% Addendum 47: Route 3 — ρ(x) as Sector Trace Generating Function
%              Spectral Analysis of L_{X_sector} and the Harmonic Diagonal
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\OO}{\mathbb{O}}
\newcommand{\JO}{J_3(\OO)}
\newcommand{\muzero}{\mu_0}
\newcommand{\muone}{\mu_1}
\newcommand{\MU}{\mathrm{MU}}
\newcommand{\rhoTOE}{\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x}
\newcommand{\Tr}{\mathrm{Tr}}
\newcommand{\Xsec}{X_{\mathrm{sec}}}
\newcommand{\Xone}{X_1}

\title{Route~3 Resolved: $\rho(x)$ as Sector Trace Generating Function\\
Spectral Analysis of $L_{X_{\mathrm{sec}}}$ and the Harmonic Diagonal\\[0.5em]
\large Addendum~47 to the TOE Corpus}

\author{L.\ F.\ Vlegels\\
Independent Researcher\\
\texttt{axiomofchoicepuzzle@gmail.com}}

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

\begin{abstract}
Addendum~46 identified Route~3---treating $\rho(x)$ as the spectral density
of $\Xsec$ under a $\JO$ probability measure---as the most natural path toward
closing Conjectures~B and~D.  The present addendum executes Route~3 in full.

\medskip\noindent\textbf{Negative result.}
$\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$ is \emph{not} the spectral
density of $L_{\Xsec}$ in any standard operator sense.  The Peirce spectrum
of $L_{\Xsec}$ is a discrete set in $[\pi, 4\pi^3]$, entirely disjoint from
$[0,1]$; the spectral trace equals $9\mu_0$, not $\mu_0$; and the Gauss
quadrature nodes of $\rho$ do not match the Peirce eigenvalues under any
natural normalization.

\medskip\noindent\textbf{Positive result (new theorem).}
$\rho(x)$ is the \emph{sector trace generating polynomial} of $\Xsec$.  For
all $k \geq 0$:
\[
  \mu_k = \int_0^1 x^k\rho(x)\,dx
        = \Tr_{\JO}(\Xsec \circ D_k),
  \qquad
  D_k = \mathrm{diag}\!\left(\frac{2}{k+2},\frac{3}{k+3},\frac{4}{k+4}\right).
\]
The matrix $D_k$ is the \emph{harmonic diagonal of order $k$}.  At $k=0$,
$D_0 = I$ and $\mu_0 = \Tr(\Xsec)$.  At $k=1$, $D_1 = K$ is the Peirce
center matrix of Addendum~46.  The generating function $F(z) = \Tr(\Xsec
\circ D_z)$ is rational with poles at $z = -2,-3,-4$.  A further exact
identity: $\Tr_{\mathrm{spec}}(L_{\Xsec})/\dim\JO = \mu_0/3$.

\medskip\noindent\textbf{Status.}
Conjectures~B and~D are not closed.  The trace identity $\mu_0 = \Tr(\Xsec)$
is tautological.  The structural gap (OP-B) survives.
\end{abstract}

\section{Introduction}
\label{sec:intro}

\subsection{Context}

Addendum~46 reformulated Conjectures~B and~D as $\JO$ trace identities and
listed three routes toward structural derivations.  Route~3 was declared ``most
natural'' with the informal description: treat $\rho(x)$ as the spectral
density of $\Xsec$ under a $\JO$ probability measure, so that its normalization
$\int_0^1\rho(x)\,dx = \mu_0 = \Tr(\Xsec)$ is the trace formula for the
spectral measure.

This addendum makes Route~3 precise and resolves both questions:
\begin{enumerate}[noitemsep]
  \item Do the Gauss quadrature nodes of $\rho$ match Peirce eigenvalues of
    $L_{\Xsec}$?
  \item What is the correct $\JO$ algebraic object encoding $\rho$ and its
    full moment hierarchy?
\end{enumerate}

\subsection{Constants and notation}

\begin{align}
  \muzero &= 4\pi^3 + \pi^2 + \pi = 137.036\,304, \label{eq:mu0}\\
  \muone  &= \tfrac{16\pi^3}{5}+\tfrac{3\pi^2}{4}+\tfrac{2\pi}{3}
            = 108.716\,684, \label{eq:mu1}\\
  \MU     &= \muone/\muzero = 0.793\,342. \label{eq:MU}
\end{align}
The sector element $\Xsec = \mathrm{diag}(\pi, \pi^2, 4\pi^3) \in V_1
\subset \JO$ has eigenvalues $\alpha_1 = \pi$, $\alpha_2 = \pi^2$,
$\alpha_3 = 4\pi^3$ with sector powers $n_1 = 1$, $n_2 = 2$, $n_3 = 3$.
The monad density is $\rho(x) = 2\pi x + 3\pi^2 x^2 + 16\pi^3 x^3$
with moments $\mu_k = 16\pi^3/(k+4) + 3\pi^2/(k+3) + 2\pi/(k+2)$.

\section{Spectral analysis of $L_{\Xsec}$}
\label{sec:spectrum}

\subsection{Peirce spectrum}

The Jordan multiplication operator $L_{\Xsec}: \JO \to \JO$ acts by
$L_{\Xsec}(Y) = \Xsec \circ Y$.  For a diagonal element
$\Xsec = \sum_i \alpha_i e_i$ with distinct eigenvalues, the Peirce
decomposition gives (McCrimmon~\cite{McCrimmon}):
\begin{itemize}[noitemsep]
  \item Eigenvalue $\alpha_i$ on $\R\,e_i \subset V_1$, multiplicity~$1$;
  \item Eigenvalue $(\alpha_i+\alpha_j)/2$ on $J_{ij} \cong \OO$,
    multiplicity~$8$, for each $i < j$.
\end{itemize}
The six distinct Peirce eigenvalues are given in Table~\ref{tab:spectrum}.

\begin{table}[ht]
\centering
\caption{Peirce spectrum of $L_{\Xsec}$ on $\JO$.}
\label{tab:spectrum}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{llr}
\toprule
Eigenvalue & Multiplicity & Numerical value \\
\midrule
$\alpha_1 = \pi$                  & $1$ & $3.14159$ \\
$\alpha_2 = \pi^2$                & $1$ & $9.86960$ \\
$\alpha_3 = 4\pi^3$               & $1$ & $124.025$ \\
$(\alpha_1+\alpha_2)/2$           & $8$ & $6.5056$  \\
$(\alpha_1+\alpha_3)/2$           & $8$ & $63.583$  \\
$(\alpha_2+\alpha_3)/2$           & $8$ & $66.947$  \\
\bottomrule
\end{tabular}
\end{table}

Total weight: $1+1+1+8+8+8 = 27 = \dim\JO$.  The spectrum lies in
$[\pi,\,4\pi^3] \approx [3.14,\,124.0]$, entirely disjoint from $[0,1]$.

\begin{proposition}[Spectral trace identity]\label{prop:spectral-trace}
\begin{equation}\label{eq:spectral-trace}
  \Tr_{\mathrm{spec}}(L_{\Xsec}) = \sum_i w_i\lambda_i = 9\,\muzero,
  \qquad
  \frac{\Tr_{\mathrm{spec}}(L_{\Xsec})}{\dim\JO} = \frac{\muzero}{3}.
\end{equation}
\end{proposition}

\begin{proof}
Direct expansion using Table~\ref{tab:spectrum}:
\begin{align*}
  \Tr_{\mathrm{spec}}
  &= (\alpha_1+\alpha_2+\alpha_3)
    + 8\cdot\tfrac{\alpha_1+\alpha_2}{2}
    + 8\cdot\tfrac{\alpha_1+\alpha_3}{2}
    + 8\cdot\tfrac{\alpha_2+\alpha_3}{2}\\
  &= (\alpha_1+\alpha_2+\alpha_3) + 4(\alpha_1+\alpha_2)
    + 4(\alpha_1+\alpha_3) + 4(\alpha_2+\alpha_3)\\
  &= 9(\alpha_1+\alpha_2+\alpha_3) = 9\,\muzero.
\end{align*}
Dividing by $\dim\JO = 27$ gives the second identity.
\end{proof}

\begin{remark}
The spectral trace $9\mu_0 \approx 1233$ differs from the Jordan trace
$\mu_0 \approx 137$ by a factor of $9$.  The Jordan trace is the
rank-$3$ trace (sum of eigenvalues of $\Xsec$ as a $3\times 3$ matrix);
the spectral trace is the full-$27$-dimensional operator trace.  Neither
equals $\mu_0$ alone, though their ratio $\Tr_{\mathrm{spec}}/9 =
\Tr_{\JO}$ relates them precisely.
\end{remark}

\subsection{Gauss quadrature nodes: comparison with Peirce eigenvalues}

The Gauss quadrature nodes for the weight $\rho(x)/\mu_0$ on $[0,1]$ are
computed by the Stieltjes--Gram-Schmidt algorithm applied to the moment
sequence.  The leading nodes are:

\medskip
\noindent
\textit{3-node:} $0.33521,\quad 0.68662,\quad 0.93548$.

\medskip
The Peirce eigenvalues normalized by $\mu_0$ are
$\pi/\mu_0 \approx 0.0229$, $\pi^2/\mu_0 \approx 0.0720$,
$4\pi^3/\mu_0 \approx 0.9051$.  The 3-node Gauss points do not match
these (nor do they match the Peirce center eigenvalues $2/3, 3/4, 4/5$).

\begin{proposition}[No spectral identification]\label{prop:no-match}
$\rho(x)$ is not the spectral density of $L_{\Xsec}$:
\begin{enumerate}[noitemsep, label=(\alph*)]
  \item The supports are disjoint: $\mathrm{supp}(\rho) = [0,1]$,
    $\mathrm{spec}(L_{\Xsec}) \subset [\pi, 4\pi^3]$.
  \item Gauss quadrature nodes of $\rho$ do not match Peirce eigenvalues
    under normalization by $\mu_0$, $4\pi^3$, or $\log(4\pi^3)$.
  \item The spectral $k$-th moment $\sum_i w_i\lambda_i^k$ grows as
    $(4\pi^3)^k$; the density moment $\mu_k \to 0$ as $k\to\infty$.
\end{enumerate}
\end{proposition}

\section{The sector trace generating function}
\label{sec:generating}

\subsection{Decomposition of $\rho$}

From the sector construction (Addendum~36), the coefficient of $x^{n_i}$
in $\rho$ is $c_i = \alpha_i(n_i+1)$:
\begin{equation}\label{eq:rho-decomp}
  \rho(x)
  = \sum_{i=1}^{3}\alpha_i\,(n_i+1)\,x^{n_i}
  = 2\pi x + 3\pi^2 x^2 + 16\pi^3 x^3.
\end{equation}
(Check: $c_3 = 4\pi^3 \cdot 4 = 16\pi^3$.)
Each normalized sector density $\rho_i^{(\mathrm{n})}(x) = (n_i+1)\,x^{n_i}$
integrates to $1$:
\[
  \rho(x) = \sum_i \alpha_i\,\rho_i^{(\mathrm{n})}(x),
  \qquad
  \int_0^1\rho_i^{(\mathrm{n})}(x)\,dx = 1.
\]

\subsection{Harmonic diagonal}

\begin{definition}[Harmonic diagonal]\label{def:Dk}
For $k \geq 0$, the \emph{harmonic diagonal of order $k$} is
\begin{equation}\label{eq:Dk}
  D_k = \mathrm{diag}\!\left(\frac{2}{k+2},\;\frac{3}{k+3},\;\frac{4}{k+4}\right)
      = \mathrm{diag}\!\left(\frac{n_1+1}{k+n_1+1},\;\frac{n_2+1}{k+n_2+1},\;
        \frac{n_3+1}{k+n_3+1}\right) \in V_1.
\end{equation}
\end{definition}

Properties: $D_0 = I_{V_1}$; $D_1 = K = \mathrm{diag}(2/3, 3/4, 4/5)$
(the Peirce center matrix of Addendum~46); $D_k \to 0$ monotonically as
$k\to\infty$; $(D_k)_i = 1/(1 + k/(n_i+1))$ (harmonic damping).

\begin{lemma}[Integral representation]\label{lem:integral}
$(D_k)_i = \int_0^1 x^k\,\rho_i^{(\mathrm{n})}(x)\,dx = (n_i+1)/(k+n_i+1)$.
\end{lemma}

\begin{proof}
Direct: $\int_0^1 x^k(n_i+1)x^{n_i}\,dx = (n_i+1)/(k+n_i+1)$.
\end{proof}

\subsection{Main theorem}

\begin{theorem}[Moment hierarchy as $\JO$ trace hierarchy]\label{thm:main}
For all $k \geq 0$:
\begin{equation}\label{eq:main}
  \mu_k = \int_0^1 x^k\rho(x)\,dx = \Tr_{\JO}(\Xsec \circ D_k).
\end{equation}
\end{theorem}

\begin{proof}
Using \eqref{eq:rho-decomp} and Lemma~\ref{lem:integral}:
\begin{equation*}
  \int_0^1 x^k\rho(x)\,dx
  = \sum_{i=1}^{3}\alpha_i\int_0^1 x^k\rho_i^{(\mathrm{n})}(x)\,dx
  = \sum_{i=1}^{3}\alpha_i(D_k)_i
  = \Tr_{\JO}(\Xsec \circ D_k).\qquad\square
\end{equation*}
\end{proof}

\begin{corollary}\label{cor:special}
$\mu_0 = \Tr(\Xsec\circ D_0) = \Tr(\Xsec) = \muzero$;
$\mu_1 = \Tr(\Xsec\circ D_1) = \Tr(\Xsec\circ K) = \muone$.
The Peirce center $K = D_1$ is the harmonic diagonal at $k=1$.
\end{corollary}

\begin{table}[ht]
\centering
\caption{Harmonic diagonal $D_k$ and corresponding moments.  Entries verified
  numerically to $10^{-10}$ precision.}
\label{tab:Dk}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{rllr}
\toprule
$k$ & $D_k$ & $\Tr(\Xsec\circ D_k)$ & $\mu_k$ \\
\midrule
$0$ & $\mathrm{diag}(1,\;1,\;1)$                     & $137.036\,304$ & $137.036\,304$ \\
$1$ & $\mathrm{diag}(\frac{2}{3},\frac{3}{4},\frac{4}{5})$ & $108.716\,684$ & $108.716\,684$ \\
$2$ & $\mathrm{diag}(\frac{1}{2},\frac{3}{5},\frac{2}{3})$ & $90.175\,963$ & $90.175\,963$ \\
$3$ & $\mathrm{diag}(\frac{2}{5},\frac{1}{2},\frac{4}{7})$ & $77.062\,929$ & $77.062\,929$ \\
$4$ & $\mathrm{diag}(\frac{1}{3},\frac{3}{7},\frac{1}{2})$ & $67.289\,581$ & $67.289\,581$ \\
\bottomrule
\end{tabular}
\end{table}

\section{The generating function as a Stieltjes series}
\label{sec:stieltjes}

\begin{proposition}[Rational generating function]\label{prop:rational}
The function $F(z) = \Tr_{\JO}(\Xsec \circ D_z)$, defined for $z \notin \{-2,-3,-4\}$, is
\begin{equation}\label{eq:Fz}
  F(z) = \frac{2\pi}{z+2} + \frac{3\pi^2}{z+3} + \frac{16\pi^3}{z+4},
\end{equation}
with: $(i)$ $\mu_k = F(k)$ for all $k\geq 0$;
$(ii)$ poles at $z = -(n_i+1) = -2,-3,-4$ with residues
$\alpha_i(n_i+1) = 2\pi,\,3\pi^2,\,16\pi^3$;
$(iii)$ $F(0) = \muzero$, $F(1) = \muone$, $F(z)\to 0$ as $z\to+\infty$.
\end{proposition}

\begin{proof}
$F(z) = \sum_i\alpha_i(n_i+1)/(z+n_i+1)$ by Definition~\ref{def:Dk}.
Substituting $n_i+1 \in \{2,3,4\}$ and $\alpha_i(n_i+1) \in \{2\pi,3\pi^2,16\pi^3\}$
gives \eqref{eq:Fz}.  Items (i)--(iii) are direct.
\end{proof}

\begin{remark}[Poles and sector structure]
The poles at $z = -2,-3,-4$ encode the sector monomial degrees $n_i$:
each sector density $\rho_i^{(\mathrm{n})} \propto x^{n_i}$ integrates
against $x^z$ as $(z+n_i+1)^{-1}$, producing a simple pole at
$z = -(n_i+1)$.  The residues $\alpha_i(n_i+1)$ are the sector energies
weighted by the sector dimension coefficient.  This is the exact Stieltjes
structure of the three-sector $V_1$ decomposition.
\end{remark}

\begin{remark}[$G_0$--$G_1$ as Stieltjes ratio]
$\MU = F(1)/F(0) = \muone/\muzero$.  The $G_0$--$G_1$ gap of Addendum~46 is:
\[
  G_0 - G_1 = \frac{\ln F(0) - \ln F(1)}{\MU}
            = \frac{-\ln\MU}{\MU}
            = \frac{-\ln(F(1)/F(0))}{F(1)/F(0)},
\]
a function of consecutive Stieltjes values.  Identifying $-\ln\MU/\MU$ as
a $\JO$ intrinsic invariant remains open (OP-D).
\end{remark}

\section{Consequences for Conjectures~B and~D}
\label{sec:conjectures}

\subsection{What is established}

\begin{theorem}[Route~3 summary]\label{thm:route3}
The following exact identities hold, all following from
Theorem~\ref{thm:main} and Proposition~\ref{prop:spectral-trace}:
\begin{align}
  \mu_k &= \Tr_{\JO}(\Xsec \circ D_k) \quad\forall k\geq 0,
           \label{eq:R3a}\\
  \mu_0 &= \Tr_{\JO}(\Xsec) = \muzero,
           \label{eq:R3b}\\
  \mu_1 &= \Tr_{\JO}(\Xsec \circ K) = \muone,
           \label{eq:R3c}\\
  \frac{\Tr_{\mathrm{spec}}(L_{\Xsec})}{\dim\JO}
        &= \frac{\muzero}{3}.
           \label{eq:R3d}
\end{align}
The Peirce center matrix $K = D_1$ is the $k=1$ case of the universal
harmonic family $D_k$.
\end{theorem}

\subsection{What is not established}

\begin{proposition}[Tautology and surviving gap]\label{prop:gap}
\begin{enumerate}[label=(\alph*)]
  \item Identity \eqref{eq:R3b} is tautological: $\rho$ was defined so that
    $\int_0^1\rho\,dx = \alpha_1+\alpha_2+\alpha_3$, which by construction
    equals both $\mu_0$ and $\Tr(\Xsec)$.
  \item Conjecture~B requires deriving $m_t/m_c = \Tr(\Xsec)$ without
    invoking the mass formula.  This requires identifying $\Xsec$ as the
    canonical $\JO$ generation mass element from $G_2$-orbit structure,
    not by postulation.  Route~3 provides no new input for this.
  \item The spectral density identification fails (Proposition~\ref{prop:no-match});
    the tautological trace identity survives but does not close OP-B.
\end{enumerate}
\end{proposition}

\subsection{Updated open problem status}

\begin{description}
  \item[OP-B (surviving)]
    Derive $m_t/m_c = \Tr_{\JO}(\Xsec) = \mu_0$ without using the mass
    formula in the forward direction.  The exact $\JO$ algebraic form is
    identified.  Missing: a $G_2$-orbit derivation of $\Xsec$ as canonical.

  \item[OP-D (surviving, reduces to OP-B)]
    Given OP-B, Conjecture~D is structural iff $-\ln\MU/\MU =
    -\ln(F(1)/F(0))/(F(1)/F(0))$ admits a $\JO$-intrinsic derivation.
    The Stieltjes form of $F$ provides additional structure but no
    identification of $-\ln\MU/\MU$ as a Casimir or Killing invariant.
\end{description}

\section{Summary}
\label{sec:summary}

\begin{enumerate}[noitemsep]
  \item \textbf{$\rho(x)$ is not the spectral density of $L_{\Xsec}$}
    (Proposition~\ref{prop:no-match}).  Route~3 as stated in Addendum~46
    does not hold.

  \item \textbf{$\rho(x)$ is the sector trace generating polynomial}
    (Theorem~\ref{thm:main}):
    $\mu_k = \Tr_{\JO}(\Xsec \circ D_k)$ for all $k\geq 0$,
    with harmonic diagonal $D_k = \mathrm{diag}(2/(k+2), 3/(k+3), 4/(k+4))$.

  \item \textbf{The Peirce center $K = D_1$ is the $k=1$ harmonic diagonal},
    extending the $G_0$--$G_1$ link of Addendum~46 to a full
    rational generating function $F(z)$ with poles at $z = -2,-3,-4$.

  \item \textbf{Spectral mean identity}:
    $\Tr_{\mathrm{spec}}(L_{\Xsec})/27 = \mu_0/3$ exactly
    (Proposition~\ref{prop:spectral-trace}).

  \item \textbf{Conjectures~B and~D are not closed.}  The trace identity
    $\mu_0 = \Tr(\Xsec)$ is tautological.  OP-B survives unchanged.
\end{enumerate}

\begin{table}[ht]
\centering
\caption{Status of Conjectures~B and~D after Addenda~46--47.}
\label{tab:status}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{p{3.0cm}p{5.7cm}p{5.7cm}}
\toprule
 & \textbf{Conjecture~B} & \textbf{Conjecture~D} \\
\midrule
$\JO$ form
  & $m_t/m_c = \Tr(\Xsec\circ D_0) = \Tr(\Xsec)$
  & $E_d = \ln(\Tr(\Xsec\circ D_1))/\MU$ \\
Exact identity
  & $\Tr(\Xsec\circ D_0) = \mu_0$ (tautological)
  & $\Tr(\Xsec\circ D_1) = \mu_1$ (tautological) \\
Route~3 verdict
  & Spectral identification fails; generating function holds
  & Same \\
New structure
  & Harmonic family $D_k$; rational $F(z)$
  & $\MU = F(1)/F(0)$ Stieltjes ratio \\
Remaining input
  & $\Xsec$ as $G_2$-orbit canonical (not postulated)
  & $-\ln\MU/\MU$ as $\JO$ invariant \\
Status
  & OP-B open
  & OP-D open (reduces to OP-B) \\
\bottomrule
\end{tabular}
\end{table}

\begin{thebibliography}{99}

\bibitem{P36} L.~F.\ Vlegels, \emph{Addendum~36: Sector-Energy Mass Formula---Charged Leptons}, TOE Corpus, 2026.

\bibitem{P45} L.~F.\ Vlegels, \emph{Addendum~45: Quark Orbit Matching---$G_2$ Subgroup Actions on $\JO$ and Structural Classification of Conjectures A--E, I, J}, TOE Corpus, 2026.

\bibitem{P46} L.~F.\ Vlegels, \emph{Addendum~46: Conjectures~B and~D as $J_3(\mathbb{O})$ Trace Identities and the $G_0$--$G_1$ Link}, TOE Corpus, 2026.

\bibitem{McCrimmon} K.~McCrimmon, \emph{A Taste of Jordan Algebras}, Springer, 2004.

\bibitem{Faraut} J.~Faraut and A.~Kor\'{a}nyi, \emph{Analysis on Symmetric Cones}, Oxford University Press, 1994.

\bibitem{PDG24} R.~L.\ Workman et al.\ (Particle Data Group), \emph{Review of Particle Physics}, Prog.\ Theor.\ Exp.\ Phys.\ \textbf{2022}, 083C01 (2022), updated 2024.

\end{thebibliography}

\end{document}
