% Addendum 162: Correction to Addendum 87 — Second Spectral Moment M_2
%   Identifies and corrects two exponent errors in P87 Proposition 2.1.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

\documentclass[11pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{mathtools}
\usepackage{geometry}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\usepackage{booktabs}
\geometry{margin=1in}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{remark}[theorem]{Remark}

\newcommand{\muzero}{\mu_0}
\newcommand{\Tr}{\mathrm{Tr}}
\newcommand{\diag}{\mathrm{diag}}
\newcommand{\JO}{J_3(\mathbb{O})}

\title{Correction to Addendum~87: Exponent Errors in the Second Spectral Moment\\[0.5em]
\large Addendum~162 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle

\begin{abstract}
Addendum~87, Proposition~2.1 states the second spectral moment of the sector
density as $M_2 = (\pi^3 + \pi^5 + 16\pi^7)/\muzero \approx 354$.  Two exponent
errors are present: the term $x_2^2 w_2$ should be $\pi^6/\muzero$ (written as
$\pi^5/\muzero$) and the term $x_3^2 w_3$ should be $64\pi^9/\muzero$ (written as
$16\pi^7/\muzero$).  The correct second moment is
$M_2 = (\pi^3 + \pi^6 + 64\pi^9)/\muzero \approx 13\,929$, a factor of $\approx 39$
larger.  Downstream results in P87 that depend on $\sqrt{M_2}$ as a spectral energy
are correspondingly corrected.  The near-identity $E_H \approx \sqrt{\pi\,\Tr(X^2)/\muzero}$
established in P87 Section~5 is unaffected: it involves the separate functional
$\pi M_1^{(\text{disc})}$, not $M_2$.
\end{abstract}

%==========================================================================
\section{P87 setup and the erroneous Proposition~2.1}
\label{sec:setup}

Addendum~87 defines the sector operator $X_{\text{sector}} = \diag(\pi,\pi^2,4\pi^3)
\in \JO$ with eigenvalues $x_1 = \pi$, $x_2 = \pi^2$, $x_3 = 4\pi^3$, trace
$\muzero = \pi + \pi^2 + 4\pi^3 = \alpha^{-1} \approx 137.036$, and spectral weights
\begin{equation}\label{eq:weights}
  w_i = \frac{x_i}{\muzero},\qquad
  w_1 = \frac{\pi}{\muzero},\quad
  w_2 = \frac{\pi^2}{\muzero},\quad
  w_3 = \frac{4\pi^3}{\muzero}.
\end{equation}
The $k$-th moment is $M_k = \sum_{i=1}^3 x_i^k w_i$.  For $k=2$ the three
terms are $x_1^2 w_1$, $x_2^2 w_2$, $x_3^2 w_3$.  P87 Proposition~2.1 states
\begin{equation}\label{eq:wrong}
  M_2^{(\text{P87})} \;=\; \frac{\pi^3 \;+\; \pi^5 \;+\; 16\pi^7}{\muzero}
  \;\approx\; 354. \quad\text{\emph{(incorrect)}}
\end{equation}

%==========================================================================
\section{Identification of the two errors}
\label{sec:errors}

\begin{proposition}[Error in $x_2^2 w_2$]
\label{prop:err1}
With $x_2 = \pi^2$ and $w_2 = \pi^2/\muzero$:
\[
  x_2^2\,w_2 = (\pi^2)^2 \cdot \frac{\pi^2}{\muzero} = \frac{\pi^4 \cdot \pi^2}{\muzero}
             = \frac{\pi^6}{\muzero}.
\]
P87 Proposition~2.1 writes $\pi^5/\muzero$.  The exponent is $6$, not $5$.
\end{proposition}

\begin{proposition}[Error in $x_3^2 w_3$]
\label{prop:err2}
With $x_3 = 4\pi^3$ and $w_3 = 4\pi^3/\muzero$:
\[
  x_3^2\,w_3 = (4\pi^3)^2 \cdot \frac{4\pi^3}{\muzero}
             = \frac{16\pi^6 \cdot 4\pi^3}{\muzero}
             = \frac{64\pi^9}{\muzero}.
\]
P87 Proposition~2.1 writes $16\pi^7/\muzero$.  The correct coefficient is $64$
(not $16$) and the correct exponent is $9$ (not $7$).  The error conflates the
factor of $16\pi^6$ from $x_3^2$ with the weight $w_3 = 4\pi^3/\muzero$,
effectively using $w_3^{*} = \pi^3/\muzero$ instead of $4\pi^3/\muzero$.
\end{proposition}

%==========================================================================
\section{Corrected second moment}
\label{sec:correct}

\begin{theorem}[Correct $M_2$]
\label{thm:M2}
The second spectral moment of the sector density is
\begin{equation}\label{eq:correct}
  M_2 \;=\; x_1^2 w_1 + x_2^2 w_2 + x_3^2 w_3
       \;=\; \frac{\pi^3 + \pi^6 + 64\pi^9}{\muzero}.
\end{equation}
Numerically ($\pi^3 = 31.006$, $\pi^6 = 961.39$, $64\pi^9 = 64 \times 29\,809.1
= 1\,907\,780$, $\muzero = 137.036$):
\[
  M_2 \;=\; \frac{31.006 + 961.39 + 1\,907\,780}{137.036}
       \;\approx\; \frac{1\,908\,772}{137.036}
       \;\approx\; \mathbf{13\,929}.
\]
The standard deviation of the sector measure is $\sigma = \sqrt{M_2 - (M_1^{(\mathrm{disc})})^2}
\approx \sqrt{13\,929 - 12\,769} = \sqrt{1\,160} \approx 34.1$ spectral units.
\end{theorem}

\begin{corollary}[Factor-of-39 error]
The ratio of the correct value to the value claimed in P87 Proposition~2.1 is
\[
  \frac{M_2}{M_2^{(\mathrm{P87})}} \;=\; \frac{13\,929}{354} \;\approx\; 39.3.
\]
The P87 abstract and Proposition~2.1 understate $M_2$ by a factor of $\approx 39$.
\end{corollary}

\begin{center}
\fbox{$\displaystyle
  M_2 \;=\; \frac{\pi^3 + \pi^6 + 64\pi^9}{\muzero} \;\approx\; 13\,929
$}
\end{center}

%==========================================================================
\section{Downstream consequences}
\label{sec:downstream}

\paragraph{$\sqrt{M_2}$ as a spectral energy.}
P87 Section~4 uses $\sqrt{M_2^{(\mathrm{P87})}} \approx 18.84$ as a spectral
coordinate in the mass formula and finds a $\approx 0.33\%$ near-coincidence
with the Higgs coordinate $E_H \approx 18.78$.  With the correct $M_2 \approx 13\,929$,
the square root is $\sqrt{13\,929} \approx 118.0$, which inserted into the mass
formula yields a mass far above the Planck scale (P87 Proposition~4.1).  No
near-identification of $\sqrt{M_2}$ with a Standard Model scale survives.

\paragraph{Higgs near-identity (unaffected).}
The near-identity established in P87 Theorem~4.10,
\[
  E_H \;\approx\; \sqrt{\frac{\pi\,\Tr(X_{\mathrm{sector}}^2)}{\muzero}}
                = \sqrt{\pi\,M_1^{(\mathrm{disc})}}, \qquad \text{residual }0.33\%,
\]
involves the quantity $\pi M_1^{(\mathrm{disc})} = (\pi^3 + \pi^5 + 16\pi^7)/\muzero
\approx 355$, which P87 Section~5 correctly identifies as $\pi\,\Tr(X^2)/\muzero$
--- not the second moment $M_2$.  This near-identity is structurally correct and
is unchanged by the present correction.

\paragraph{Variance.}
The corrected variance is $\sigma^2 = M_2 - (M_1^{(\mathrm{disc})})^2
\approx 13\,929 - 12\,769 = 1\,160$, replacing the value $\approx 354.5$
stated in P87 Proposition~2.4 (which used the erroneous $M_2$).

%==========================================================================
\begin{thebibliography}{9}
\bibitem{A87}
L.\ F.\ Vlegels, \textit{Addendum~87: The Second Spectral Moment and
$\Tr(X_{\mathrm{sector}}^2)$}, TOE Corpus Addendum, May~2026.
\bibitem{A80}
L.\ F.\ Vlegels, \textit{Addendum~80: $\mathrm{MU}$ as the Spectral Mean},
TOE Corpus Addendum, May~2026.
\end{thebibliography}

\end{document}
