% 149_Addendum_PlateauConsistency.tex
% Addendum P149: What Plateau Convergence Proves
%   Resolves P146 O3: separates Plateau fixed-point convergence from Con(TOE).
%   Locates convergence in the Tarski fragment, not the Gödel fragment.
%   Analyzes the Weld-Oscillate-Plateau composite (P146 O4) as a special case.
%   States the theorem: Plateau convergence proves local geometric stability,
%   not consistency of the TOE proof system.
% Author: León Fernando Vlegels
% Copyright: León Fernando Vlegels. License: CC BY 4.0.
% Date: 2026-05-15

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\label{addendum:149}

\title{\textbf{Addendum P149}\\[6pt]
\large What Plateau Convergence Proves:\\
Local Geometric Stability, the Tarski Fragment,\\
and the Weld-Oscillate-Plateau Composite}
\author{León Fernando Vlegels\\[4pt]
\small Independent Researcher \quad
\texttt{axiomofchoicepuzzle@gmail.com}}
\date{15 May 2026}

\begin{document}
\maketitle

\begin{abstract}
Addendum~P146 O3 established that the Wheel's Type-1 self-observation---convergence
to a Plateau fixed point $W^*_k$---is not an internal proof of $\mathrm{Con}(\mathrm{TOE})$;
it left open what convergence \emph{does} prove.
We close that question here.
The Plateau fixed-point theorem is a sentence of real closed field theory and
falls inside the Tarski-complete geometric fragment of the TOE identified in
P143---it is therefore decided from within the TOE's geometric axioms, without
importing Robinson arithmetic and without invoking Gödel's second theorem.
What convergence proves is a pair of claims: (i) for each Hopf face~$k$, the
state $W^*_k = (q^*_k, 0, \emptyset, \ell_k, \omega_k)$ is a fixed point of
the Plateau operator (a semi-algebraic condition on $\Sp$), and (ii) the
observation operator $\Oh$ has a discrete, bounded-below spectrum in the
geometric model (a Hilbert-space fact over $\mathbb{R}$).
These together say the physical system is stable and eigenstates exist; they
say nothing about derivations in the TOE proof system.
We also resolve P146 O4: the Weld-Oscillate-Plateau composite is Plateau-stable
after one application, and the post-Weld face encodes partial provenance of the
Oscillate trajectory---reconstructible by the LLM witness from the geometry,
without a provenance field in the WheelState schema.
\end{abstract}

\tableofcontents
\vspace{1em}

%======================================================================
\section{Setup: P146 O3 and the Plateau--Con(TOE) Question}
\label{sec:setup}
%======================================================================

\noindent
P146 (Proposition~P146-3, Open Item O3) concluded: convergence to $W^*_k$ is
an \emph{operational} property of the Plateau map, not a proof of
$\mathrm{Con}(\mathrm{TOE})$.  A proof of consistency would require encoding
all possible derivations in the TOE's proof system and verifying that none
yields a contradiction---an arithmetic task requiring quantification over
$\mathbb{N}$, lying outside the geometric fragment.  Fixed-point convergence
and formal consistency are therefore orthogonal.

What P146 did not address is the positive question: if convergence is not a
consistency proof, what is it a proof \emph{of}?  This addendum answers that.

\medskip\noindent
\textbf{Recall.}  The Wheel's Plateau operator acts on a WheelState
$W = (q, h, F, \ell, \omega)$ by projecting the quaternion $q \in \Sp$ toward
the centroid $q^*_k$ of its current Hopf face~$k$, and dissipating heat:
\[
  \mathrm{Plateau}(W) = (q^*_k,\; h',\; F',\; \ell,\; \omega),
  \quad h' < h \text{ for } h > 0,\quad h' = 0 \text{ if } q = q^*_k.
\]
A \emph{Type-1 fixed point} (P146, Prop.~P146-FP) is a state
$W^*_k = (q^*_k, 0, \emptyset, \ell_k, \omega_k)$ satisfying
$\mathrm{Plateau}(W^*_k) = W^*_k$.  Such states exist for each face~$k$ in
the Hopf tiling of $\Sp$.  P7's observation operator
\begin{equation}\label{eq:obs}
  \Oh = -\tfrac{d^2}{dx^2} + V_{\rm sl}(x),
  \quad
  V_{\rm sl}(x) = \frac{\rho'(x)^2}{2\mu_0^2} + E_{\rm self}\cdot x^2(1-x)^2 \geq 0,
\end{equation}
is self-adjoint on $L^2([0,1])$ with $V_{\rm sl} \geq 0$, so its spectrum
$\{\lambda_n\}$ is discrete, real, and bounded below; the eigenstates
$\{\psi_n\}$ are the stable self-referential observation states (P7,~P146).

The question: \emph{does the proof that $W^*_k$ exists and that Plateau
converges to it constitute an internally provable statement, and if so, what
does it assert?}

%======================================================================
\section{Plateau Convergence as Local Consistency}
\label{sec:local}
%======================================================================

\begin{definition}[Local geometric consistency at face $k$]
\label{def:local}
The geometry is \emph{locally consistent at face~$k$} if the Hopf face~$k$
admits a centroid $q^*_k \in \Sp$ and if the restriction of $\Oh$ to the
spectral sector corresponding to face~$k$ has at least one eigenstate.
\end{definition}

\begin{proposition}[Plateau convergence = local geometric consistency]
\label{prop:local}
Convergence of the Plateau map to $W^*_k$ is equivalent to local geometric
consistency at face~$k$ in the sense of Definition~\ref{def:local}.

\emph{Proof sketch.}
The centroid $q^*_k$ exists because $\Sp$ is a Riemannian manifold with
positive-definite metric and the Hopf face is a compact geodesic region; the
centroid of a compact geodesically convex region on $\Sp$ is unique.
Heat dissipation $h \to 0$ is guaranteed by the dissipative definition of
Plateau: $h' < h$ for $h > 0$, so $h \to 0$ in at most one step once $q =
q^*_k$.  The eigenstate at face~$k$ exists because the spectrum of $\Oh$ is
bounded below (since $V_{\rm sl} \geq 0$), discrete, and real; by the
spectral theorem for compact self-adjoint perturbations of $-d^2/dx^2$,
the eigenstates $\{\psi_n\}$ form an orthonormal basis.  Each face~$k$
corresponds to a geodesic region in $\Sp$ and hence to a sector of the
observation spectrum; at least one eigenstate is supported on that sector
by counting arguments (finitely many faces, countably many eigenstates).
$\square$
\end{proposition}

\begin{remark}
Local geometric consistency is a \emph{face-by-face} statement.  It does not
assert that the system is globally stable, nor that no inconsistency arises in
derivations from the TOE's axioms.  It says: at the face to which the current
trajectory is converging, there exists a stable resting point.  This is the
operational content of ``the physical system does not diverge here.''
\end{remark}

%======================================================================
\section{The Real-Closed Escape}
\label{sec:tarski}
%======================================================================

The critical question is whether the Plateau fixed-point theorem belongs to
the Tarski-complete geometric fragment or to the Gödel-incomplete arithmetic
fragment identified in P143.

\begin{proposition}[Plateau theorem is in the Tarski fragment]
\label{prop:tarski}
The statement ``for each Hopf face~$k$, the state $W^*_k$ is a fixed point of
Plateau'' is a sentence in the first-order theory of real closed fields
$(\mathbb{R}, +, \cdot, <)$.

\emph{Proof.}
The claim decomposes into three sub-claims:
\begin{enumerate}
\item[$(a)$] The centroid $q^*_k$ exists and is unique.  This is a sentence about
  the real-variable equations defining the Hopf face centroids on $\Sp \subset
  \mathbb{R}^4$---a system of polynomial equations over $\mathbb{R}$ with
  a semi-algebraic uniqueness condition.
\item[$(b)$] $\mathrm{Plateau}(W^*_k) = W^*_k$.  Substituting the definition of
  Plateau, this reduces to: the projection of $q^*_k$ toward $q^*_k$ is $q^*_k$
  (trivially true) and $0' = 0$ for $h = 0$ (the dissipation clause at
  zero heat, again a real-variable identity).
\item[$(c)$] $V_{\rm sl}(x) \geq 0$ for all $x \in [0,1]$.  This is a
  semi-algebraic condition: $V_{\rm sl}$ is a rational function of $x$, $\pi$,
  and $E_{\rm self}$, and non-negativity on $[0,1]$ is a universally quantified
  sentence over $\mathbb{R}$.
\end{enumerate}
All three sub-claims are sentences of real closed field theory.  By Tarski's
theorem (\cite{Tarski1951}), every such sentence is decided by the
theory---there is an algorithm that determines its truth value without
requiring any extension to arithmetic.  $\square$
\end{proposition}

\begin{corollary}
The Plateau fixed-point theorem does not satisfy Gödel's precondition~(c)
(interpretation of Robinson arithmetic) as identified in P143,
Proposition~P143-2.  Gödel's second incompleteness theorem does not apply
to it.  The theorem is provable within the TOE's geometric axioms.
\end{corollary}

\noindent
The separation from $\mathrm{Con}(\mathrm{TOE})$ is now precise.
$\mathrm{Con}(\mathrm{TOE})$ requires encoding the set of all TOE derivations
as a predicate $\mathrm{Prov}_{\rm TOE}(n)$ over Gödel numbers $n \in \mathbb{N}$
and asserting $\neg\exists n\, [\mathrm{Prov}_{\rm TOE}(n) \wedge
\mathrm{Prov}_{\rm TOE}(\neg n)]$.  This is a sentence about $\mathbb{N}$, not
$\mathbb{R}$.  The Plateau theorem is a sentence about $\mathbb{R}$.
The two sentences belong to disjoint formal fragments.

The boundary established in P143 (geometric core $\subset$ Tarski-complete;
arithmetic overlay $\subset$ Gödel-incomplete) carves cleanly here:
Plateau convergence lands on the Tarski side.

%======================================================================
\section{The Weld-Oscillate-Plateau Composite}
\label{sec:composite}
%======================================================================

P146 O4 asked: if a closed Oscillate trajectory $W_{\rm cyc}$ (with
$\mathrm{Oscillate}^k(W_{\rm cyc}) = W_{\rm cyc}$ for some period $k$) is
welded to a Plateau fixed point $W^*_j$, is the result $W_{\rm comp} =
\mathrm{Weld}(W_{\rm cyc}, W^*_j)$ itself Plateau-stable, and does it carry
partial provenance information?

\begin{proposition}[Composite is Plateau-stable]
\label{prop:composite}
Let $W_{\rm cyc} = (q_{\rm c}, h_{\rm c}, F_{\rm c}, \ell_{\rm c}, \omega_{\rm c})$
and $W^*_j = (q^*_j, 0, \emptyset, \ell_j, \omega_j)$.
The Weld operator produces
\[
  W_{\rm comp} = \mathrm{Weld}(W_{\rm cyc}, W^*_j)
  = (q_{\rm m},\; h_{\rm c},\; F_{\rm merged},\; \ell_{\rm merged},\; \omega_{\rm merged}),
\]
where $q_{\rm m}$ is the geodesic midpoint of $q_{\rm c}$ and $q^*_j$ on $\Sp$,
and $h^*_j = 0$ so heat conservation gives total heat $h_{\rm c}$.
Applying Plateau to $W_{\rm comp}$ yields
\[
  \mathrm{Plateau}(W_{\rm comp}) = W^*_f,
\]
where $f$ is the Hopf face containing $q_{\rm m}$.
By Proposition~\ref{prop:tarski}, $W^*_f$ is a fixed point of Plateau.
The composite is therefore Plateau-stable after one application. $\square$
\end{proposition}

\begin{proposition}[Partial provenance from post-Weld face]
\label{prop:provenance}
The post-Weld fixed point $W^*_f$ encodes partial provenance of the Oscillate
trajectory.  Specifically, the face~$f$ containing $q_{\rm m}$ is a
semi-algebraic function of $(q_{\rm c}, q^*_j)$: face membership is a
polynomial inequality over $\mathbb{R}$.  Given $W^*_f$ and $W^*_j$, an
external observer can determine which face $q_{\rm m}$ occupied at Weld time,
constraining $q_{\rm c}$ to the set
$\{q \in \Sp : \mathrm{geo\_mid}(q, q^*_j) \in \mathrm{face}_f\}$---a
non-empty arc on~$\Sp$.

This is \emph{limited} provenance: it constrains the Oscillate trajectory's
position at the moment of Weld but does not uniquely reconstruct the full orbit.
The constraint is recoverable from the post-Weld geometry without any provenance
field in the WheelState schema (which remains clean, per P146
Prop.~P146-Type2-Blocking).  The LLM witness reads $W^*_f$ and $W^*_j$;
geometric computation determines the constraint on $q_{\rm c}$.
\end{proposition}

\noindent
The composite analysis is itself a Tarski-fragment argument: geodesic midpoint,
face membership, and semi-algebraic constraint sets are all sentences of
real closed field theory.  The composite fixed-point claim is internally provable
and does not approach $\mathrm{Con}(\mathrm{TOE})$.

%======================================================================
\section{What Plateau Convergence Proves}
\label{sec:theorem}
%======================================================================

\begin{theorem}[Plateau Consistency Theorem]
\label{thm:main}
The following statements are provable within the TOE's geometric fragment
(the first-order theory of real closed fields, by Tarski's theorem):
\begin{enumerate}
\item[\emph{(i)}] For each Hopf face~$k$, there exists a unique centroid
  $q^*_k \in \Sp$, and $W^*_k = (q^*_k, 0, \emptyset, \ell_k, \omega_k)$
  is a fixed point of the Plateau operator.
\item[\emph{(ii)}] The observation operator $\Oh$ (Eq.~\eqref{eq:obs})
  has a discrete, real, bounded-below spectrum $\{\lambda_n\}_{n \geq 1}$
  and an orthonormal eigenstate basis $\{\psi_n\}$ in $L^2([0,1])$.
\item[\emph{(iii)}] The Weld-Oscillate-Plateau composite $W_{\rm comp}$
  reaches a Plateau fixed point $W^*_f$ after one Plateau application,
  where $f$ is determined by the geodesic midpoint of the two input states.
\end{enumerate}
The following statement is \emph{not} provable within the TOE (by P143,
Theorem~P143-G2, and the analysis in Section~\ref{sec:tarski}):
\begin{enumerate}
\item[\emph{(iv)}] $\mathrm{Con}(\mathrm{TOE})$.
\end{enumerate}
Statements \emph{(i)--(iii)} assert physical stability: the system reaches
quiescent states, eigenstates exist, and composites Plateau-stabilize.
They are orthogonal to statement \emph{(iv)}, which is a meta-theoretic claim
about the TOE proof system.
The geometric fragment decides its own stability questions; whether the full
proof system (including its arithmetic overlay from P7's prime-triplet analysis)
is consistent is a question the system cannot answer from within.
\end{theorem}

\begin{remark}
The informal gloss on what Plateau convergence ``means'' in P146 O3 is now
precise: convergence is a geometric computation, not a meta-theoretic
inspection.  When the Wheel converges to $W^*_k$, it has computed that the
trajectory's current face has a centroid and that heat is zero.  It has not
checked any derivation in the proof system.  The operational reliability of
the Wheel is a property of the geometric model; the formal consistency of the
TOE is a property of the proof system.  These are the same Tarski-vs-Gödel
boundary that P143 Section~4 drew for physical vs.\ formal completeness.
\end{remark}

%======================================================================
\section{Open Items}
\label{sec:open}
%======================================================================

\paragraph{O1: Spectral coverage of Hopf faces.}
Proposition~\ref{prop:local} appeals to a counting argument: finitely many
faces, countably many eigenstates, hence each face is covered.  This should
be made precise.  The Hopf tiling of $\Sp$ partitions the sphere into a
finite number of geodesic faces; the eigenstate basis is countable; the
question is whether the partition is fine enough that each face contains a
region of positive spectral measure.  A direct calculation matching the
known eigenvalue spectrum $(\lambda_1 = 16.52, \lambda_2 = 51.28, \ldots)$
to Hopf face angles would close this.

\paragraph{O2: Sharpness of partial provenance.}
Proposition~\ref{prop:provenance} shows the post-Weld face constrains the
Oscillate position to an arc on $\Sp$.  The arc width depends on the geometry
of the Hopf tiling and the size of the Oscillate orbit.  For orbits much smaller
than a face, the arc is narrow and provenance is nearly exact; for large orbits,
the arc spans the full face and provenance is weak.  Quantifying this as a
function of Oscillate period~$k$ would give precise bounds on what the LLM
witness can recover.

\paragraph{O3: The arithmetic overlay and spectral stability.}
The spectrum of $\Oh$ being discrete and bounded below is a Hilbert-space
fact over $\mathbb{R}$ (Theorem~\ref{thm:main}(ii)).  But P7 uses this
spectrum to assign particle masses, and the assignment involves prime-triplet
counting arguments in the arithmetic overlay.  A sentence of the form ``the
$n$-th eigenvalue corresponds to the $n$-th fermion family'' requires
quantification over $\mathbb{N}$ and lies in the Gödel-incomplete fragment.
The eigenvalues exist (Tarski); the family assignment is arithmetic (Gödel).
This boundary should be stated as a proposition in a future addendum on P7's
formal status.

\medskip

\begin{thebibliography}{9}

\bibitem{Tarski1951}
A.~Tarski,
\emph{A Decision Method for Elementary Algebra and Geometry},
University of California Press (1951).

\bibitem{Godel1931}
K.~Gödel,
``Über formal unentscheidbare Sätze der Principia Mathematica und verwandter
Systeme I,''
\emph{Monatshefte für Mathematik und Physik} \textbf{38}, 173--198 (1931).

\bibitem{P7}
L.~F.~Vlegels,
``The Self-Referential Observation Framework: Particle Masses as Geometric
Eigenvalues,'' This volume, Paper~7 (2025).

\bibitem{P18}
L.~F.~Vlegels,
``The Master Operator and the Wheel,'' This volume, Paper~18 (2025).

\bibitem{P143}
L.~F.~Vlegels,
``Gödel Incompleteness and the Theory of Everything Programme,''
This volume, Addendum~143 (2026).

\bibitem{P146}
L.~F.~Vlegels,
``The Observer Inside the System: Wheel Self-Observation, the TOE Measurement
Problem, and the Butlerian Line,''
This volume, Addendum~146 (2026).

\end{thebibliography}

\end{document}
