% 145_Addendum_ScopeTheorem.tex
% Addendum P145: The Scope Theorem — What the TOE Can and Cannot Predict
%   Derives a formal characterisation of the TOE's predictive domain as the
%   algebraic skeleton of gauge theory determined by the J3(O) spectrum.
%   Classifies observables as algebraic, dynamical, or hybrid; derives the
%   boundary criterion from P141 Peirce block orthogonality; states and proves
%   the Scope Theorem; applies it to muon g-2; argues the boundary is a feature.
% Author: León Fernando Vlegels
% Copyright: León Fernando Vlegels. License: CC BY 4.0.
% Date: 2026-05-15

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\newcommand{\JO}{J_3(\mathbb{O})}
\newcommand{\SJ}{\mathcal{S}^{\JO}_{\rm had}}
\newcommand{\Piow}{\Pi_{\rm had}^{\rm TOE}}
\newcommand{\Pism}{\Pi_{\rm had}^{\rm SM}}
\newcommand{\PEW}{\mathbf{P}_{\rm EW}}
\newcommand{\Gtwo}{G_2}
\newcommand{\TOE}{\mathrm{TOE}}
\newcommand{\muconf}{\mu_{\rm conf}}
\newcommand{\Dahad}{\Delta\alpha_{\rm had}}

\label{addendum:145}

\title{\textbf{Addendum P145}\\[6pt]
\large The Scope Theorem: What the TOE Predicts and Why}
\author{L\'eon Fernando Vlegels\\[4pt]
\small Independent Researcher \quad
\texttt{axiomofchoicepuzzle@gmail.com}}
\date{15 May 2026}

\begin{document}
\maketitle

\begin{abstract}
Addendum P141 established that $85.1\%$ of the PDG hadronic vacuum polarisation
$\Dahad^{\rm PDG}$ is decoupled from the $\JO$ eigenvalue spectrum by construction.
This raises the structural question: what, precisely, is the TOE a theory of?
We derive a \emph{Scope Theorem}: the TOE is the theory of the \emph{algebraic
skeleton} of gauge physics, comprising exactly those observables whose hadronic
corrections are expressible as linear functionals of the discrete pole spectrum
$\SJ$.
The boundary criterion is the $\PEW$-projectability of the hadronic dispersion
kernel: an observable is TOE-algebraic if and only if its kernel $K_O(s)$ is
effectively supported at scales $s \gtrsim M_\rho^2$ where the $\JO$ poles
dominate.
Algebraic observables are mutually correlated (all functions of the same $\JO$
structure); dynamical ones are independent inputs.
The non-derivability of QCD dynamics from $\JO$ is itself a theorem, not an
observation: it follows from a dimension argument on $L^2(\mathbb{R}_+)$.
Applied to the muon $g{-}2$: any tension between theory and experiment is
structurally confined to the dynamical sector and cannot originate in the
algebraic EW precision observables.
\end{abstract}

\tableofcontents
\vspace{1em}

%==============================================================
\section{Setup}
\label{sec:setup}
%==============================================================

Addenda P121--P141 completed the TOE electroweak precision programme.
The quantities $\alpha^{-1}$, $M_W$, $M_Z$, $G_F$, $\sin^2\theta_W$,
$\alpha_s(M_Z)$, the five resonance masses $\{M_\rho, M_\omega, M_\phi,
M_{J/\psi}, M_\Upsilon\}$, and the CKM Wolfenstein parameters were all
derived from the $\JO$ discrete spectrum and the master operator $\hat{O}$
(P18, P20) to per-mille accuracy or better.
In every case the prediction rests on the same structure: eigenvalues and
Peirce-block residues of $\JO$, with no QCD Fock-space input.

Simultaneously, P136--P141 showed that the multi-pion continuum, the
charm--bottom hadronic running, and the hadronic light-by-light contributions
are \emph{structurally outside} this framework by Peirce block orthogonality
(P141, Theorem~1).
The TOE accounts for $14.9\%$ of $\Dahad^{\rm PDG}$; the remaining $85.1\%$
is annihilated by $\PEW$.

The pattern raises the deepest structural question: is this a limitation of the
TOE, or a precise characterisation of what the TOE claims to be?
The answer is the latter.
The present addendum derives it formally and states it as a theorem.

%==============================================================
\section{Observable Classification}
\label{sec:classification}
%==============================================================

Let $O$ be an observable whose hadronic correction takes the dispersive form
\begin{equation}\label{eq:disp}
  \Delta O_{\rm had} = \int_{4m_\pi^2}^{\infty} K_O(s)\,
  \frac{\mathrm{Im}\,\Pi_{\rm had}(s+i0^+)}{\pi\, s^n}\,ds,
\end{equation}
for a non-negative kernel $K_O$ and integer $n \geq 1$.

\begin{definition}[Observable classification]
\label{def:class}
An observable $O$ admitting the representation~\eqref{eq:disp} is:
\begin{itemize}
  \item \emph{TOE-algebraic} if $\Delta O_{\rm had}$ is well-approximated by
    replacing $\mathrm{Im}\,\Pi_{\rm had}$ with $\mathrm{Im}\,\Piow$, the
    $\PEW$-projected correlator of P141; error is $O(\alpha_s/\pi)$.
  \item \emph{TOE-dynamical} if the dominant support of
    $K_O(s)\,\mathrm{Im}\,\Pi_{\rm had}(s)/s^n$ lies in the branch-cut
    sector annihilated by $\PEW$ (Lemma~1, P141).
  \item \emph{Hybrid} if both sectors contribute at the $\gtrsim 10\%$ level.
\end{itemize}
\end{definition}

\noindent
Table~\ref{tab:class} gives canonical examples.

\begin{table}[h]
\centering
\caption{Classification of observables by TOE-accessibility.
$Q_{\rm eff}$ is the effective scale at which $K_O(s)$ is peaked.}
\label{tab:class}
\smallskip
\begin{tabular}{llcc}
\toprule
Observable & Physical content & $Q_{\rm eff}$ & Class \\
\midrule
$\alpha^{-1} = 4\pi^3+\pi^2+\pi$ & $\int_0^1\rho\,dx$ (P3) & --- & Algebraic \\
$M_W,\,M_Z$ & EW Peirce eigenvalues (P49, P93) & $M_Z$ & Algebraic \\
$G_F$ & Fermi self-consistency (P123) & $M_W$ & Algebraic \\
$\sin^2\theta_W$ & $SU(2)/U(1)$ dimension ratio (P42) & $M_Z$ & Algebraic \\
$\alpha_s(M_Z)$ & $\Gtwo$-sector matching (P86, P90) & $M_Z$ & Algebraic \\
$M_\rho, M_\omega, M_\phi$ & $\JO$ resonance poles (P136) & $M_V$ & Algebraic \\
CKM angles & Wolfenstein from $\mathbb{Z}_3$ (P82) & --- & Algebraic \\
$a_\mu^{\rm HVP,LO}$ & Hadronic dispersion at $\mu$-scale & $m_\mu$ & Dynamical \\
$a_\mu^{\rm HLbL}$ & Four-photon loop at $m_\mu$ & $m_\mu$ & Dynamical \\
$|F_\pi(s)|^2$, $s < M_\rho^2$ & Sub-$\rho$ pion form factor (P141) & $< M_\rho$ & Dynamical \\
$\alpha_s(\mu)$, $\mu < M_Z$ & QCD running below EW scale & $\mu$ & Dynamical \\
$\muconf = 313\,\mathrm{MeV}$ & Confinement onset (P120) & $\muconf$ & Hybrid \\
\bottomrule
\end{tabular}
\end{table}

\noindent
The confinement scale $\muconf$ is hybrid: its \emph{value} is algebraically
determined as the scale at which $\alpha_s = O(1)$ by $\Gtwo$-matching (P120),
but the running of $\alpha_s$ between $\muconf$ and $M_Z$ is a $\Gtwo$-sector
dynamical process not yet derived from $\JO$ alone.

%==============================================================
\section{The Boundary Criterion}
\label{sec:criterion}
%==============================================================

\begin{definition}[$\PEW$-projectability]
\label{def:proj}
An observable $O$ is \emph{$\PEW$-projectable} if
\begin{equation}\label{eq:proj}
  \int K_O(s)\,\frac{\mathrm{Im}\,\Pism(s)}{\pi s^n}\,ds
  \;\approx\;
  \int K_O(s)\,\frac{\mathrm{Im}\,\Piow(s)}{\pi s^n}\,ds,
\end{equation}
up to $O(\alpha_s/\pi)$ corrections.  Since
$\mathrm{Im}\,\Piow(s) = \pi\sum_{V \in \SJ} C_V\,\delta(s - M_V^2)$
(P141, eq.~(3)), the right side reduces to
\begin{equation}\label{eq:alg}
  \Delta O_{\rm had}^{\TOE} = \sum_{V \in \SJ}
  \frac{K_O(M_V^2)\,C_V}{M_V^{2n}}.
\end{equation}
This formula is valid when $K_O$ is slowly varying on the scale $\Gamma_V \ll M_V$.
\end{definition}

\begin{proposition}[Scale criterion]
\label{prop:scale}
An observable $O$ is TOE-algebraic if and only if $K_O(s)$ has effective peak
scale $Q_{\rm eff}^2 \gtrsim M_\rho^2$.
The natural boundary is $Q_* \approx M_\rho \approx 770\,\mathrm{MeV}$,
set by the lightest algebraic pole in $\SJ$.
\end{proposition}

\begin{proof}
For $Q_{\rm eff}^2 \gtrsim M_\rho^2$, the integrand in~\eqref{eq:disp} has
dominant support at $s \gtrsim M_\rho^2$ where $\mathrm{Im}\,\Pism \approx
\mathrm{Im}\,\Piow$ by Theorem~1(a) of P141 (no branch cuts in $\Piow$).
For $Q_{\rm eff}^2 \ll M_\rho^2$, dominant support lies in the branch-cut sector
$s \in [4m_\pi^2, M_\rho^2)$, which is orthogonal to $\PEW$ by Lemma~1 of P141.
\end{proof}

\noindent
\textit{Illustration.}
The Fermi hadronic running uses kernel $K_\alpha(s) = (1 - s/M_Z^2)^{-1}$,
peaked at $s \sim M_Z^2 \gg M_\rho^2$: algebraic (P135).
The muon HVP uses $K_\mu(s) \approx m_\mu^2/(3s)$, peaked at $s \sim m_\mu^2$,
a factor $\sim 53$ below $M_\rho^2$: dynamical.

%==============================================================
\section{Scope Theorem}
\label{sec:theorem}
%==============================================================

\begin{theorem}[Scope Theorem]
\label{thm:scope}
Let $\mathcal{A}$ be the set of TOE-algebraic observables.
\begin{enumerate}
  \item[\rm (a)] \textbf{Algebraic completeness.}
    Every $O \in \mathcal{A}$ is determined by the ten real numbers
    $\{M_V, C_V : V \in \SJ\}$ together with the $\JO$ group-theoretic
    data via equation~\eqref{eq:alg} and the master operator $\hat{O}$.
  \item[\rm (b)] \textbf{Dynamical independence.}
    Let $\mathcal{D}$ be the dynamical complement.  The hadronic spectral
    function $\rho_{\rm had}(s) \in L^2(\mathbb{R}_+)$ has infinite-dimensional
    function space; the algebraic data $\{M_V, C_V\}$ occupy a
    $10$-dimensional parameter space.
    No map $\mathbb{R}^{10} \to L^2(\mathbb{R}_+)$ is surjective onto any open
    set; hence generic QCD dynamical observables are not determined by $\JO$ alone.
  \item[\rm (c)] \textbf{Algebraic correlations.}
    Any two $O_1, O_2 \in \mathcal{A}$ satisfy an algebraic relation
    $R(O_1, O_2, \{M_V\}, \{C_V\}) = 0$ derivable from $\JO$.
    No such universal relation exists between any element of $\mathcal{A}$ and
    any element of $\mathcal{D}$.
  \item[\rm (d)] \textbf{Non-derivability of dynamics is a theorem.}
    The non-derivability in (b) is not an empirical observation about the
    TOE's current state of development; it is a consequence of the
    dimension argument applied to the finite-rank map $\mathbb{R}^{10}
    \hookrightarrow L^2(\mathbb{R}_+)$.
    Completing the TOE corpus cannot change this: any extension that adds
    new algebraic content adds finitely many parameters, while
    $L^2(\mathbb{R}_+)$ remains infinite-dimensional.
\end{enumerate}
\end{theorem}

\begin{proof}[Proof sketch]
(a) follows from Theorem~1 of P141 together with P82--P135.
(b) and (d): $\rho_{\rm had}(s)$ is an arbitrary non-negative function subject
only to QCD unitarity; its function space is infinite-dimensional.
A finite-rank map $\mathbb{R}^{10} \to L^2(\mathbb{R}_+)$ has image of Lebesgue
measure zero; it cannot represent generic hadronic spectral functions.
(c): All $O \in \mathcal{A}$ are rational functions of $\{M_V^2, C_V\}$ via
\eqref{eq:alg}; eliminating the pole parameters gives an algebraic relation.
Elements of $\mathcal{D}$ depend on the $\PEW$-orthogonal branch-cut sector
(P141, Lemma~1), which is independent of $\{M_V, C_V\}$.
\end{proof}

\begin{corollary}[Falsifiability structure]
\label{cor:falsify}
All algebraic observables lie on a $10$-dimensional algebraic submanifold of
the EW precision data space.
A single measurement placing two algebraic observables outside this submanifold
falsifies the TOE.
Measurements of dynamical observables, however, cannot falsify the algebraic
sector: they probe independent degrees of freedom.
\end{corollary}

\noindent
The Scope Theorem resolves the apparent tension between the TOE's precision
in the EW sector and its silence on hadronic dynamics.
The silence is not a gap to be filled by future addenda; it is a structural
boundary with a mathematical proof.

%==============================================================
\section{Muon $g{-}2$ as Test Case}
\label{sec:gm2}
%==============================================================

The leading hadronic contribution to the muon anomalous magnetic moment is
\begin{equation}\label{eq:hvp}
  a_\mu^{\rm HVP,LO}
  = \frac{\alpha}{\pi}
  \int_{4m_\pi^2}^\infty K_\mu(s)\,
  \frac{\mathrm{Im}\,\Pi_{\rm had}(s)}{\pi s}\,ds,
\end{equation}
with $K_\mu(s)$ peaked at $s \sim m_\mu^2 = (105.7\,\mathrm{MeV})^2$.
Since $m_\mu^2/M_\rho^2 \approx 1/53$, Proposition~\ref{prop:scale} classifies
$a_\mu^{\rm HVP,LO}$ as \emph{TOE-dynamical}.
The $\rho$-pole contributes $\approx 28\%$ of the total via vector meson dominance;
the remaining $\approx 72\%$ arises from the sub-threshold pion form factor, which
is outside $\SJ$.
The hadronic light-by-light piece $a_\mu^{\rm HLbL}$ is also entirely dynamical:
the relevant four-point loop momentum is $\sim m_\mu$, sub-confinement.

\paragraph{What the TOE predicts about the tension.}
By Theorem~\ref{thm:scope}(c), algebraic and dynamical observables are
independent.
If experiment finds $a_\mu^{\rm exp} \neq a_\mu^{\rm SM}$, the discrepancy
\emph{cannot} originate in the algebraic sector:
$\alpha$, $M_W$, $M_Z$, and $\sin^2\theta_W$ are independently verified to
lie on the $\JO$ algebraic submanifold, and these are the inputs to the
leptonic and electroweak corrections in $a_\mu^{\rm SM}$.
The discrepancy must therefore be located in either (i) the dynamical hadronic
sector (HVP or HLbL) or (ii) a BSM contribution.
The TOE provides no constraint between these two options; both are in $\mathcal{D}$.

\paragraph{Dispersive vs.\ lattice.}
The ongoing disagreement between lattice QCD evaluations of $a_\mu^{\rm HVP,LO}$
and dispersive evaluations from $e^+e^-$ cross-section data is a dispute
\emph{within} $\mathcal{D}$: both compute the same dynamical observable from
different methods.
Their disagreement is a QCD systematics question, not an algebraic-sector question.
The TOE neither adjudicates nor contributes to its resolution.

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

\paragraph{O1: $\Gtwo$ renormalisation group.}
The confinement scale $\muconf = 313\,\mathrm{MeV}$ is algebraically fixed (P120),
but $\alpha_s(\mu)$ for $\muconf \leq \mu < M_Z$ requires the $\Gtwo$-sector
$\beta$-function, not yet derived from $\JO$ first principles.
Completing the $\Gtwo$ RG would promote $\alpha_s(\mu)$ from hybrid to fully
algebraic, closing the largest hybrid observable in Table~\ref{tab:class}.

\paragraph{O2: Pion form factor from $\mathcal{P}_{12}$.}
Deriving $|F_\pi(s)|^2$ via Omnès unitarisation from the $\mathcal{P}_{12}$
Peirce sector (P113) would extend the algebraic boundary below the $\rho$ peak
and reduce the dynamical fraction of $a_\mu^{\rm HVP,LO}$ from $72\%$ toward a
calculable residual.
This would be the first closure of a currently dynamical observable.

\paragraph{O3: Explicit algebraic relations.}
Corollary~\ref{cor:falsify} predicts algebraic relations among
$\{M_W, M_Z, \sin^2\theta_W, \alpha^{-1}, G_F, \alpha_s(M_Z)\}$ beyond the SM
tree-level identities.
The TOE corrections are parametrised by the Peirce residues $C_V$; deriving
these relations explicitly would provide a new class of falsifiability tests
distinct from individual-observable precision.

\paragraph{O4: Gravity in the scope picture.}
$G$ (Newton) is algebraically determined (P17); the bulk $B^4$ carries dynamical
gravitational degrees of freedom.
The Scope Theorem structure should extend to gravity, with boundary $S^3$
observables algebraic and bulk dynamical observables in a gravitational
$\mathcal{D}$.
Making this precise would complete the scope picture across all four forces.

\medskip

\begin{thebibliography}{9}

\bibitem{P17}
L.~F.~Vlegels,
``Gravity as Bulk Propagation,''
This volume, Paper~17 (2025).

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

\bibitem{P20}
L.~F.~Vlegels,
``Standard Physics Embedding,''
This volume, Paper~20 (2025).

\bibitem{P82}
L.~F.~Vlegels,
``Addendum P82: CKM Complete,''
LumenOS corpus (2026).

\bibitem{P120}
L.~F.~Vlegels,
``Addendum P120: Confinement Scale from $G_2$ Matching,''
LumenOS corpus (2026).

\bibitem{P123}
L.~F.~Vlegels,
``Addendum P123: Fermi Coupling Closure,''
LumenOS corpus (2026).

\bibitem{P141}
L.~F.~Vlegels,
``Addendum P141: Multi-Pion Continuum Decoupling,''
LumenOS corpus (2026).

\bibitem{P143}
L.~F.~Vlegels,
``Addendum P143: G\"odel Incompleteness and the TOE Programme,''
LumenOS corpus (2026).

\bibitem{BMW2021}
S.~Borsanyi et al.\ (BMW),
``Leading hadronic contribution to the muon magnetic moment from lattice QCD,''
\emph{Nature} \textbf{593}, 51--55 (2021).

\bibitem{Davier2020}
M.~Davier, A.~Hoecker, B.~Malaescu, Z.~Zhang,
``Reevaluation of the hadronic vacuum polarisation contributions to the
Standard Model predictions of the muon $g{-}2$,''
\emph{Eur.\ Phys.\ J.\ C} \textbf{80}, 241 (2020).

\end{thebibliography}

\end{document}
