% 139_Addendum_CharmBottomHVP.tex
% Addendum P139: Charm, Bottom, and Continuum Contributions to the Hadronic
%   Vacuum Polarisation — Sector Decomposition of Delta_alpha_had(M_Z^2).
%   P136 (NWA) and P137 (GS lineshape) established the light-meson sum and
%   bracketed the Fermi target; J/psi and Upsilon contributions were left as
%   open items (PDG-anchored, not TOE-derived).
%   This addendum closes those open items: derives f_{J/psi}^2 from the J_3(O)
%   Peirce-weight formula and from PDG, quantifies and explains the factor-2.78
%   discrepancy (NLO QCD suppression of Gamma_ee at alpha_s(m_c)~0.39),
%   computes the Upsilon(1S,2S,3S) contributions in detail, estimates the
%   five-flavour pQCD continuum from the Upsilon to M_Z, and assembles a
%   complete sector decomposition showing that resonances account for ~20% and
%   the quark-level continuum ~80% of the PDG value 0.02750.
% Author: Leon Fernando Vlegels
% Copyright: Leon Fernando Vlegels. License: CC BY 4.0.
% Date: 2026-05-14

\documentclass[11pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{geometry}
\geometry{margin=2.5cm}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\usepackage{booktabs}
\usepackage{array}
\usepackage{xcolor}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}

\newcommand{\JO}{J_3(\mathbb{O})}
\newcommand{\Da}{\Delta\alpha}
\newcommand{\Dalep}{\Delta\alpha_{\rm lep}}
\newcommand{\Dahad}{\Delta\alpha_{\rm had}}
\newcommand{\Dahlm}{\Delta\alpha_{\rm had}^{\rm lm}}
\newcommand{\MZ}{M_Z}
\newcommand{\MW}{M_W}
\newcommand{\MU}{\mathrm{MU}}
\newcommand{\PDG}{\mathrm{PDG}}
\newcommand{\TOE}{\mathrm{TOE}}
\newcommand{\NWA}{\mathrm{NWA}}
\newcommand{\GS}{\mathrm{GS}}
\newcommand{\Dahrho}{\Delta\alpha_{\rm had}^\rho}

\title{\textbf{Addendum P139}\\[6pt]
\large Charm, Bottom, and Continuum Contributions\\
to the Hadronic Vacuum Polarisation:\\
Sector Decomposition of $\Delta\alpha_{\rm had}(M_Z^2)$}
\author{L\'eon Fernando Vlegels\\[4pt]
\small Independent Researcher \quad
\texttt{axiomofchoicepuzzle@gmail.com}}
\date{14 May 2026}

\begin{document}
\maketitle

\begin{abstract}
Addenda P136 and P137 derived the hadronic vacuum polarisation from the
$J_3(\mathbb{O})$ spectral masses, establishing that the three light vector
mesons ($\rho$, $\omega$, $\phi$) account for
$\Delta\alpha_{\rm had}^{\rm lm} \in [3.731,\,4.522]\times10^{-3}$ and bracket the Fermi route target
$0.00411$ from both sides.
Two sectors were flagged as open: the $J/\psi$ and $\Upsilon$ resonances, whose
PDG-anchored entries in the P136 table rested on the admission that the
$J_2$-sector Peirce-weight formula predicts $f_{J/\psi}^2 = 9\pi/\MU^2 = 44.9$,
a factor $2.78$ below the PDG value $124.5$.
This addendum closes those open items.
We show that the factor-$2.78$ discrepancy is the expected signature of
the NLO QCD annihilation correction
$\delta\Gamma_{ee}/\Gamma_{ee} = -16\alpha_s(m_c)/(3\pi) \approx -66\%$
at the charm scale where $\alpha_s(m_c)\approx 0.39$: the TOE Peirce-weight
formula gives the algebraically correct Born-level (quenched) result;
the physical leptonic width is suppressed by the large QCD dressing, which
drives $f_{J/\psi}^2 = M_{J/\psi}/(C\,\Gamma_{ee})$ upward by the reciprocal
factor.
We compute the $J/\psi$, $\psi(2S)$, and $\Upsilon(1S,2S,3S)$ dispersive
contributions using PDG decay constants, confirming the P136 table entries,
and we estimate the five-flavour perturbative QCD continuum from $M_\Upsilon$
to $M_Z$.
The resulting sector decomposition accounts for the full PDG value $0.02750$
within the pQCD systematic uncertainty, with resonances contributing
$\sim20\%$ and the quark-level continuum $\sim80\%$.
\end{abstract}

\tableofcontents
\vspace{1em}

%======================================================================
\section{Setup: the Residual Gap After P136 and P137}
\label{sec:setup}
%======================================================================

The hadronic contribution to the running of the fine-structure constant is
defined through the dispersive relation
\begin{equation}\label{eq:dispersion}
  \Dahad(M_Z^2) = \frac{\alpha}{3\pi}
  \int_{4m_\pi^2}^{\infty} \frac{R(s)}{s}\,\frac{M_Z^2}{M_Z^2 - s}\,ds,
  \quad
  R(s) = \frac{\sigma(e^+e^-\to\mathrm{hadrons})}{\sigma(e^+e^-\to\mu^+\mu^-)}.
\end{equation}
For a narrow resonance $V$, the NWA collapses equation~\eqref{eq:dispersion}
to the closed-form contribution
\begin{equation}\label{eq:NWA}
  \Delta\alpha_{\rm had}^V = \frac{4\pi\alpha\,\mathrm{Br}_{\rm had}}{f_V^2}
  \cdot \frac{M_Z^2}{M_Z^2 - M_V^2},
\end{equation}
where the VMD decay constant satisfies
$f_V^2 = 4\pi\alpha^2 M_V/(3\Gamma_{ee})$ (P108).

P136 applied equation~\eqref{eq:NWA} to the light neutral vector mesons with
fully-derived $\JO$ decay constants (P117--P119), finding
$\Delta\alpha_{\rm had}^{\rm lm,NWA} = 4.522\times10^{-3}$.
It also listed charmonium and bottomonium contributions, but using PDG $f_V^2$
values because the $J_2$-sector formula for charmonium gives
$f_{J/\psi}^2|_{\TOE} = 9\pi/\MU^2 = 44.9$, which is a factor $2.78$ below
the PDG value $124.5$ extracted from the measured leptonic width.
The full NWA resonance sum, including $J/\psi$, $\psi(2S)$, and
$\Upsilon(1S,2S,3S)$, is $\Dahad^{\rm res} = 5.523\times10^{-3}$.
P137 then replaced the $\rho$ NWA entry with the Gounaris-Sakurai lineshape
integral ($\Delta\alpha_{\rm had}^{\rho,\GS} = 2.943\times10^{-3}$), giving a GS light-meson
sum of $3.731\times10^{-3}$ and bracketing the Fermi target from below.

The PDG dispersive value is $\Dahad^{\PDG} = 0.02750$.
The gap between the full NWA resonance sum and the PDG total is
$0.02750 - 0.00552 = 0.02198$, or $79.9\%$ of the PDG total.
This addendum addresses the gap systematically: first by resolving the
$f_{J/\psi}^2$ discrepancy, then by assembling a sector-by-sector budget
including the perturbative QCD continuum.

%======================================================================
\section{The $J/\psi$ Decay Constant: TOE Prediction and QCD Dressing}
\label{sec:jpsi}
%======================================================================

\subsection{$f_{J/\psi}^2$ from PDG data}

The $J/\psi$ has $M_{J/\psi} = 3.097\,\mathrm{GeV}$, $\Gamma_{ee} = 5.55\,\mathrm{keV}$,
and $\Gamma_{\rm tot} = 92.9\,\mathrm{keV}$ (PDG 2024).
The hadronic branching ratio is
$\mathrm{Br}(J/\psi\to\mathrm{had}) = 1 - 2\times(5.55/92.9) = 0.881$.
By the VMD definition,
\begin{equation}\label{eq:fJpsi_PDG}
  f_{J/\psi}^2\!\bigl|_{\PDG}
  = \frac{4\pi\alpha^2 M_{J/\psi}}{3\,\Gamma_{ee}}
  = \frac{4\pi\times(1/137.036)^2\times 3097\,\mathrm{MeV}}
         {3\times 5.55\times10^{-3}\,\mathrm{MeV}}
  = 124.5,
\end{equation}
consistent with the P136 tabulated value $124.916$.
Substituting into equation~\eqref{eq:NWA} gives
\begin{equation}\label{eq:DaJpsi}
  \Delta\alpha_{\rm had}^{J/\psi}\!\bigl|_{\PDG}
  = \frac{4\pi\alpha\times0.881}{124.5}
  = \frac{0.09166\times0.881}{124.5}
  = 6.47\times10^{-4},
\end{equation}
reproducing the P136 entry.
The kinematic factor $M_Z^2/(M_Z^2 - M_{J/\psi}^2) = 1.00115$ contributes
less than $0.12\%$ and is negligible.

\subsection{$f_{J/\psi}^2$ from the $\JO$ Peirce-weight formula}

The $J_3(\mathbb{O})$ Peirce decomposition assigns a weight $\MU = \mu_1/\mu_0 = 0.79334$
to the $J_2$-sector (P117).
For the $\phi(1020)$, a $s\bar s$ state with $|Q_s| = 1/3$, P117 found
$f_\phi^2 = 36\pi/\MU^2 = 179.7$.
Since the VMD leptonic width scales as $\Gamma_{ee} \propto Q_q^2$, the decay
constant satisfies $f_V^2 \propto M_V/\Gamma_{ee} \propto 1/Q_q^2$.
The $J/\psi$ carries charm-quark charge $Q_c = +2/3$, giving $Q_c^2 = 4/9$,
compared to $Q_s^2 = 1/9$ for the strange quark.
The $J_2$-sector formula for charmonium is therefore
\begin{equation}\label{eq:fJpsi_TOE}
  f_{J/\psi}^2\!\bigl|_{\TOE}
  = \frac{Q_s^2}{Q_c^2}\,f_\phi^2
  = \frac{1/9}{4/9}\times\frac{36\pi}{\MU^2}
  = \frac{9\pi}{\MU^2}
  = \frac{9\pi}{(0.79334)^2}
  = 44.9.
\end{equation}
The ratio to the PDG value is $124.5/44.9 = 2.77$, reproducing the P136
flag to within rounding.

\subsection{Structural origin: NLO QCD suppression of $\Gamma_{ee}$}

The Peirce-weight formula~\eqref{eq:fJpsi_TOE} is the algebraically
exact Born-level prediction for a ${}^3S_1$ $c\bar c$ state embedded in the
$J_2$-sector conformal geometry.
It contains no QCD loop corrections.
The leptonic annihilation of a vector quarkonium state receives a
leading radiative correction (van Royen-Weisskopf formula at NLO in QCD):
\begin{equation}\label{eq:NLO}
  \Gamma_{ee}^{\rm NLO}
  = \Gamma_{ee}^{\rm LO}\left(1 - \frac{16\alpha_s(m_c)}{3\pi}
    + O(\alpha_s^2)\right).
\end{equation}
The charm mass from the TOE (P84, Proposition~1) is $m_c = 1287\,\mathrm{MeV}$,
within $1.3\%$ of the PDG value $1270\,\mathrm{MeV}$.
Running the P106 value $\alpha_s(M_Z) = 0.1186$ down to the charm scale using
three-loop QCD gives $\alpha_s(m_c) \approx 0.39$.
Substituting into equation~\eqref{eq:NLO}:
\begin{equation}
  \frac{16\alpha_s(m_c)}{3\pi} \approx \frac{16\times0.39}{9.425} = 0.663.
\end{equation}
The NLO correction is $-66\%$ of the leading-order leptonic width.
Since $f_{J/\psi}^2 \propto 1/\Gamma_{ee}$, this suppression of $\Gamma_{ee}$
induces an enhancement of $f_{J/\psi}^2$:
\begin{equation}\label{eq:enhancement}
  \frac{f_{J/\psi}^2|_{\rm physical}}{f_{J/\psi}^2|_{\TOE}}
  \approx \frac{1}{1 - 16\alpha_s(m_c)/(3\pi)}
  = \frac{1}{1 - 0.663} = 2.97.
\end{equation}
This is consistent with the empirical ratio $2.77$: the $\sim7\%$ residual is
within the $O(\alpha_s^2)$ corrections and the relativistic $v^2/c^2 \sim 0.25$
corrections to the non-relativistic wavefunction at the origin.
The qualitative conclusion is unambiguous.
The factor-2.78 tension is not a failure of the $\JO$ geometric formula.
It is the expected consequence of a strong coupling $\alpha_s(m_c) \approx 0.39$
that is large enough to make the NLO correction comparable to the LO result.
The TOE Peirce-weight formula gives the correct conformal skeleton; QCD dressing
at the charm scale inflates the physical $f_{J/\psi}^2$ by the observed factor.

A self-consistency check comes from the $\Upsilon$.
At the bottom scale $\alpha_s(m_b) \approx 0.22$, so $16\alpha_s/(3\pi) \approx
0.37$, predicting a correction factor $\sim 1/(1-0.37) \approx 1.59$.
One therefore expects the third-generation Peirce-weight formula (when derived)
to underpredict $f_\Upsilon^2$ by a factor $\sim 1.6$ rather than $\sim 2.8$,
because $\alpha_s$ is smaller at the bottom scale.
This is a testable structural prediction of the TOE.

The practical consequence for the HVP computation is clear.
Because PDG $\Gamma_{ee}$ values are used as input to equation~\eqref{eq:NWA},
the charmonium and bottomonium entries in the P136 table are already
physically correct.
The factor-2.78 discrepancy lives in the wavefunction and does not propagate
into $\Dahad$ when empirical leptonic widths are the anchor.

%======================================================================
\section{$\Upsilon$ Contributions}
\label{sec:upsilon}
%======================================================================

The three narrow $\Upsilon$ resonances contribute through equation~\eqref{eq:NWA}
with PDG leptonic widths.
For $\Upsilon(1S)$: $M_\Upsilon = 9.460\,\mathrm{GeV}$,
$\Gamma_{ee} = 1.340\,\mathrm{keV}$, $\Gamma_{\rm tot} = 54.02\,\mathrm{keV}$,
$\mathrm{Br}_{\rm had} = 1 - 2\times(1.340/54.02) = 0.950$ (using equal
leptonic partial widths approximation; PDG gives $0.965$ when muonic and
tauonic modes are separately accounted).
Using the PDG hadronic branching ratio directly,
\begin{equation}
  f_{\Upsilon(1S)}^2\!\bigl|_{\PDG}
  = \frac{4\pi\alpha^2\times 9460\,\mathrm{MeV}}
         {3\times 1.340\times10^{-3}\,\mathrm{MeV}}
  = 1575,
\end{equation}
and the NWA contribution is
\begin{equation}
  \Delta\alpha_{\rm had}^{\Upsilon(1S)}
  = \frac{4\pi\alpha\times0.965}{1575}
  = \frac{0.09166\times0.965}{1575}
  = 5.62\times10^{-5}.
\end{equation}
Table~\ref{tab:upsilon} lists all three states.

\begin{table}[h]
\centering
\caption{$\Upsilon(1S,2S,3S)$ contributions to $\Dahad(M_Z^2)$ via
equation~\eqref{eq:NWA} with PDG inputs.
$\mathrm{Br}_{\rm had}$ from PDG 2024.
The kinematic factor $M_Z^2/(M_Z^2-M_V^2)$ differs from unity by between
$1.1\%$ and $1.4\%$ across the three states, negligible at the precision of
the NWA, and is absorbed into the quoted values.}
\label{tab:upsilon}
\begin{tabular}{lrrrrc}
\toprule
State & $M_V$ (GeV) & $\Gamma_{ee}$ (keV) & $f_V^2$ &
$\mathrm{Br}_{\rm had}$ & $\Delta\alpha_{\rm had}^V$ \\
\midrule
$\Upsilon(1S)$ & $9.460$ & $1.340$ & $1575$ & $0.965$ & $5.62\times10^{-5}$ \\
$\Upsilon(2S)$ & $10.023$ & $0.612$ & $3652$ & $0.962$ & $2.41\times10^{-5}$ \\
$\Upsilon(3S)$ & $10.355$ & $0.443$ & $5211$ & $0.957$ & $1.68\times10^{-5}$ \\
\midrule
\multicolumn{5}{l}{\textit{$\Upsilon$ sum}} & $\mathbf{9.71\times10^{-5}}$ \\
\bottomrule
\end{tabular}
\end{table}

The $\Upsilon$ contributions are strongly suppressed by the larger $f_V^2$
values at the bottom scale: the Upsilon tower contributes $9.71\times10^{-5}$,
approximately $1.8\%$ of the $J/\psi$ contribution ($6.47\times10^{-4}$) and
$0.35\%$ of the PDG $\Dahad$.
The three-resonance sum agrees with the P136 table entries to within rounding,
confirming that those PDG-anchored values are correct.

%======================================================================
\section{Perturbative QCD Continuum Above the $\Upsilon$}
\label{sec:continuum}
%======================================================================

\subsection{The five-flavour region ($M_\Upsilon$ to $M_Z$)}

Above the open-bottom threshold and above the $\Upsilon$ resonances, the
hadronic $R(s)$ is well described by five-flavour perturbative QCD,
\begin{equation}
  R_5(s) = N_c\sum_{q=u,d,s,c,b} Q_q^2
    \left(1 + \frac{\alpha_s(s)}{\pi} + \cdots\right)
  = \frac{11}{3}\left(1 + \frac{\alpha_s(s)}{\pi} + \cdots\right).
\end{equation}
The charge sum $\sum Q_q^2 = 4/9+1/9+1/9+4/9+1/9 = 11/9$ with $N_c = 3$.
With $\bar\alpha_s \approx 0.15$ averaged over $\sqrt{s}\in[M_\Upsilon, M_Z]$
(running from $\alpha_s(m_b)\approx0.22$ to $\alpha_s(M_Z) = 0.1186$ from P106),
the NLO-improved ratio is $R_5^{\rm NLO} \approx 3.667\times1.048 = 3.84$.

Approximating $M_Z^2/(M_Z^2-s) \approx 1$ over the range
$\sqrt{s}\in[M_\Upsilon, 0.95\,M_Z]$ where the approximation holds to better
than $50\%$ (the Z-pole itself is excluded from the hadronic dispersive integral),
\begin{equation}\label{eq:Da_cont_5f}
  \Delta\alpha_{\rm had}^{{\rm cont},5f}
  \approx \frac{\alpha}{3\pi}\,R_5^{\rm NLO}\,\ln\frac{M_Z^2}{M_\Upsilon^2}.
\end{equation}
With $\alpha/(3\pi) = 7.74\times10^{-4}$ and
$\ln(M_Z^2/M_\Upsilon^2) = 2\ln(91.19/9.460) = 4.532$:
\begin{equation}\label{eq:Da_cont_5f_result}
  \Delta\alpha_{\rm had}^{{\rm cont},5f}
  \approx 7.74\times10^{-4}\times3.84\times4.532
  = 1.35\times10^{-2}.
\end{equation}
Including the kinematic enhancement from $M_Z^2/(M_Z^2-s)$ (which grows from
$1.01$ at $\sqrt{s} = M_\Upsilon$ to $\sim3$ at $\sqrt{s} = 0.95\,M_Z$,
averaging $\sim1.1$ over most of the range), the corrected estimate is
$\sim1.49\times10^{-2}$, with a systematic uncertainty of $\sim20\%$ from
the ambiguity in the upper cutoff and the pQCD matching scale.

\subsection{Sub-$\Upsilon$ continuum sectors}

Between the $J/\psi$ family and the $\Upsilon$ (approximately $3.7$--$10.5\,\mathrm{GeV}$),
open-charm and open-bottom meson pair production contributes through
the four-flavour $R$ ratio.
With $R_4 = (10/3)(1+\alpha_s/\pi)\approx3.57$ at $\bar\alpha_s\approx0.22$:
\begin{equation}
  \Delta\alpha_{\rm had}^{{\rm cont},4f}
  \approx \frac{\alpha}{3\pi}\times3.57\times\ln\frac{(10.5\,\mathrm{GeV})^2}
  {(3.7\,\mathrm{GeV})^2}
  = 7.74\times10^{-4}\times3.57\times1.878
  \approx 5.2\times10^{-3}.
\end{equation}
Below the open-charm threshold, from the end of the $\rho$-dominated region
($\sim1\,\mathrm{GeV}$) to the $J/\psi$, the three-flavour continuum
($R_3 = 2$, light quarks $u,d,s$) contributes:
\begin{equation}
  \Delta\alpha_{\rm had}^{{\rm cont},3f}
  \approx \frac{\alpha}{3\pi}\times2\times\ln\frac{(3.7\,\mathrm{GeV})^2}
  {(1\,\mathrm{GeV})^2}
  = 7.74\times10^{-4}\times2\times2.634
  \approx 4.1\times10^{-3}.
\end{equation}
These pQCD estimates carry $\sim25\%$ systematic uncertainty: neither region
is purely perturbative (the open-charm threshold has significant $D$-meson
resonance structure; the $1$--$2\,\mathrm{GeV}$ range is where data-driven
and pQCD methods diverge most).
Nevertheless, both estimates are consistent in sign and order of magnitude
with the data-driven dispersive calculation used by the PDG.

%======================================================================
\section{Sector Decomposition and Comparison to PDG}
\label{sec:decomposition}
%======================================================================

Table~\ref{tab:budget} assembles the complete $\Dahad(M_Z^2)$ budget.

\begin{table}[h]
\centering
\caption{Sector decomposition of $\Dahad(M_Z^2)$.
Resonance entries use equation~\eqref{eq:NWA} (NWA).
Light-meson $f_V^2$ are TOE-native (P117--P119); heavy-quarkonium $f_V^2$
are PDG-anchored (see Section~\ref{sec:jpsi}).
Continuum entries are pQCD estimates with $\sim20$--$25\%$ systematic uncertainty.}
\label{tab:budget}
\begin{tabular}{llcc}
\toprule
Sector & Source & Contribution & Fraction of PDG \\
\midrule
$\rho(770)$ NWA         & TOE (P119) & $3.735\times10^{-3}$ & $13.6\%$ \\
$\omega(782)$ NWA       & TOE (P118) & $2.810\times10^{-4}$ & $1.0\%$ \\
$\phi(1020)$ NWA        & TOE (P117) & $5.073\times10^{-4}$ & $1.8\%$ \\
$J/\psi$ NWA            & PDG-anchored & $6.47\times10^{-4}$ & $2.4\%$ \\
$\psi(2S)$ NWA          & PDG-anchored & $2.55\times10^{-4}$ & $0.9\%$ \\
$\Upsilon(1S,2S,3S)$ NWA & PDG-anchored & $9.71\times10^{-5}$ & $0.4\%$ \\
\midrule
\textit{Resonance subtotal} & & $\mathbf{5.52\times10^{-3}}$ & $\mathbf{20.1\%}$ \\
\midrule
3f continuum ($1$--$3.7$ GeV) & pQCD ($R_3\approx2$) &
  $\sim4.1\times10^{-3}$ & $\sim14.9\%$ \\
4f continuum ($3.7$--$10.5$ GeV) & pQCD ($R_4\approx3.57$) &
  $\sim5.2\times10^{-3}$ & $\sim18.9\%$ \\
5f continuum ($M_\Upsilon$--$M_Z$) & pQCD ($R_5\approx3.84$) &
  $\sim1.49\times10^{-2}$ & $\sim54.2\%$ \\
\midrule
\textit{Continuum subtotal} & & $\sim2.42\times10^{-2}$ & $\sim88.0\%$ \\
\midrule
\textbf{Total} & & $\sim2.97\times10^{-2}$ & $\sim108\%$ \\
\textbf{PDG} & Data-driven dispersion & $0.02750$ & $100\%$ \\
\bottomrule
\end{tabular}
\end{table}

The total $2.97\times10^{-2}$ exceeds the PDG value by $\sim8\%$, well within
the combined systematic uncertainties of the pQCD continuum estimates
($\sim20$--$25\%$ per sector).
The sector decomposition confirms the structural picture established in P136:
narrow resonances account for $\sim20\%$ of $\Dahad$, with the $\rho$ alone
responsible for $13.6\%$; the quark-level continuum supplies the remaining
$\sim80\%$.
This is entirely consistent with the $\JO$ discrete spectrum being a complete
description of the resonance sector and having no autonomous continuum degrees
of freedom between the spectral eigenvalues.

The separation between TOE-native and PDG-anchored entries is also instructive.
The three light-meson contributions are fully derived from $\JO$ first principles.
The heavy-quarkonium entries are PDG-anchored because the TOE Born-level formula
underpredicts $f_{J/\psi}^2$ by the QCD dressing factor $\sim2.78$, which is
external to the algebraic structure.
The continuum contributions are pQCD estimates derived from the colour group
structure and the running coupling, neither of which requires external data
but which depend on the perturbative QCD framework rather than the $\JO$
spectral geometry directly.
Closing all three of these dependencies from first principles is the long-range
programme; the present addendum fixes the logical status of each.

%======================================================================
\section{Conclusion}
\label{sec:conclusion}
%======================================================================

The two open items flagged by P136 and P137 are resolved.

The factor-$2.78$ discrepancy between the TOE Peirce-weight prediction
$f_{J/\psi}^2|_{\TOE} = 44.9$ and the PDG value $124.5$ is the expected
consequence of NLO QCD suppression of the leptonic width at the charm scale.
The formula $\delta\Gamma_{ee}/\Gamma_{ee} = -16\alpha_s(m_c)/(3\pi) \approx -66\%$
at $\alpha_s(m_c)\approx0.39$ predicts an enhancement of $f_{J/\psi}^2$ by
a factor $\sim2.97$, consistent with the observed $2.78$ within $O(\alpha_s^2)$.
The TOE Peirce-weight formula is the Born-level (quenched) result; the QCD
dressing is external.
Because PDG $\Gamma_{ee}$ values are used as input to the dispersive formula,
the charmonium and bottomonium entries in the P136 table are already correct:
the factor-2.78 discrepancy does not propagate into $\Dahad$.

The $\Upsilon(1S,2S,3S)$ contributions ($9.71\times10^{-5}$ total) are small
relative to the charmonium sector and are confirmed against the P136 table.
The five-flavour pQCD continuum from $M_\Upsilon$ to $M_Z$ contributes
$\sim1.49\times10^{-2}$, more than twice the entire resonance sum, and is the
dominant source of the gap between the NWA resonance total ($5.52\times10^{-3}$)
and the PDG value ($0.0275$).
Together with the sub-$\Upsilon$ continuum sectors, the complete budget accounts
for the PDG total within the pQCD systematic uncertainty.

The Fermi route to $M_W$ uses the discrete $\JO$ resonance sector
($\Dahad\approx0.004$--$0.005$), not the full PDG dispersive value, because
the $\JO$ spectral structure is discrete.
The quark-level continuum, which carries $\sim80\%$ of the PDG integral,
enters the electroweak sector through a channel consistent with the $E_6$
off-diagonal Peirce structure, rather than additively through the photon
polarisation appearing in the Fermi constraint.
Deriving that pathway from first principles remains the central open item in
the HVP programme.

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

\paragraph{1. TOE-native $f_{J/\psi}^2$ with QCD dressing.}
The Born-level formula $9\pi/\MU^2 = 44.9$ is structurally complete within
the $J_2$-sector geometry.
A TOE-native derivation of the NLO correction requires connecting the
$\JO$ strong-coupling structure (P106 derives $\alpha_s(M_Z) = 0.1186$) to the
renormalisation-group flow of the charmonium short-distance coefficient.
This is a two-step derivation: the $J_2$-sector one-loop coefficient in the
NRQCD factorisation formula, and the three-loop RG running from $M_Z$ to $m_c$.
Both steps are tractable within the corpus.

\paragraph{2. Third-generation Peirce weights for $\Upsilon$.}
The $\phi$ and $J/\psi$ share $\MU = 0.79334$ because both are $J_2$-sector
states.
The $\Upsilon$ requires the third-generation Peirce weight $\MU_3 = \mu_2/\mu_0$,
which has not been derived.
Computing $\MU_3$ from $\JO$ spectral data would yield a TOE-native $f_\Upsilon^2$
prediction and test whether the discrepancy factor follows the predicted
$\alpha_s$-scaling from Section~\ref{sec:jpsi} ($\sim1.6$ at the bottom scale
vs.\ $\sim2.8$ at the charm scale).

\paragraph{3. Continuum derivation from the $E_6$ exterior structure.}
The $\sim80\%$ of $\Dahad$ from the quark-level continuum has no counterpart
in the $\JO$ discrete spectrum.
P136 proposed that this continuum enters electroweak observables through the
$E_6$ off-diagonal Peirce blocks $P_{12}, P_{13}, P_{23}$.
A positive derivation --- showing from the $E_6$ supergravity sector that
multiparticle hadronic production does not contribute additively to the photon
polarisation in the Fermi constraint --- is the principal open derivation in
the entire HVP programme.

\paragraph{4. Sub-threshold pion form factor.}
As noted in P137, the GS lineshape understates $R(s)$ below
$\sqrt{s}\approx0.6\,\mathrm{GeV}$.
A TOE-native derivation of the charged-pion form factor from the $\JO$ charged
pion sector (P113) would supply the Omnès factor geometrically and close the
$\rho$-sub-threshold gap without external data input.

\end{document}
