label{prop:p0-characters-cannot-see-tearing}
Let $f:\mathbb R^2\to X$ be any function whatsoever of the charge endpoint, and
let $\gamma,\delta$ be charge-compatible add--scale histories with common charge
$(A_*,M_*)$.  Then
\[
  f(A_\gamma,M_\gamma)=f(A_\delta,M_\delta),
\]
while their evaluation defect is the relative torsion
\[
  \nu_x(\gamma)-\nu_x(\delta)=\tau(\gamma,\delta),
\]
which is independent of $x$ and need not vanish.  Consequently no observable
that factors through the charge endpoint can detect tearing.
\end{proposition}

\begin{proof}
The first display is immediate from charge compatibility.  The second is
Proposition~\ref{prop:relative-torsion-independent}.  Nonvanishing is witnessed
by the elementary rectangle: for $\gamma=(\mathsf A_p,\mathsf M_q)$ and
$\delta=(\mathsf M_q,\mathsf A_p)$ one has $\tau(\gamma,\delta)=p(e^{q}-1)$,
which is nonzero whenever $p\ne0$ and $q\ne0$.
\end{proof}

Characters of the charge group are the special case $X=S^1$ with $f$
multiplicative, so Proposition~\ref{prop:p0-characters-cannot-see-tearing}
subsumes them; but the statement is deliberately stated for arbitrary functions,
because its content is not about representation theory.  It is about a
forgetting map: the charge endpoint is the image of the history under the
map \eqref{eq:direct-acs-evaluation}'s charges, and any observable of the image
loses whatever the map forgot.  Two consequences should be stated together to
avoid overreading the proposition.

First, the proposition does \emph{not} say that no linear description can see
tearing.  On the contrary, the affine matrix
$\bigl(\begin{smallmatrix}\Phi&\xi\\0&1\end{smallmatrix}\bigr)$ of
Section~\ref{sec:p0-affine-cocycles} is faithful on the affine sector, the
target-frame translation $\xi$ is precisely the coordinate in which the
two-history defect is measured, and $\xi$ is \emph{not} a function of the charge
endpoint: the charge endpoint gives the scale coordinate alone.  What is
excluded is a description that first passes to the commutative charges and then
reads a (scalar or vectorial) observable of them.

Second, the measure of what such a description loses is exactly the ACS area of
Theorem~\ref{thm:torsion-stokes}: for charge-compatible pairs the defect equals
the weighted area between the two charge paths.  So the correct summary is not
that characters fail, but that the commutativization has a price, and the price
is computable.

\subsection{What process geometry proposes instead}
\label{subsec:p0-beyond-linear-proposals}

The companion programme process geometry develops a vocabulary for exactly this
situation, and the vocabulary is quotable \cite{Yuan2026ProcessGeometry}.  Its
central methodological statement is that geometry is not supplied in advance but
generated by the process and by what its observers can distinguish; and that no
familiar structure may be imported into a \emph{native} description merely because it
is familiar:

\begin{itemize}
  \item geometry enters "because process generation and observer-relative
  distinguishability acquire stable relational, local, and compositional
  structure---not because a manifold, metric, coordinate system, or arithmetic
  model was supplied in advance";
  \item "no level should be imported merely because the next level is
  mathematically familiar";
  \item the public vocabulary deliberately contains \emph{no} group or Lie-group
  hierarchy, \emph{no} general representation hierarchy, and \emph{no} spectrum or
  eigenmode ontology; those "may become justified later, but only after an
  executable vignette requires it";
  \item where a classical mechanism (power series, ordinary polynomial algebra,
  matrix linearization, Fourier or spectral methods, generic computer algebra)
  is used, it must be declared as such and accompanied by a \emph{lowering witness}
  scoped to the exact task and lane, rather than silently acquiring the status
  of native structure.
\end{itemize}

Two features of that programme are relevant to AEG, and only two are claimed
here.

\emph{First, it introduces structure without presupposing a linear carrier.}  A
process is a generation rule; a history is what that rule produces; a task
selects which distinctions matter; and the resulting quotient, when it is
stable, may be \emph{objectified} into a new primitive of the next rank.  The
vocabulary is deliberately non-linear: it speaks of presentations, semantic
compression, rank raising, and rank lowering, and its stated obligations are
compositional (a higher-rank composition must remain interpretable at the lower
rank) rather than algebraic in the module-theoretic sense.

\emph{Second, it records an obstruction of the same shape as
Proposition~\ref{prop:p0-characters-cannot-see-tearing}.}  Its finite-family
layer introduces families with a parameter composition law, together with
one-dimensional response objects, and then states plainly that such
one-dimensional scalar response is \emph{not} sufficient to encode the additive
--multiplicative interaction while retaining a nontrivial translation response;
the layer stops at that obstruction instead of enlarging the ambient linear
structure.  The shape is the same as ours: a \emph{scalar} observable of a
commutative label fails, while a non-scalar description succeeds.  In our
setting the non-scalar description is the affine matrix of
Section~\ref{sec:p0-affine-cocycles}; the finite-family layer's declaration
that its object "does not require inverses, topology, Haar measure, Lie algebra,
commutativity, or even a group structure" is a statement about what that layer
deliberately does \emph{not} assume, not evidence that such structures are
unavailable.

\subsection{Exact calibrations that meet this paper's laws}

The proposals above would be of little use if they did not meet the identities
proved here.  They do, and in one case the match is literal.

\begin{enumerate}
  \item \emph{The discrete transport law.}  In the calibration of the companion
  programme, the objectified multiplication acts on translation by
  \[
    D_kT_a=T_{ka}D_k ,
  \]
  and the mixed language generated by both is the positive affine monoid
  $\mathbb Z\rtimes\mathbb N_{>0}$.  This is exactly the relation
  \eqref{eq:p0-bs-relation} of Section~\ref{sec:p0-aes}, read in the equivalent
  direction convention:
  \[
    \mathsf Y_k\,\mathsf X_s\,\mathsf Y_k^{-1}=\mathsf X_{ks},
  \]
  with $D_k=\mathsf Y_k$ and $T_a=\mathsf X_a$.  The two programmes therefore
  meet at the same discrete law, one derived from the arithmetic grid and the
  other from a rank-lowering obligation.
  \item \emph{The infinitesimal transport law.}  In the continuous form the same
  structure appears as $S_sT_t=T_{e^st}S_s$ together with the algebra relation
  $[A,M]=A$ for $A=\partial_a$ and $M=\partial_v+a\partial_a$.  This is the
  affine structure equation already used in Section~\ref{sec:p0-aes}.  It must
  not be confused with the horizontal bracket
  \eqref{eq:horizontal-noncommutation}: the latter is the curvature of the
  horizontal distribution in the three-dimensional contact model of
  Section~\ref{sec:p0-contact}, whereas $[A,M]=A$ is a bracket on the base
  algebra of the two arithmetic motions.  The two are related --- both have the
  same coefficient $\mu\lambda$ in the arithmetic frame --- but they are
  different objects, and the identification is not made here.
  \item \emph{Non-integrability in a second calibration.}  The companion
  programme also records the exact local obstruction
  \[
    \vartheta=dz-x\,dy,
    \qquad
    X=\partial_x,
    \qquad
    Y=\partial_y+x\partial_z,
    \qquad
    [X,Y]=\partial_z,
    \qquad
    \vartheta([X,Y])=1 .
  \]
  Since $d\vartheta=-dx\wedge dy$, one has
  $\vartheta\wedge d\vartheta=\pm\,dx\wedge dy\wedge dz\ne0$: $\vartheta$ is a
  contact form whose kernel does not close, exactly as for the arithmetic
  contact form \eqref{eq:arithmetic-contact-form}.  The two forms are not the
  same object --- in $\alpha$ the multiplier of the multiplicative direction is
  the assignment $a$ itself, whereas in $\vartheta$ the coefficient $x$ is an
  independent coordinate --- but the non-integrability pattern is identical:
  a Pfaffian form on a three-dimensional space whose kernel is spanned by two
  fields that fail to close, with a nonzero constant bracket component.  The
  same pattern therefore appears in a calibration where no arithmetic assignment
  is present at all, which is evidence that the phenomenon belongs to the two
  directions rather than to the assignment.
\end{enumerate}

These three items are exact computations, not proposals.  Their status is
nevertheless local to their own calibration: they are recorded in a research
repository whose own maturity labels are, for the items cited, of the kind
"precise conjecture" and "research calibration", and this paper does not promote
them.

\subsection{Status and non-claims}

For the avoidance of doubt, this section claims the following and nothing more.

\begin{itemize}
  \item Proposition~\ref{prop:p0-arithmetic-generators} and
  Proposition~\ref{prop:p0-characters-cannot-see-tearing} are proved above, by
  elementary computation, in this paper.
  \item The vocabulary of Subsection~\ref{subsec:p0-beyond-linear-proposals} is a
  \emph{structural proposal}: it is a way of organizing the construction of
  non-linear descriptions, imported from a companion programme and used here as
  a language, not as a theorem about AEG.
  \item This section does not claim that manifolds, metrics, Lie groups,
  modules, or characters are illegitimate in AEG.  Sections~\ref{sec:p0-aes},
  \ref{sec:p0-acs-tearing}, and~\ref{sec:p0-contact} use all of them, and use
  them with proofs.  What is refused is their status as ontology: no linear
  structure is imported into a foundational claim without a derivation of its
  transport at the arithmetic rank in question.
  \item This section does not claim any universality of arithmetic-generated
  geometry, and it does not import the companion programme's own open
  conjectures.  Where that programme states that a claim is not established, the
  same claim is not established here.
  \item No efficiency, complexity, or approximation statement is made.  The two
  propositions above are structural obstructions, and an obstruction is not a
  lower bound: no conclusion about computational cost follows from them, in
  either direction.
\end{itemize}