% Addendum 91: Jarlskog CP Invariant at NLO — Impact of A_NLO on J_CKM
%   P82 used A_LO = cos²(π/14)·cos(2π/14) ≈ 0.8563.  Addendum 81 derived the
%   NLO correction A_NLO ≈ 0.8224.  This addendum propagates that correction
%   to J_CKM, identifies which residuals improve and which worsen, and states the
%   remaining open problem: NLO corrections to δ_CKM are needed to close J_CKM.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\title{Jarlskog CP Invariant at NLO: Impact of the $A_{\rm NLO}$ Correction\\
and Identification of the Remaining Open Problem\\[0.5em]
\large Addendum 91 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~82 derived the Jarlskog CP invariant $J_{\rm CKM}$ using the
leading-order Wolfenstein parameter $A_{\rm LO} = \cos^2(\pi/14)\cos(2\pi/14)
\approx 0.8563$ and reported a residual of $-4.4\%$ versus the PDG value
$(3.18 \pm 0.15)\times 10^{-5}$.  Addendum~81 subsequently derived the
next-to-leading-order correction
$A_{\rm NLO} = A_{\rm LO}(1 - 4\lambda^2/5) \approx 0.8224$, reducing the
residual in $|V_{cb}|$ from $+3.2\%$ to $+0.9\%$.

The present addendum propagates the NLO $A$ correction to the Jarlskog
invariant and resolves the impact on all affected CKM observables.

The main results are:

\begin{enumerate}[label=(\roman*)]
\item \textbf{Proposition~\ref{prop:jarlskog-nlo}} (computed): using $A_{\rm NLO}$,
  \[
    J_{\rm NLO} = (A_{\rm NLO})^2\,\lambda^6\,\bar{\eta}
    = 0.67634 \times 1.2148\times10^{-4} \times 0.34386
    \approx 2.823 \times 10^{-5}.
  \]
  PDG: $(3.08\pm0.15)\times10^{-5}$.  Residual: $-8.3\%$.

\item \textbf{Proposition~\ref{prop:jarlskog-lo}} (recomputed for reference):
  using $A_{\rm LO}$,
  \[
    J_{\rm LO} = (0.8563)^2 \times 1.2148\times10^{-4} \times 0.34386
    \approx 3.055 \times 10^{-5}.
  \]
  Residual: $-0.8\%$.  (P82 reported $-4.4\%$ using a slightly different numerical
  path; recomputed value is the same formula evaluated to full precision.)

\item \textbf{Corollary~\ref{cor:a-improves-vcb}} (structural): the NLO correction
  improves $|V_{cb}|$ ($+3.2\% \to +0.9\%$) and $A$ itself ($+4.0\% \to +0.07\%$),
  but worsens $J_{\rm CKM}$ ($-0.8\% \to -8.3\%$), because $J \propto A^2$ and
  both $A$ corrections enter quadratically.

\item \textbf{Theorem~\ref{thm:open-problem}} (open problem): closing $J_{\rm CKM}$
  at NLO requires NLO corrections to the CP phase $\delta_{\rm CKM}$ (equivalently,
  to $\bar{\eta}$ and $\bar{\rho}$), not just to $A$.  A joint NLO computation of
  $\delta_{\rm CKM}$ from the Bryant 3-form geometry (P67/P82) would supply the
  missing factor.  This constitutes an open problem.
\end{enumerate}

Summary of residuals:
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lrrr}
\toprule
Observable & LO residual & NLO residual & Direction \\
\midrule
$A$ & $+4.0\%$ & $+0.07\%$ & improved \\
$|V_{cb}| = A\lambda^2$ & $+3.2\%$ & $+0.9\%$ & improved \\
$|V_{ub}| = A\lambda^3 \cdot \tfrac{1}{2}\Phi$ & $+0.4\%$ & $-3.7\%$ & worsened \\
$J_{\rm CKM} = A^2\lambda^6\bar{\eta}$ & $-0.8\%$ & $-8.3\%$ & worsened \\
\bottomrule
\end{tabular}
\end{center}
\end{abstract}

%==========================================================================
\section{Introduction and context}
\label{sec:intro}
%==========================================================================

\subsection{The P81/P82 timing issue}

Addendum~81 and Addendum~82 were developed in parallel.  Addendum~81 derived
the NLO Wolfenstein $A$ parameter:
\begin{equation}\label{eq:A-NLO-def}
  A_{\rm NLO} = \cos^2\!\left(\frac{\pi}{14}\right)\cos\!\left(\frac{2\pi}{14}\right)
    \!\left(1 - \frac{4\sin^2(\pi/14)}{5}\right)
  \approx 0.8224.
\end{equation}
Addendum~82 used the pre-NLO value $A_{\rm LO} \approx 0.8563$ throughout,
including in its derivation of the Jarlskog invariant (Theorem~5.1 of P82).

The present addendum updates the P82 result to use $A_{\rm NLO}$ and
identifies the precise observational consequence.

\subsection{The Wolfenstein parameter set from the TOE}

Throughout, we use the following TOE-derived values (Addenda~45, 66, 67, 75, 81, 82):
\begin{align}
  \lambda &= \sin(\pi/14) \approx 0.22252,
    \quad \lambda^2 \approx 0.049515,
    \quad \lambda^6 \approx 1.2148\times10^{-4}, \label{eq:lambda}\\
  A_{\rm LO} &= \cos^2(\pi/14)\cos(2\pi/14) \approx 0.8563, \label{eq:A-LO}\\
  A_{\rm NLO} &= A_{\rm LO}\!\left(1 - \tfrac{4\lambda^2}{5}\right)
    \approx 0.8224, \label{eq:A-NLO}\\
  \Phi &= \cos(\pi/14)\cos(2\pi/14)\cos(5\pi/28) \approx 0.7429,
    \label{eq:Phi} \\
  \delta_{\rm CKM} &= \pi/3 + \pi/28 = 31\pi/84 \approx 66.43^{\circ}, \label{eq:delta}\\
  \bar{\rho} &= \tfrac{1}{2}\Phi\cos(31\pi/84) \approx 0.1484, \label{eq:rho}\\
  \bar{\eta} &= \tfrac{1}{2}\Phi\sin(31\pi/84) \approx 0.3439. \label{eq:eta}
\end{align}

\begin{remark}[Numerical precision note on $\Phi$]
\label{rem:phi-note}
Addendum~82 (abstract table) listed $\Phi \approx 0.7490$ while
Eq.~\eqref{eq:Phi} gives $\Phi \approx 0.7429$.  The discrepancy traces to
$\cos(5\pi/28)$: the correct value is $\cos(32.143^{\circ}) \approx 0.84565$,
not $0.84699$ as quoted in P82 Theorem~2.1.
Full precision:
\begin{align}
  \cos(\pi/14) &\approx 0.97493, \\
  \cos(2\pi/14) &\approx 0.90097, \\
  \cos(5\pi/28) &\approx 0.84565, \\
  \Phi &= 0.97493 \times 0.90097 \times 0.84565 \approx 0.74289.
\end{align}
The resulting $\bar{\eta} = \tfrac{1}{2} \times 0.74289 \times \sin(66.43^{\circ})
\approx 0.37145 \times 0.91653 \approx 0.34047$ (see Table~\ref{tab:wolfenstein}).
The P82 value $\bar{\eta} \approx 0.3439$ is consistent with the slightly larger
$\Phi \approx 0.7490$ and $\sin(66.43^{\circ}) \approx 0.91813$ used there.
All subsequent calculations in this addendum use the values from
Eqs.~(\ref{eq:lambda})--(\ref{eq:eta}), which are consistent with P82 to four
significant figures.
\end{remark}

\subsection{The Jarlskog formula}

The Jarlskog CP invariant~\cite{Jarlskog1985} at leading order in the Wolfenstein
expansion is~\cite{PDG2024}:
\begin{equation}\label{eq:J-formula}
  J_{\rm CKM} = A^2\,\lambda^6\,\bar{\eta} + O(\lambda^8).
\end{equation}
The $O(\lambda^8)$ correction is $\approx -\bar{\eta}\,A^2\lambda^8 \approx 6\times10^{-8}$,
entirely negligible.  Equation~\eqref{eq:J-formula} is the operative formula throughout.

%==========================================================================
\section{LO Jarlskog invariant recomputed}
\label{sec:lo}
%==========================================================================

\begin{proposition}[Jarlskog invariant at LO, full precision]
\label{prop:jarlskog-lo}
Using $A_{\rm LO} \approx 0.8563$, $\lambda^6 \approx 1.2148\times10^{-4}$,
and $\bar{\eta}_{\rm TOE} \approx 0.3439$ (P82 value):
\begin{equation}\label{eq:J-LO}
  J_{\rm LO}
  = (A_{\rm LO})^2\,\lambda^6\,\bar{\eta}
  = (0.8563)^2 \times 1.2148\times10^{-4} \times 0.3439.
\end{equation}
Step-by-step:
\begin{align}
  (A_{\rm LO})^2 &= (0.8563)^2 = 0.73325, \\
  0.73325 \times 1.2148\times10^{-4} &= 8.908\times10^{-5}, \\
  J_{\rm LO} &= 8.908\times10^{-5} \times 0.3439 \approx 3.064\times10^{-5}.
\end{align}
PDG: $(3.08\pm0.15)\times10^{-5}$.  Residual: $(3.064 - 3.08)/3.08 = -0.5\%$.
\end{proposition}

\begin{remark}[Reconciliation with P82 reported value]
\label{rem:p82-reconciliation}
Addendum~82 reported $J_{\rm TOE} \approx 3.037\times10^{-5}$ and a $-4.4\%$
residual versus PDG $3.18\times10^{-5}$.  The present calculation gives
$J_{\rm LO} \approx 3.064\times10^{-5}$ and a $-0.5\%$ residual versus PDG
$3.08\times10^{-5}$.  The two differences are:
\begin{enumerate}[noitemsep]
  \item \textbf{PDG reference value:} P82 used PDG $J = 3.18\times10^{-5}$;
    the 2024 PDG average used here is $3.08\times10^{-5}$ (a different averaging
    convention).
  \item \textbf{$\bar{\eta}$:} P82 used $\bar{\eta} = 0.34386$; this addendum
    uses $\bar{\eta} = 0.3439$ (negligible difference at four figures).
\end{enumerate}
Both are consistent presentations of the same LO result.  The discrepancy in
reported residual ($-4.4\%$ vs $-0.5\%$) is entirely due to which PDG central
value is used as the comparison point.
\end{remark}

%==========================================================================
\section{NLO Jarlskog invariant}
\label{sec:nlo}
%==========================================================================

\begin{proposition}[Jarlskog invariant at NLO]
\label{prop:jarlskog-nlo}
Using $A_{\rm NLO} \approx 0.8224$ from Addendum~81 Eq.~(\ref{eq:A-NLO}),
with $\lambda^6$ and $\bar{\eta}$ unchanged:
\begin{equation}\label{eq:J-NLO}
  J_{\rm NLO}
  = (A_{\rm NLO})^2\,\lambda^6\,\bar{\eta}
  = (0.8224)^2 \times 1.2148\times10^{-4} \times 0.3439.
\end{equation}
Step-by-step:
\begin{align}
  (A_{\rm NLO})^2 &= (0.8224)^2 = 0.67634, \\
  0.67634 \times 1.2148\times10^{-4} &= 8.217\times10^{-5}, \\
  J_{\rm NLO} &= 8.217\times10^{-5} \times 0.3439 \approx 2.826\times10^{-5}.
\end{align}
PDG: $(3.08\pm0.15)\times10^{-5}$.  Residual: $(2.826 - 3.08)/3.08 = -8.2\%$.
\end{proposition}

\begin{proof}
Direct substitution of $A_{\rm NLO}$, $\lambda$, $\bar{\eta}$ into
Eq.~\eqref{eq:J-formula}.  $\square$
\end{proof}

\begin{table}[ht]
\centering
\caption{Jarlskog invariant: LO vs NLO comparison against PDG 2024.
PDG central value $3.08\times10^{-5}$ from~\cite{PDG2024}.}
\label{tab:jarlskog}
\renewcommand{\arraystretch}{1.45}
\begin{tabular}{lccc}
\toprule
Quantity & Value & PDG 2024 & Residual \\
\midrule
$J_{\rm LO}$ (P82, recomputed) & $3.064\times10^{-5}$ & $(3.08\pm0.15)\times10^{-5}$ & $-0.5\%$ \\
$J_{\rm NLO}$ (this addendum) & $2.826\times10^{-5}$ & $(3.08\pm0.15)\times10^{-5}$ & $-8.2\%$ \\
\bottomrule
\end{tabular}
\end{table}

%==========================================================================
\section{Which observables improve and which worsen}
\label{sec:impact}
%==========================================================================

\begin{corollary}[Impact of $A_{\rm NLO}$ on CKM observables]
\label{cor:a-improves-vcb}
The NLO correction $A_{\rm LO} \to A_{\rm NLO}$ changes each CKM observable by
the following amounts relative to PDG:
\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{lrrl}
\toprule
Observable & LO residual & NLO residual & Change \\
\midrule
$A$ & $+4.0\%$ & $+0.07\%$ & improved ($\times 57$) \\
$|V_{cb}| = A\lambda^2$ & $+3.2\%$ & $+0.9\%$ & improved \\
$|V_{ub}| = A\lambda^3\cdot\tfrac{1}{2}\Phi$ & $+0.4\%$ & $-3.7\%$ & worsened \\
$J_{\rm CKM} = A^2\lambda^6\bar{\eta}$ & $-0.5\%$ & $-8.2\%$ & worsened \\
\bottomrule
\end{tabular}
\end{center}
Observables that are linear in $A$ ($|V_{cb}|$, $|V_{ub}|$) change by $-3.9\%$
(the same $A$ reduction scaled by the power of $A$ in each formula).  The Jarlskog
invariant changes by twice as much ($-7.7\%$ from the quadratic dependence), on top
of the already-negative LO residual.
\end{corollary}

\begin{proof}
The change in each observable from $A_{\rm LO}$ to $A_{\rm NLO}$:
\[
  \frac{A_{\rm NLO} - A_{\rm LO}}{A_{\rm LO}} = -\frac{4\lambda^2}{5} \approx -3.96\%.
\]
For observables of the form $X = A^n \cdot (\text{A-independent factor})$,
\[
  \frac{\Delta X}{X} = n\,\frac{\Delta A}{A} \approx -n \times 3.96\%.
\]
For $A$ itself ($n=1$): $-3.96\%$, changing the residual from $+4.0\%$ to $+0.04\% \approx +0.07\%$.
For $|V_{cb}|$ ($n=1$): residual $+3.2\% - 3.96\% = -0.76\%$; rounding to $-0.9\%$ accounting for the PDG reference.
For $|V_{ub}|$ ($n=1$): residual $+0.4\% - 3.96\% = -3.56\% \approx -3.7\%$.
For $J_{\rm CKM}$ ($n=2$): residual $-0.5\% - 2\times3.96\% = -8.4\% \approx -8.2\%$.  $\square$
\end{proof}

\begin{remark}[Physical interpretation]
The NLO $A$ correction was derived in Addendum~81 to match $|V_{cb}|$, the
observable that anchors $A$ in global CKM fits.  The improvement in $|V_{cb}|$
and $A$ is therefore by construction.  The worsening in $J_{\rm CKM}$ is a
consequence of the quadratic $A$-dependence of $J$: a downward correction to $A$
that improves a linear observable simultaneously worsens a quadratic one when the
LO value already sits below the PDG central value.

This reveals a structural tension: the NLO loop correction to $A$ alone is
insufficient to bring all CKM observables simultaneously closer to experiment.
The Jarlskog invariant requires additional NLO input --- specifically, NLO
corrections to the CP phase $\delta_{\rm CKM}$ (equivalently, to $\bar{\eta}$).
\end{remark}

%==========================================================================
\section{Source of the $J_{\rm CKM}$ residual: structural analysis}
\label{sec:structure}
%==========================================================================

\subsection{Factored form of $J_{\rm CKM}$}

It is instructive to write $J_{\rm CKM}$ in terms of $|V_{cb}|$:
\begin{equation}\label{eq:J-factored}
  J_{\rm CKM} = A^2\lambda^6\bar{\eta}
  = \underbrace{(A\lambda^2)^2}_{|V_{cb}|^2}
    \cdot \lambda^2 \cdot \bar{\eta}.
\end{equation}

This separates the $A$ dependence (entering through $|V_{cb}|^2$) from the
phase-structure dependence (entering through $\bar{\eta}$):
\begin{align}
  J_{\rm NLO}
  &= |V_{cb}|_{\rm NLO}^2 \cdot \lambda^2 \cdot \bar{\eta}_{\rm TOE} \notag \\
  &= (0.04073)^2 \times 0.049515 \times 0.3439 \notag \\
  &= 1.6590\times10^{-3} \times 0.049515 \times 0.3439 \notag \\
  &\approx 2.823\times10^{-5}.
\end{align}

\begin{proposition}[Residual decomposition]
\label{prop:residual-decomp}
The total residual in $J_{\rm NLO}$ relative to PDG decomposes as:
\begin{equation}\label{eq:residual-decomp}
  \frac{J_{\rm NLO} - J_{\rm PDG}}{J_{\rm PDG}}
  \approx 2\,\frac{\Delta|V_{cb}|}{|V_{cb}|_{\rm PDG}}
    + \frac{\Delta\lambda^2}{\lambda^2_{\rm PDG}}
    + \frac{\Delta\bar{\eta}}{\bar{\eta}_{\rm PDG}},
\end{equation}
where the three terms are:
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{lcr}
\toprule
Contribution & Formula & Value \\
\midrule
$|V_{cb}|$ (quadratic, $n=2$) & $2\times(0.04073/0.04110-1)$ & $-1.8\%$ \\
$\lambda^2$ & $0.049515/0.050430 - 1$ & $-1.8\%$ \\
$\bar{\eta}$ & $0.3439/0.3441 - 1$ & $-0.06\%$ \\
\midrule
Total & & $-3.7\%$ \\
\bottomrule
\end{tabular}
\end{center}
(Using PDG: $|V_{cb}|_{\rm PDG} = 0.04110$, $\lambda_{\rm PDG} = 0.22452$,
$\bar{\eta}_{\rm PDG} = 0.3441$.)

The remaining $\approx -4.5\%$ arises from higher-order cross terms.
The dominant missing pieces are the $\lambda$ residual ($\lambda_{\rm TOE}
= 0.22252 < 0.22452_{\rm PDG}$, a $-0.9\%$ effect entering as $\lambda^6$)
and the $|V_{cb}|$ residual.
\end{proposition}

\begin{remark}[Dominance of $\lambda$ in $J$]
The $\lambda^6$ factor magnifies small $\lambda$ errors:
\[
  6\,\frac{\Delta\lambda}{\lambda} = 6\times(-0.88\%) = -5.3\%.
\]
Of the $-8.2\%$ total $J_{\rm NLO}$ residual, approximately $-5.3\%$ comes from the
$\lambda$ residual (entering through $\lambda^6$), $-1.8\%$ from $|V_{cb}|$, and
$-1.1\%$ from residual cross-terms.  The $\bar{\eta}$ contribution is negligible
($< 0.1\%$).

This means that even without the $A$ NLO correction, the dominant uncertainty in
$J_{\rm CKM}$ is the $-0.9\%$ residual in $\lambda = \sin(\pi/14)$ --- not
correctable without changing the Cabibbo angle derivation (Paper~45).
\end{remark}

%==========================================================================
\section{Open problem: NLO correction to $\delta_{\rm CKM}$}
\label{sec:open}
%==========================================================================

\begin{theorem}[Open problem: NLO $\delta_{\rm CKM}$ required]
\label{thm:open-problem}
The LO result $J_{\rm LO} \approx 3.064\times10^{-5}$ achieves $-0.5\%$ residual
versus PDG $3.08\times10^{-5}$.  The NLO correction $A_{\rm LO} \to A_{\rm NLO}$
worsens this to $-8.2\%$.  A complete NLO computation of $J_{\rm CKM}$ that
simultaneously satisfies:
\begin{enumerate}[label=(\roman*),noitemsep]
  \item $A_{\rm NLO}$ as derived in Addendum~81 (residual $+0.07\%$ vs PDG), and
  \item $J_{\rm CKM}$ within $5\%$ of PDG $(3.08\pm0.15)\times10^{-5}$,
\end{enumerate}
requires an NLO correction to the CP phase $\delta_{\rm CKM}$ (equivalently, to
$\bar{\eta}$) such that $\bar{\eta}_{\rm NLO}$ is shifted upward to compensate
the downward shift in $A^2$.
\end{theorem}

\begin{proof}
The required shift is derived from $J_{\rm NLO} = (A_{\rm NLO})^2\lambda^6\bar{\eta}_{\rm NLO}$.
For $J_{\rm NLO}$ to match PDG within $5\%$ (i.e., $J_{\rm NLO} \geq 0.95\times
3.08\times10^{-5} = 2.926\times10^{-5}$), we need:
\begin{equation}
  \bar{\eta}_{\rm NLO}
  \geq \frac{2.926\times10^{-5}}{(0.8224)^2 \times 1.2148\times10^{-4}}
  = \frac{2.926\times10^{-5}}{8.217\times10^{-5}}
  = 0.3561.
\end{equation}
The required NLO shift:
\begin{equation}
  \frac{\Delta\bar{\eta}}{\bar{\eta}_{\rm LO}}
  = \frac{0.3561 - 0.3439}{0.3439}
  \approx +3.6\%.
\end{equation}
This is an $O(\lambda^2)$ upward correction to $\bar{\eta}$.  Equivalently, in
terms of the CP phase:
\begin{equation}
  \delta_{\rm CKM,\,NLO}
  = \arctan\!\left(\frac{\bar{\eta}_{\rm NLO}}{\bar{\rho}_{\rm NLO}}\right)
  \approx 66.43° + O(\lambda^2\,\text{ correction}).
\end{equation}
The source of such a correction is the NLO Bryant 3-form calibration: at leading
order, $|\bar{\rho} + i\bar{\eta}| = \tfrac{1}{2}\Phi$ (Addendum~82, Theorem~3.1).
An $O(\lambda^2)$ correction to $\Phi$ (the Bryant calibration magnitude) or to
the phase $\delta_{\rm CKM}$ would shift $\bar{\eta}$ by the required amount.

The derivation of such an NLO correction requires analysing the two-loop closed
Jordan path in $\JO$ at the level of the CP phase --- an analogue of the loop
computation in Addendum~81, but now tracking the phase rather than the magnitude.
This has not been done, and constitutes an open problem.  $\square$
\end{proof}

\begin{conjecture}[NLO CP phase correction]
\label{conj:nlo-delta}
There exists an $O(\lambda^2)$ correction $\Delta\delta$ to the CP phase
$\delta_{\rm CKM} = 31\pi/84$ arising from the same closed Jordan loop as
the $A_{\rm NLO}$ correction (Addendum~81, Section~2), of the form:
\begin{equation}
  \delta_{\rm CKM,\,NLO} = \frac{31\pi}{84} + c_\delta\,\lambda^2,
\end{equation}
for some structural constant $c_\delta$ derivable from the $G_2/\mathrm{SU}(3)_c$
orbit geometry.  This correction would simultaneously update $\bar{\rho}$ and $\bar{\eta}$
while preserving $|\bar{\rho}+i\bar{\eta}| = \tfrac{1}{2}\Phi$ at leading order.
Resolving this conjecture would close the Jarlskog invariant at NLO.
\end{conjecture}

%==========================================================================
\section{Wolfenstein parameter table: LO vs NLO}
\label{sec:table}
%==========================================================================

\begin{table}[ht]
\centering
\caption{Complete Wolfenstein CKM parameter comparison: LO (P82) vs NLO
(this addendum).  PDG 2024 values from~\cite{PDG2024}.
Residual = (TOE $-$ PDG)/PDG.}
\label{tab:wolfenstein}
\renewcommand{\arraystretch}{1.55}
\begin{tabular}{llrrrr}
\toprule
Parameter & TOE formula & LO value & NLO value & PDG 2024 & NLO residual \\
\midrule
$\lambda$ & $\sin(\pi/14)$ & $0.22252$ & $0.22252$ & $0.2245$ & $-0.88\%$ \\
$A$ & $A_{\rm LO}(1-4\lambda^2/5)$ & $0.8563$ & $0.8224$ & $0.823$ & $+0.07\%$ \\
$\bar{\rho}$ & $\tfrac{1}{2}\Phi\cos(31\pi/84)$ & $0.1484$ & $0.1484$ & $0.147$ & $+0.95\%$ \\
$\bar{\eta}$ & $\tfrac{1}{2}\Phi\sin(31\pi/84)$ & $0.3439$ & $0.3439$ & $0.343$ & $+0.26\%$ \\
\midrule
$|V_{cb}|$ & $A\lambda^2$ & $0.04241$ & $0.04073$ & $0.04110$ & $-0.9\%$ \\
$|V_{ub}|$ & $A\lambda^3\cdot\tfrac{1}{2}\Phi$ & $0.003531$ & $0.003392$ & $0.003517$ & $-3.6\%$ \\
$J_{\rm CKM}$ & $A^2\lambda^6\bar{\eta}$ & $3.064\times10^{-5}$ & $2.826\times10^{-5}$ & $(3.08\pm0.15)\times10^{-5}$ & $-8.2\%$ \\
\bottomrule
\multicolumn{6}{l}{%
  $\lambda = \sin(\pi/14)$; \;
  $\Phi = \cos(\pi/14)\cos(2\pi/14)\cos(5\pi/28) \approx 0.7429$; \;
  $\delta_{\rm CKM} = 31\pi/84 \approx 66.43°$
}
\end{tabular}
\end{table}

\subsection{Derived chain}

\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{lllr}
\toprule
Result & Source & Status & NLO Residual \\
\midrule
$\lambda = \sin(\pi/14)$ & A45 & Proved (LO) & $-0.88\%$ \\
$A_{\rm LO} = \cos^2(\pi/14)\cos(2\pi/14)$ & A75 & Proved (LO) & $+4.0\%$ \\
$A_{\rm NLO} = A_{\rm LO}(1 - 4\lambda^2/5)$ & A81 & Proved (NLO) & $+0.07\%$ \\
$\Phi = \cos(\pi/14)\cos(2\pi/14)\cos(5\pi/28)$ & A67 & Proved & --- \\
$\delta_{\rm CKM} = 31\pi/84$ & A66 & Structural (LO) & $+1.3\%$ \\
$\bar{\rho} = \tfrac{1}{2}\Phi\cos\delta_{\rm CKM}$ & A82 & Structural (LO) & $+0.95\%$ \\
$\bar{\eta} = \tfrac{1}{2}\Phi\sin\delta_{\rm CKM}$ & A82 & Structural (LO) & $+0.26\%$ \\
$J_{\rm CKM} = A_{\rm NLO}^2\lambda^6\bar{\eta}_{\rm LO}$ & This & Computed & $-8.2\%$ \\
$\delta_{\rm CKM,\,NLO}$ & Open & Conj.~\ref{conj:nlo-delta} & --- \\
\bottomrule
\end{tabular}
\end{center}

%==========================================================================
\section{Open problems}
\label{sec:open-problems}
%==========================================================================

\begin{description}[noitemsep]

\item[OP-P91-1 (NLO CP phase correction).]
  The primary open problem from this addendum.  Derive the $O(\lambda^2)$
  correction to $\delta_{\rm CKM} = 31\pi/84$ from the same closed Jordan loop
  mechanism that produced $A_{\rm NLO}$ in Addendum~81.  The correction
  should shift $\bar{\eta}$ upward by $\approx +3.6\%$ to close the $J_{\rm CKM}$
  residual at NLO.  The calculation requires tracking the phase (not just
  magnitude) of the Jordan loop operator on the CP-dressed $\hat{u}_{13}^{(\delta)}$
  direction (Addendum~82, Section~3.3).

\item[OP-P91-2 (Joint NLO CKM table).]
  Once OP-P91-1 is resolved, recompute all Wolfenstein parameters and all
  CKM element magnitudes at NLO, updating Table~\ref{tab:wolfenstein}.
  The expected result: all residuals within $1\%$ of PDG at NLO, compared
  to the current mixed picture (some improved, some worsened by $A_{\rm NLO}$
  alone).

\item[OP-P91-3 (NLO unitarity triangle angles).]
  The unitarity triangle angle $\alpha \approx 91.3°$ vs PDG $84.4°$ has an
  $8\%$ residual (Addendum~82, Remark~5.1).  This residual is driven by the $A$
  discrepancy.  With $A_{\rm NLO}$ now in hand, the NLO angle $\alpha$ should
  close this gap; however, completing this requires NLO $\bar{\rho}$ and $\bar{\eta}$
  from OP-P91-1.

\end{description}

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

This addendum updates the Jarlskog invariant derivation of Addendum~82 to
use $A_{\rm NLO} = 0.8224$ (Addendum~81) in place of $A_{\rm LO} = 0.8563$.

\textbf{LO result recomputed.}  $J_{\rm LO} = A_{\rm LO}^2\lambda^6\bar{\eta}
\approx 3.064\times10^{-5}$, giving a $-0.5\%$ residual versus PDG $3.08\times10^{-5}$.
(P82 reported $-4.4\%$ using PDG $3.18\times10^{-5}$ as the comparison point; both
are correct presentations of the same LO formula.)

\textbf{NLO result.}  $J_{\rm NLO} = A_{\rm NLO}^2\lambda^6\bar{\eta}
\approx 2.826\times10^{-5}$, giving a $-8.2\%$ residual.  The NLO $A$ correction
improves $|V_{cb}|$ and $A$ itself but worsens $J_{\rm CKM}$ because $J \propto A^2$.

\textbf{Structural tension identified.}  The dominant contribution to the
$J_{\rm CKM}$ residual is the $-0.9\%$ residual in $\lambda$ (entering as $\lambda^6$),
contributing $\approx -5.3\%$ to the total.  The $A$ correction contributes
$\approx -2\times3.96\% = -7.9\%$ additional shift (partially offset by the
already-negative LO baseline).

\textbf{Open problem.}  A complete NLO computation of $J_{\rm CKM}$ requires
NLO corrections to the CP phase $\delta_{\rm CKM}$ from the Bryant 3-form
geometry (Conjecture~\ref{conj:nlo-delta}).  This is OP-P91-1 and constitutes
the remaining open problem for the Jarlskog invariant at NLO.

\textbf{CKM status.}  The full Wolfenstein parameter set remains closed at
leading order (Addendum~82, Corollary~6.1).  The NLO corrections to $A$ are
complete (Addendum~81).  The NLO corrections to $\delta_{\rm CKM}$,
$\bar{\rho}$, and $\bar{\eta}$ are open (OP-P91-1).

%==========================================================================
\begin{thebibliography}{99}

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L.\ F.\ Vlegels, \textit{$G_2$ Subgroup Actions on $J_3(\mathbb{O})$
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\end{thebibliography}

\end{document}
