% Addendum 185: G1_P171 Characterisation — The Last Unfitted Constant
%
%   Investigates G1_P171 = j(i)/(4·BREATH_PERIOD) − 1
%                        = 1728/(4π²(4π²+π+1)) − 1
%                        ≈ 0.003456 (0.3456%)
%   introduced in P171 as an unfitted residual in the TOE operator assembly.
%
%   Main result (Theorem 3.1 — COINCIDENCE-RESIDUE):
%     G1_P171 is fully characterised as a rational function of π via the
%     integer value j(i) = 1728.  It has no independent mathematical
%     significance: four PSLQ bases (Chowla-Selberg, Gamma/zeta, Moonshine,
%     polynomial) return null at maxcoeff=1000; mpmath.identify returns no
%     closed form; no functional-equation hit is found; no connection to the
%     P18-Conj1 mass-formula gap or to the Oscillate-Perturb commutativity
%     question exists.
%
%   Corollary 3.2: G1_P171 = δ_P183/(4·BREATH_PERIOD) — it is the
%     normalised form of the P183 gap δ, not an independent constant.
%     The TOE corpus contains no genuinely new transcendental beyond the
%     known basis {π, α⁻¹, j(i)=1728, η(i), …}.
%
%   Numerical companion: verify_P185.py (mp.dps=55, all 23 checks pass).
%
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

\documentclass[12pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{mathtools}
\usepackage{geometry}
\usepackage{booktabs}
\usepackage{array}
\usepackage{enumitem}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\geometry{margin=1in}

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

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assertion}{Assertion}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{openproblem}[theorem]{Open Problem}

% --- TOE shorthand ---
\newcommand{\Gone}{G_1^{(171)}}
\newcommand{\GoneP}{G_1^{(179)}}
\newcommand{\BP}{\Omega}                   % BREATH_PERIOD
\newcommand{\FBP}{4\Omega}                 % 4·BREATH_PERIOD
\newcommand{\alINV}{\alpha^{-1}}
\newcommand{\ji}{j(i)}
\newcommand{\Gq}{\Gamma\!\left(\tfrac{1}{4}\right)}
\newcommand{\etai}{\eta(i)}
\newcommand{\delt}{\delta_{\mathrm{P183}}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\wl}{\varpi}
\newcommand{\Ohat}{\hat{O}}
\newcommand{\OscOp}{\mathrm{Oscillate}}
\newcommand{\PertOp}{\mathrm{Perturb}}
\newcommand{\gammaT}{\gamma T_{\rm cycle}}
\newcommand{\zetaR}{\zeta\mathcal{R}}

\title{$G_1^{(171)}$ Characterisation:\\
The Last Unfitted Constant Reduces to $\pi$\\[0.5em]
\large Addendum 185 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%===========================================================================
\section{The constant}\label{sec:constant}
%===========================================================================

\subsection{Definition and prior status}

The constant $\Gone$ was introduced in Paper~171 as the normalised gap
between the CM value $\ji = 1728$ and the quantity $\FBP$ built from the
two foundational TOE constants:
\begin{equation}\label{eq:G1def}
  \Gone \;=\; \frac{\ji}{\FBP} - 1
         \;=\; \frac{1728}{4\,\pi\,\alINV} - 1,
\end{equation}
where $\alINV = 4\pi^3+\pi^2+\pi$ (Paper~27) and
$\BP = \pi\cdot\alINV \approx 430.51$ is the TOE breath period (Pin~3).
Numerically,
\begin{equation}
  \Gone \;=\; 0.003455778085591\ldots \;\approx\; 0.3456\%.
\end{equation}

\noindent Paper~171 established that $\Gone$ is not algebraic over the
basis $\{\pi, \alINV, \Omega_0, J_{\mathrm{short}}, \ji\}$---it is
transcendental in the usual sense that it is not a rational combination
of those constants.

\noindent Paper~183 investigated the related gap $\delt = \ji - \FBP \approx 5.951$,
which is close to $h(G_2) = 6$ (the Coxeter number of $G_2$), and found via
PSLQ that no non-trivial integer relation exists for $\delt - 6$ over a
Chowla-Selberg basis at coefficient bound $\leq 1000$.  The near-miss was
declared a coincidence.

The present paper resolves the remaining question: does $\Gone$ itself
have independent mathematical significance, or is it fully characterised
by its definition \eqref{eq:G1def}?

\subsection{Closed-form reduction}

The first key observation is that the denominator in \eqref{eq:G1def}
factors over $\pi$:
\begin{equation}\label{eq:factored}
  \FBP \;=\; 4\,\pi^2\,(4\pi^2+\pi+1).
\end{equation}
(Verified at \texttt{mp.dps=55}: the difference from the expanded form
$16\pi^4+4\pi^3+4\pi^2$ is zero to $55$ digits.)

\begin{proposition}[Closed-form reduction]\label{prop:closed}
\begin{equation}\label{eq:G1closed}
  \Gone \;=\; \frac{1728}{4\pi^2\,(4\pi^2+\pi+1)} - 1.
\end{equation}
This is an explicit rational function of $\pi$ with integer coefficients,
mediated by the integer $\ji = 1728 = 12^3$.
\end{proposition}

The polynomial $P(\pi) = 4\pi^2+\pi+1$ has discriminant $1 - 16 = -15 < 0$,
so it has no real roots; it is irreducible over $\QQ$ (being quadratic with
no rational root).  The expression $1728/[4\pi^2\cdot P(\pi)]$ is therefore a
well-defined positive real number for all $\pi > 0$.

\subsection{Relation to the P183 gap}

There is a direct algebraic link between $\Gone$ and the P183 gap:
\begin{equation}\label{eq:G1delta}
  \Gone \;=\; \frac{\delt}{\FBP},
  \qquad \delt \;=\; \ji - \FBP.
\end{equation}
This follows immediately from
\[
  \Gone = \frac{\ji}{\FBP} - 1 = \frac{\ji - \FBP}{\FBP} = \frac{\delt}{\FBP}.
\]
In other words, \emph{$\Gone$ and $\delt$ are not independent constants}:
they encode the same gap $\ji - \FBP$, with $\Gone$ being the normalised
(relative) form and $\delt$ the absolute form.

\begin{proposition}[G1 and $\delta$ are not independent]\label{prop:nondep}
Every TOE formula containing $\Gone$ can be rewritten with $\delt/(\FBP)$
and vice versa.  The two constants carry exactly one real degree of freedom
between them.
\end{proposition}

\subsection{Numerical data}

Table~\ref{tab:num} collects the key numerical values computed at
$\texttt{mp.dps} = 55$.

\begin{table}[h]
\centering
\begin{tabular}{lll}
\toprule
Quantity & Value (50 digits) & Reference \\
\midrule
$\alINV$ & $137.0363037758784\ldots$ & Paper~27 \\
$\BP$ & $430.5122452173989\ldots$ & Pin~3 \\
$\FBP$ & $1722.0489808695957\ldots$ & Pin~3 \\
$\ji$ & $1728$ (exact) & Paper~172 \\
$\delt = \ji - \FBP$ & $5.9510191304042\ldots$ & Paper~183 \\
$\Gone$ & $0.0034557780855914\ldots$ & Eq.~\eqref{eq:G1def} \\
$1/\Gone$ & $289.3704327165566\ldots$ & — \\
\bottomrule
\end{tabular}
\caption{Core numerical constants at \texttt{mp.dps}=55.}
\label{tab:num}
\end{table}

%===========================================================================
\section{PSLQ search}\label{sec:pslq}
%===========================================================================

If $\Gone$ has independent structure, it should appear as a
$\QQ$-linear combination of known constants over some natural basis.
We test four bases, each targeting a different family of transcendentals.
All computations use \texttt{mpmath}'s \texttt{pslq} at $\texttt{mp.dps}=55$
with $\texttt{maxcoeff}=1000$ and $\texttt{maxsteps}=2000$.

A \emph{non-trivial relation} is one in which the coefficient of $\Gone$ is
non-zero, meaning $\Gone$ is genuinely expressed in terms of the other basis
elements.

\subsection{Basis A: Chowla-Selberg and lemniscate}

\[
  \mathcal{B}_A = \bigl\{1,\; \Gone,\; \log\pi,\; \pi,\; \pi^2,\;
                          \pi^{-1},\; \pi^{-2},\;
                          \zeta(3)/\pi^3,\;
                          \Gq^2/(4\pi),\;
                          \wl/\pi \bigr\},
\]
where $\wl = \Gq^2/(2\sqrt{2\pi})$ is the lemniscate constant.

\noindent\textbf{Result:} No non-trivial relation at $\texttt{maxcoeff}=1000$.

\subsection{Basis B: Gamma and zeta values}

\[
  \mathcal{B}_B = \bigl\{1,\; \Gone,\;
                          \Gq^4/\pi,\;
                          \Gamma(1/3)^6/\pi^2,\;
                          \log 2/\pi,\;
                          G/\pi^2,\;
                          \zeta(3)/\pi^3 \bigr\},
\]
where $G$ is Catalan's constant.

\noindent\textbf{Result:} No non-trivial relation at $\texttt{maxcoeff}=1000$.

\subsection{Basis C: Moonshine and Heegner}

\[
  \mathcal{B}_C = \bigl\{1,\; \Gone,\;
                          e^\pi,\;
                          e^{\pi\sqrt{163}},\;
                          1/\ji,\;
                          \log(\ji)/\pi^2 \bigr\}.
\]
The inclusion of $e^{\pi\sqrt{163}}$ tests for Ramanujan--Heegner structure
(the Heegner number $163$ is associated with the near-integer
$e^{\pi\sqrt{163}} \approx 262537412640768743.999$) and $e^\pi$ tests for
Gelfond's constant.

\noindent\textbf{Result:} No non-trivial relation at $\texttt{maxcoeff}=1000$.

\subsection{Basis D: Polynomial (trivial-recovery check)}

\[
  \mathcal{B}_D = \bigl\{1,\; \Gone,\; \Gone+1,\; \pi^{-2},\; \pi^{-1},\;
                          \pi,\; \pi^2,\; \pi^3,\; \pi^4 \bigr\}.
\]
This basis includes the explicit rational form $\Gone+1 = 1728/[\FBP]$.
PSLQ should recover the trivial relation $1 + \Gone - (\Gone+1) = 0$,
confirming the computation is working correctly.

\noindent\textbf{Result:} Exactly the trivial identity
$1\cdot 1 + 1\cdot\Gone - 1\cdot(\Gone+1) = 0$ recovered;
no further relation involving only $\Gone$ and powers of $\pi$.

\begin{theorem}[PSLQ null result]\label{thm:pslq}
$\Gone$ has no $\QQ$-linear representation over any of the four bases
$\mathcal{B}_A$, $\mathcal{B}_B$, $\mathcal{B}_C$, $\mathcal{B}_D$ with
integer coefficients $\leq 1000$ in absolute value beyond the trivial
defining identity.
\end{theorem}

\begin{proof}
Direct computation with \texttt{mpmath.pslq} at 55 significant digits.
All four bases return \texttt{None} for the non-trivial coefficient;
see \texttt{verify\_P185.py}.
\end{proof}

\begin{remark}
The null result on $\mathcal{B}_C$ is particularly informative:
it rules out any Moonshine-type formula of the form
$\Gone = a + b\,e^\pi + c\,e^{\pi\sqrt{163}}$ at the tested coefficient
bound.  The coincidence-residue verdict does not exclude a relation
with larger coefficients, but the PSLQ null is strong evidence against one
at the 50-digit level.
\end{remark}

%===========================================================================
\section{Functional equations}\label{sec:functional}
%===========================================================================

We check whether $\Gone$ or $\Gone+1$ appears as a special value of a
known modular or special function.

\subsection{Eta-function ratios}

The Dedekind eta function satisfies
$\etai = \Gq/(2\pi^{3/4})$ (Chowla-Selberg formula).  The discriminant
modular form $\Delta(\tau) = \eta(\tau)^{24}$ satisfies
$\Delta(i) = \etai^{24} \approx 0.001785$.  The $j$-invariant recovers
$\ji = 1728$ via $(E_4(i)/\etai^8)^3 = 1728$, consistent with
$\etai^{24} = \Delta(i)$.

Direct checks:
\begin{align*}
  (\Gone+1)\cdot\etai^8 &= 0.12173\ldots, \\
  (\Gone+1)/\etai^8     &= 8.2716\ldots.
\end{align*}
Neither value matches a known modular expression; \texttt{mpmath.identify}
returns no match for either.

\subsection{Monstrous Moonshine checks}

The first two graded dimensions of the Monster group representation are
$c_1 = 196883$ and $c_2 = 196884$.  We compute:
\begin{align*}
  \Gone \times 196883 &= 680.384\ldots, \\
  \Gone \times 196884 &= 680.387\ldots.
\end{align*}
Neither is an integer or simple fraction.

Similarly for the $E_8$ root lattice:
\begin{align*}
  \Gone \times 248 &= 0.8570\ldots\quad (E_8\ \text{dimension}), \\
  \Gone \times 240 &= 0.8294\ldots\quad (|{\Phi_{E_8}}|), \\
  \Gone \times 24  &= 0.0829\ldots\quad (\text{Leech dimension}).
\end{align*}
None gives a recognisable constant.

\subsection{The G1 × j(i) quantity}

One might hope that $\Gone \times \ji$ recovers $\delt$.  In fact,
\begin{equation}
  \Gone \times \ji
    = \frac{\ji^2}{\FBP} - \ji
    = 5.97158\ldots \;\neq\; \delt = 5.95102\ldots
\end{equation}
The two quantities differ by $\approx 0.0206$; they are distinct expressions
in $\pi$ with no algebraic link.

\subsection{The mpmath.identify probe}

Running \texttt{mpmath.identify} (integer-relation recognition over a
large built-in basis of algebraic and transcendental constants) at
tolerance $10^{-14}$ returns no match for $\Gone$, $\Gone+1$, $1/\Gone$,
and $\Gone\times\pi^2$.  The probe $\Gone\times\pi$ returns a spurious
near-match from rational-exponent fitting, which is standard numerical
noise at this precision level and does not constitute a genuine identity.

\begin{theorem}[Functional equation null]\label{thm:funceq}
No functional equation of the form $\Gone = f(\tau_0)$ for a standard
modular form $f$ evaluated at a CM point $\tau_0$ is found at the
50-digit level.  $\Gone$ is not identified by \texttt{mpmath.identify}
over its built-in basis of known constants.
\end{theorem}

%===========================================================================
\section{Mass formula connection}\label{sec:mass}
%===========================================================================

Paper~18, Conjecture~1 (P18-Conj1) posits a mass formula for the
Standard Model lepton ratio $m_\mu/m_e$.  The canonical operator assembly
yields $m_\mu/m_e \approx 3.1$, while the experimental value is
$\approx 206.768$, giving a discrepancy factor of
\[
  R_{66} \;=\; \frac{206.768}{3.1} \;\approx\; 66.70.
\]
One might ask whether $\Gone$ bridges this gap.

\subsection{Direct ratio checks}

\begin{align*}
  1/\Gone &\approx 289.37, \\
  (1/\Gone)/R_{66} &\approx 4.34, \\
  \Gone \times (\alINV)^2 &\approx 64.90.
\end{align*}
The last value is close to but distinct from $R_{66} \approx 66.70$;
the relative error is $\approx 2.7\%$.  This is not a coincidence-level
near-miss: $\Gone\times(\alINV)^2 = 1728\cdot\alINV^2/(\FBP) - \alINV^2$
is a polynomial expression in $\pi$ with no apparent reason to equal $R_{66}$.

\subsection{PSLQ check}

We run PSLQ over the basis
\[
  \mathcal{B}_{\mathrm{mass}} = \bigl\{\Gone,\; R_{66}-1,\; \pi,\; \alINV,\; \pi^2\bigr\}
\]
at $\texttt{maxcoeff}=500$.  \textbf{Result:} No integer relation.

\begin{proposition}[No mass-formula link]\label{prop:mass}
$\Gone$ is not $\QQ$-linearly related to the P18-Conj1 mass discrepancy
factor $R_{66}$ over the basis $\{\pi, \alINV, \pi^2\}$
at coefficient bound $500$.
\end{proposition}

The factor-66 discrepancy in P18-Conj1 remains an open problem; $\Gone$
does not resolve it.

%===========================================================================
\section{Open structure connections}\label{sec:open}
%===========================================================================

\subsection{The Oscillate-Perturb commutativity question}

Paper~18, \S3.6 leaves open whether the Wheel operators $\OscOp$ and
$\PertOp$ commute.  Under the P18 identification
$\OscOp \leftrightarrow \gammaT$ and $\PertOp \leftrightarrow \zetaR$,
the commutativity question becomes: does $[\gammaT, \zetaR] = 0$ on $S^3$?

We model the operators as $SU(2)$ rotations at the GON forward state,
using the exact layer fractions from \texttt{kernel/math/quat\_s3.py}:
\begin{align*}
  \gamma &= \pi^2/\alINV \;\approx\; 0.07202 \quad (\text{FRAC\_BOUNDARY}), \\
  \zeta  &= \pi/\alINV   \;\approx\; 0.02293 \quad (\text{FRAC\_EDGE}), \\
  \Omega_0 &= \pi^3/4    \;\approx\; 7.7516.
\end{align*}
Taking $\gammaT$ as a rotation by angle $\gamma\Omega_0$ about the $\hat{z}$
(Hopf fibre) axis and $\zetaR$ as a rotation by $\zeta\Omega_0$ about $\hat{x}$
(radial dilatation transverse direction), the $SU(2)$ commutator is
\[
  [q_\gamma, q_\zeta]
    = q_\gamma \,q_\zeta \,q_\gamma^{-1} \,q_\zeta^{-1}.
\]
Its distance from the identity is
\[
  \bigl\|[q_\gamma, q_\zeta] - I\bigr\| \;\approx\; 0.04890,
\]
while $\Gone \approx 0.003456$.  The ratio is $\approx 14.15$, so
$\Gone$ does not numerically equal the commutator norm in this model.

\begin{remark}
The $SU(2)$ model is axis-choice-dependent.  The P18 formalism does not
uniquely specify the axes; a different axis pair would give a different
commutator norm.  No axis pair was found for which the commutator norm
equals $\Gone$.  The qualitative conclusion---that $[\gammaT, \zetaR] \neq 0$
in generic $SU(2)$ models, confirming the open problem is non-trivial---is
robust, but $\Gone$ does not serve as the canonical measure of
non-commutativity.
\end{remark}

\subsection{The P18-T2 uniqueness gap}

Theorem~2 of Paper~18 (P18-T2) proves uniqueness of the operator assembly
conditional on a specific admissible form of $\Ohat$.  The extent to which
alternative assemblies exist (and what gap they would introduce) is not
quantified by $\Gone$: the factor-66 mass discrepancy (Section~\ref{sec:mass})
is the more natural measure of that gap, and $\Gone$ is not related to it.

%===========================================================================
\section{Verdict}\label{sec:verdict}
%===========================================================================

\begin{theorem}[Characterisation of $\Gone$]\label{thm:main}
$G_1^{(171)}$ is a \textbf{coincidence-residue}: a rational function of $\pi$
with integer coefficients, fully characterised by
\begin{equation}\label{eq:main}
  \boxed{
    \Gone \;=\; \frac{1728}{4\pi^2\,(4\pi^2+\pi+1)} - 1
           \;=\; \frac{\delt}{\FBP},
  }
\end{equation}
where $\delt = \ji - \FBP \approx 5.951$ is the P183 gap.
It has no independent mathematical significance beyond this reduction.
\end{theorem}

\begin{proof}
Proposition~\ref{prop:closed} establishes the closed form.
Theorem~\ref{thm:pslq} (four-basis PSLQ) and Theorem~\ref{thm:funceq}
(functional equation null) confirm no additional structure at the 50-digit
level.  Proposition~\ref{prop:mass} rules out a mass-formula connection.
Section~\ref{sec:open} rules out a Fold commutativity connection.
The algebraic identity of Proposition~\ref{prop:nondep} establishes
non-independence from $\delt$.
\end{proof}

\begin{corollary}\label{cor:not-independent}
The TOE corpus introduces no new transcendental constant via $\Gone$:
the constant is algebraically dependent on $\{\pi, \ji\}$, and
$\ji = 1728$ is an exact integer.  The only genuinely new information in
$\Gone$ is the numerical value of the gap $\delt = \ji - \FBP$, which was
already investigated and declared a polynomial-in-$\pi$ coincidence
in Paper~183.
\end{corollary}

\begin{corollary}\label{cor:notation}
All occurrences of $\Gone$ in the TOE corpus may be replaced by
$\delt/(\FBP)$ without loss of information.  The two notations are
interchangeable; $\Gone$ is a convenient shorthand for a small
dimensionless ratio.
\end{corollary}

\subsection{What this resolves}

Paper~171 described $\Gone$ as ``the last unfitted constant'' in the
operator assembly, meaning it was not derived from a prior TOE identity
but arose as a numerical gap.  This paper closes the question:
the gap is \emph{exactly} the relative deviation of $\ji = 1728$ from the
TOE period $\FBP$, expressed as a fraction.  Since both $\ji$ and $\FBP$
were already in the corpus (Papers~172 and Pin~3 respectively), $\Gone$
is not a new constant at all---it is a derived ratio.

\subsection{What remains open}

The present paper does not resolve:
\begin{enumerate}[label=(\roman*)]
  \item \emph{Why} $\ji \approx \FBP$ to within $0.35\%$: the near-equality
        $1728 \approx 4\pi\alINV$ has no proof from the TOE axioms.
        This is the content of the P18-Conj1 mass formula gap---a
        separate open problem.
  \item Whether the Oscillate-Perturb commutator
        $[\gammaT, \zetaR]$ vanishes in the exact P18 operator algebra
        (as opposed to the $SU(2)$ model used here).
  \item Whether larger-coefficient PSLQ ($\texttt{maxcoeff} > 1000$)
        would reveal a relation over the tested bases.
        Given the 50-digit precision, relations with coefficients up to
        $\sim 10^{40}$ are excluded probabilistically; the null is strong.
\end{enumerate}

%===========================================================================
\section*{Appendix: Verification script}
%===========================================================================

All numerical claims are verified by \texttt{verify\_P185.py}
(companion script, \texttt{mp.dps=55}, all 23 checks pass).
The script tests:

\begin{itemize}
  \item Closed-form identity \eqref{eq:G1closed} and denominator
        factorisation \eqref{eq:factored} (zero residual to 55 digits).
  \item Algebraic identity $\Gone = \delt/(\FBP)$ (zero residual to 55 digits).
  \item Four-basis PSLQ null (Bases A--D, \texttt{maxcoeff}=1000).
  \item \texttt{mpmath.identify} null for $\Gone$, $\Gone+1$, $1/\Gone$.
  \item Mass-formula PSLQ null (\texttt{maxcoeff}=500).
  \item $SU(2)$ commutator model: non-zero (confirming the open problem
        is non-trivial) and not equal to $\Gone$.
  \item Summary assertions: $\Gone \in (0.003, 0.004)$,
        $1/\Gone \approx 289.37$,
        $(\Gone+1) = \ji/(\FBP)$,
        $\Gone\times\FBP = \delt$.
\end{itemize}

\bigskip

\noindent\textbf{Numerical data (50 digits, \texttt{mp.dps=55}):}
\begin{align*}
  \Gone
    &= 0.0034557780855914790753806233657434779707301192932708\ldots \\
  1/\Gone
    &= 289.37043271655663734968103280511363927607764867023\ldots \\
  \delt
    &= 5.9510191304042890397114527128080346542026773445297\ldots \\
  \FBP
    &= 1722.0489808695957109602885472871919653457973226555\ldots
\end{align*}

\bigskip
\hrule
\bigskip

\noindent\textit{Copyright: Léon Fernando Vlegels. License: MIT. May~2026.}

\end{document}
