% ============================================================
% 220_Addendum_OP_rSinf100.tex
% Addendum 220 to the LumenOS TOE corpus
% Subject: Closure attempt on OP-rSinf-100
%
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0.
% ============================================================
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% Shorthand
\newcommand{\muu}{\mu}          % ALPHA_INV = 4pi^3+pi^2+pi
\newcommand{\Ee}{E_e}           % = pi
\newcommand{\Emu}{E_\mu}        % = pi^2
\newcommand{\oh}{\Omega_0}      % = pi^3/4
\newcommand{\Rinf}{R_\infty}
\newcommand{\Rsim}{\hat{R}}     % computed R_inf
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\eps}{\varepsilon}
\newcommand{\sig}{\sigma}

\title{%
  \textbf{Addendum 220}\\[4pt]
  \large Closure Attempt on Open Problem \texttt{OP-rSinf-100}:\\
  Precision Analysis of the Near-Miss $r \cdot S_\infty \approx 100\,c$\\
  and its Status under CODATA Uncertainty
}
\author{Léon Fernando Vlegels \\ \small LumenOS P18-T2 Uniqueness Programme}
\date{2026-05-18}

\begin{document}

\maketitle
\tableofcontents
\bigskip

% ============================================================
\section{Background and context}
% ============================================================

Papers P216 and P217 of the TOE corpus established that the geometric ratio
\[
  r \;=\; \frac{h\,\Emu}{\muu^2} \;=\; \frac{3\pi^2}{\muu^2}
  \;\approx\; 1.5767\times10^{-3}
\]
and the closed-form series limit
\[
  S_\infty \;=\; \frac{h\,\muu}{\Ee^3\!\left(\muu^2 + h\,\Emu\right)}
            \;=\; \frac{3\muu}{\pi^3\!\left(\muu^2+3\pi^2\right)}
  \;\approx\; 7.0494\times10^{-4}
\]
(where $\muu = 4\pi^3+\pi^2+\pi$, $h=3$, $\Ee=\pi$, $\Emu=\pi^2$) satisfy
the product identity
\begin{equation}\label{eq:closed-form}
  r\cdot S_\infty
  \;=\;
  \frac{9}{\muu\,\pi\!\left(\muu^2+3\pi^2\right)}
  \;\approx\; 1.11148\times10^{-6}.
\end{equation}
This identity was verified algebraically in P217 and is exact.

Independently, the TOE model predicts the Rydberg-analogue constant
\begin{equation}\label{eq:Rinf}
  \hat{R} \;=\; 207\!\left(1 - \frac{\oh}{\Ee^3\,\muu} + S_\infty\right),
  \qquad \oh = \frac{\pi^3}{4},
\end{equation}
giving $\hat{R} \approx 206.76828530$, which is to be compared against the
experimental target $\Rinf^{\mathrm{exp}} = 206.7682830$ (CODATA, in the
model's working units, with uncertainty $\sigma_T = 4.6\times10^{-6}$).

Define the per-prefactor residual
\begin{equation}\label{eq:c}
  c \;=\; \frac{\hat{R} - \Rinf^{\mathrm{exp}}}{207}
  \;=\; \frac{\delta_\infty}{207},
  \qquad \delta_\infty \;=\; \hat{R} - 206.7682830.
\end{equation}
At 60-digit precision (mpmath, \texttt{mp.dps=60}):
\[
  \delta_\infty \;=\; 2.30071217\ldots \times 10^{-6},
  \qquad
  c \;=\; 1.11145515\ldots \times 10^{-8}.
\]

\textbf{The open problem OP-rSinf-100} asks whether the ratio
\begin{equation}\label{eq:op}
  \frac{r\cdot S_\infty}{100\,c} \;\overset{?}{=}\; 1
\end{equation}
holds exactly, or whether the near-miss $r\cdot S_\infty \approx 100\,c$ is
merely coincidental.  The name encodes the factor 100.

\begin{remark}
  P217 phrased the near-miss as ``$r\cdot S_\infty \cdot 100 \approx c$''
  (a sign reversal and factor-of-$10^4$ transcription error carried through
  from the original notebook).  The correct reading, consistent with the
  formulas therein, is $r\cdot S_\infty \approx 100\,c$.
  The PSLQ null result in P216 was performed on the fractional deviation
  $\varepsilon$ defined below, which is unaffected by this notation issue.
\end{remark}

% ============================================================
\section{Section 1 — Precision pinning of $\varepsilon$ and CODATA uncertainty}
% ============================================================

\subsection{Full-precision value of $\varepsilon$}

Define
\begin{equation}\label{eq:eps}
  \varepsilon \;\coloneqq\; \frac{r\cdot S_\infty}{100\,c} - 1.
\end{equation}
Using \texttt{mpmath} at \texttt{mp.dps=60} one obtains
\begin{equation}\label{eq:eps-val}
  \boxed{
    \varepsilon \;=\;
    +2.20367\,25885\,93701\,88910\,63220\,44012\,48147\ldots \times 10^{-5}.
  }
\end{equation}
This is positive, confirming that $r\cdot S_\infty$ exceeds $100\,c$ by a
fractional amount of roughly $2.2\times10^{-5}$.

For reference the two sides of the near-miss are
\begin{align*}
  r\cdot S_\infty &= 1.11147964\,7459600611\,2441668291\ldots\times10^{-6},\\
  100\,c          &= 1.11145515\,4627023583\,7816110021\ldots\times10^{-6},
\end{align*}
so the absolute discrepancy is
$r\cdot S_\infty - 100\,c \approx 2.449\times10^{-11}$.

\subsection{The quantity $c$ and its experimental heritage}

Since $c = (\hat{R} - \Rinf^{\mathrm{exp}})/207$ and $\hat{R}$ is a pure
mathematical quantity (no experimental uncertainty), the only source of
uncertainty in $c$ is the CODATA uncertainty in $\Rinf^{\mathrm{exp}}$:
\[
  \sigma_c \;=\; \frac{\sigma_T}{207} \;=\; \frac{4.6\times10^{-6}}{207}
  \;\approx\; 2.222\times10^{-8}.
\]
The relative uncertainty in $c$ is therefore
\[
  \frac{\sigma_c}{c} \;=\;
  \frac{2.222\times10^{-8}}{1.111\times10^{-8}} \;\approx\; 1.999.
\]
\emph{The current CODATA measurement resolves $c$ to only about $\pm100\%$
relative precision.}  Equivalently, $\sigma_T \approx 2\,\delta_\infty$, so
the residual $\delta_\infty$ is itself uncertain to the same order as its
magnitude.

\subsection{CODATA propagation to $\sigma_\varepsilon$}

Propagating the uncertainty in TARGET through $c$ to $\varepsilon$:
\[
  \frac{\partial\varepsilon}{\partial T}
  \;=\; -\frac{r\cdot S_\infty}{100\,c^2} \cdot
         \frac{\partial c}{\partial T}
  \;=\; \frac{r\cdot S_\infty}{100\,c^2\cdot 207},
\]
giving
\begin{equation}\label{eq:sigma-eps}
  \sigma_\varepsilon \;=\;
  \frac{r\cdot S_\infty}{100\,c^2\cdot 207}\,\sigma_T
  \;\approx\; 1.999.
\end{equation}
The uncertainty in $\varepsilon$ is of order $2$ (dimensionless), roughly
$10^5$ times larger than $\varepsilon$ itself.

\subsection{Key ratio}

\begin{equation}\label{eq:ratio}
  \frac{\varepsilon}{\sigma_\varepsilon}
  \;=\;
  \frac{2.2037\times10^{-5}}{1.999}
  \;\approx\; 1.1\times10^{-5}
  \;\ll\; 1.
\end{equation}

\textbf{Conclusion (Section 1):} The near-miss $r\cdot S_\infty \approx 100\,c$
cannot be falsified with the current CODATA precision.  The deviation
$\varepsilon$ is buried approximately $10^5\,\sigma$ inside the measurement
uncertainty, so the present experimental data is entirely consistent with the
exact identity $r\cdot S_\infty = 100\,c$.  This places OP-rSinf-100 firmly in
\textbf{Case~A}.

% ============================================================
\section{Section 2 — Extended PSLQ search}
% ============================================================

P216 ran PSLQ on $\varepsilon$ using a 20-element basis of powers of $\pi$ and
$\muu$ and obtained a null result.  This addendum extends the search to three
larger bases.

\subsection{Basis 1: powers of $\pi$ and $\muu$ (20 elements)}

\textbf{Elements:} $\{\pi^k : k=1,\ldots,6\} \cup \{\muu^k : k=1,\ldots,4\}
\cup \{\pi^j\muu^k : j,k\geq1,\; j+k\leq 5\} \cup \{1, \varepsilon\}$.

\medskip
\textbf{Result:} PSLQ (maxcoeff = 1000, tol = $10^{-30}$) returned the
relation
\[
  \muu^3 - \pi\muu^2 - \pi^2\muu^2 - 4\pi^3\muu^2 = 0,
\]
with coefficient vector $(0,\ldots,0,1,0,0,-1,0,0,-1,0,0,-4,0,0)$ and
$\max|{\rm coeff}|=4$.  This is simply the defining identity
$\muu = 4\pi^3+\pi^2+\pi$, rewritten as $\muu^3 = (4\pi^3+\pi^2+\pi)\muu^2$,
which is trivially true.  The coefficient of $\varepsilon$ in this relation
is \textbf{zero}: no non-trivial relation involving $\varepsilon$ was found.

\subsection{Basis 2: algebraic elements}

\textbf{Elements:}
$\{1,\; r,\; S_\infty,\; r S_\infty,\; r^2,\; S_\infty^2,\; r+S_\infty,\;
  \oh/(\Ee^3\muu),\; h\oh/(\Ee^3\muu^2),\; \varepsilon\}$.

\medskip
\textbf{Result:} The found relation is
$-r - S_\infty + (r+S_\infty) = 0$,
i.e.\ a basis-element tautology with $\varepsilon$-coefficient \textbf{zero}.

\subsection{Basis 3: 100-centric search}

\textbf{Elements:}
$\{\varepsilon,\; r,\; S_\infty,\; 1/\muu,\; 1/\pi,\; \alpha,\; \alpha^2,\;
  r S_\infty,\; 1/100\}$, where $\alpha = 1/\muu$.

\medskip
\textbf{Result:} The relation found is $1/\muu - \alpha = 0$ (tautology,
since $\alpha\coloneqq1/\muu$). Again, $\varepsilon$-coefficient \textbf{zero}.

\subsection{Fractional-expansion checks}

As a supplementary test, we inspected $\varepsilon\cdot\muu^k$ for $k=1,\ldots,4$:
\[
\begin{array}{lll}
  k=1: & \varepsilon\cdot\muu \approx 0.003020 & \text{(distance to nearest integer: } 0.003020\text{)}\\
  k=2: & \varepsilon\cdot\muu^2 \approx 0.41383 & \text{(distance: }0.41383\text{)}\\
  k=3: & \varepsilon\cdot\muu^3 \approx 56.709  & \text{(distance from 57: }0.291\text{)}\\
  k=4: & \varepsilon\cdot\muu^4 \approx 7771.23 & \text{(distance from 7771: }0.227\text{)}\\
\end{array}
\]
None are close to an integer, ruling out a closed form $\varepsilon = n/\muu^k$
for $|n|\leq10^5$ and $k\leq 4$.

\subsection{Noise floor analysis}

The precision used is \texttt{mp.dps=60}, giving roughly 60 decimal digits.
The PSLQ tolerance is set to $10^{-30}$, well above floating-point noise but
below any physically meaningful relation.  At maxcoeff = 1000, PSLQ searches
for integer vectors with $\ell^\infty$ norm at most 1000.  The null result in
all three bases at this coefficient bound means: \emph{there is no linear
combination of the basis elements summing to zero with integer coefficients
of magnitude $\leq 1000$ that involves $\varepsilon$}.

\textbf{Conclusion (Section 2):} No closed form for $\varepsilon$ was found
in any of the three bases.  The search is consistent with P216's prior null
result and extends it to wider bases including cross-terms and algebraic
elements.  $\varepsilon$ remains algebraically unexplained in the searched
space.

% ============================================================
\section{Section 3 — Geometric interpretation of the factor 100}
% ============================================================

The factor 100 in $r\cdot S_\infty \approx 100\,c$ is suspiciously round.
This section investigates whether it has a natural combinatorial or
Lie-theoretic origin.

\subsection{The 4-simplex f-vector}

The 4-simplex (also called pentachoron or 5-cell) is the 4-dimensional
regular polytope with the fewest vertices.  Its f-vector
(counting $k$-faces for $k=0,1,2,3$) is
\[
  (f_0, f_1, f_2, f_3) \;=\; (5, 10, 10, 5).
\]
The total f-sum is
\[
  F \;\coloneqq\; f_0 + f_1 + f_2 + f_3 \;=\; 30.
\]

\begin{proposition}\label{prop:fvec}
  $100 \;=\; F^2 / h^2 \;=\; 30^2 / 9$.
\end{proposition}
\begin{proof}
  $30^2 = 900$ and $900/9 = 100$ exactly.
\end{proof}

Since the closed-form identity reads $r\cdot S_\infty = h^2/[\muu\pi(\muu^2+3\pi^2)]$
with $h=3$, the factor 100 can be written $F^2/h^2$, and the near-miss becomes
\[
  \frac{h^2}{\muu\pi(\muu^2+3\pi^2)} \;\approx\; 100\,c
  \;=\; \frac{F^2}{h^2}\cdot c,
\]
or equivalently,
\begin{equation}\label{eq:geom-statement}
  c \;\approx\;
  \frac{h^4}{\muu\pi(\muu^2+3\pi^2)\cdot F^2}
  \;=\; \frac{81}{\muu\pi(\muu^2+3\pi^2)\cdot 900}.
\end{equation}
This is a non-trivial prediction: the per-prefactor residual $c$ should equal
$81 / [900\,\muu\pi(\muu^2+3\pi^2)]$ if OP-rSinf-100 is exact.

\subsection{The coincidence $F = h^\vee(E_8)$}

The Coxeter number (dual Coxeter number for the simply-laced case) of $E_8$ is
$h^\vee(E_8) = 30 = F$.  This gives a second reading of the factor 100:
\[
  100 \;=\; \left(\frac{h^\vee(E_8)}{h^\vee(A_2)}\right)^2
  \;=\; \left(\frac{30}{3}\right)^2,
\]
where $h^\vee(A_2) = 3 = h$ is the dual Coxeter number of $A_2$, equal to the
mass quantum $h=3$ appearing in the TOE constants.  So the factor 100 can be
read as the square of the ratio of two Coxeter numbers native to the TOE
geometry.

\subsection{The $B_2$ interpretation}

A purely combinatorial observation: $100 = \dim(\mathfrak{so}(5))^2 = 10^2$,
where $\dim(\mathfrak{b}_2) = 10$.  While this is arithmetically correct, no
physical mechanism connecting $B_2$ to the near-miss has been identified.

\subsection{Negative results}

The following candidates were tested and found not to give 100:
\begin{itemize}
  \item $\dim(G_2)^2 - \dim(A_2)^2 = 196 - 64 = 132$.
  \item $h^\vee(G_2)^2 = 16$, and no simple multiple/power thereof gives 100.
  \item $h^\vee(E_8) + \dim(A_2) + h^\vee(A_2)\cdot\dim(G_2) = 30+8+42 = 80$.
\end{itemize}

\textbf{Summary (Section 3):} Two natural routes to 100 exist within the TOE
combinatorial data: (a) $F^2/h^2$ where $F=30$ is the f-sum of the 4-simplex
and $h=h^\vee(A_2)$; (b) $(h^\vee(E_8)/h^\vee(A_2))^2$.  These are the same
number expressed differently, since $F = h^\vee(E_8)$.  Whether this
coincidence lifts to a structural derivation is an open question for
Post-Addendum work; see Section~5.

% ============================================================
\section{Section 4 — Formal status}
% ============================================================

\subsection{Classification: Case A}

From Section~1 we have $|\varepsilon/\sigma_\varepsilon| \approx 1.1\times10^{-5} \ll 1$.
The deviation $\varepsilon$ is indistinguishable from zero given CODATA
uncertainty.

\begin{conjecture}[P220.1]\label{conj:P220-1}
  \textup{(OP-rSinf-100 near-miss conjecture.)}
  The following identity holds exactly:
  \[
    r \cdot S_\infty \;=\; 100\,c,
  \]
  or equivalently, in closed form,
  \[
    \frac{9}{\muu\pi(\muu^2+3\pi^2)}
    \;=\;
    \frac{100\,\bigl(\hat{R} - \Rinf^{\mathrm{exp}}\bigr)}{207},
  \]
  where $\hat{R} = 207(1 - \oh/\Ee^3/\muu + S_\infty)$ is the model
  prediction and $\Rinf^{\mathrm{exp}}$ is the CODATA experimental value.
\end{conjecture}

\subsection{Experimental test condition}

Conjecture~\ref{conj:P220-1} can be tested when the CODATA uncertainty on
$\Rinf^{\mathrm{exp}}$ is reduced below the threshold at which $\varepsilon$
becomes distinguishable from zero:
\[
  \sigma_T^{\mathrm{required}}
  \;=\; \frac{|\varepsilon|\cdot 100\,c^2\cdot 207}{r\cdot S_\infty}
  \;\approx\; 2.5\times10^{-11},
\]
which is approximately $2\times10^{5}$ times smaller than the current
$\sigma_T = 4.6\times10^{-6}$.  This is far beyond current measurement
technology, so the conjecture is not experimentally falsifiable in the near
term by direct measurement of $\Rinf$.

\subsection{Theoretical test condition}

Alternatively, a purely mathematical proof would require showing that
\[
  \frac{9}{\muu\pi(\muu^2+3\pi^2)}
  \;=\;
  \frac{100}{207}
  \left[
    207\!\left(1 - \frac{\pi^3}{4\pi^3\muu} + S_\infty\right)
    - \Rinf^{\mathrm{exp}}
  \right],
\]
which involves equating a pure-math expression to a quantity containing the
experimental constant $\Rinf^{\mathrm{exp}}$.  This is structurally analogous
to proving that a TOE prediction equals an observed constant, and would
constitute a derivation of $\Rinf$ from first principles — a major open
problem in the broader programme.

\subsection{PSLQ closure of OP-rSinf-100}

Since no closed form for $\varepsilon$ was found in any extended basis
(Section~2), \emph{Case~C} (a closed-form identity for $\varepsilon$ that
reveals the structure) does not apply at this stage.  The absence of a PSLQ
relation in bases up to 20 elements with $|\mathrm{coeff}|\leq10^3$ provides
evidence — though not proof — that $\varepsilon$ is not a simple
algebraic expression in $\pi$ and $\muu$.

% ============================================================
\section{Post-addendum questions and programme implications}
% ============================================================

\subsection{What a proof of Conjecture~P220.1 would require}

Conjecture~P220.1 equates a purely algebraic expression
$9/[\muu\pi(\muu^2+3\pi^2)]$ (provably equal to $r\cdot S_\infty$) to
$100\,c$, a quantity tied to the experimental Rydberg value.
There are two routes:

\paragraph{Route 1 (derive $\Rinf$ from TOE).}
If a future paper derives the exact value of $\Rinf^{\mathrm{exp}}$ from
TOE geometry, then $c$ becomes a computable algebraic number, and
Conjecture~P220.1 reduces to checking a specific Diophantine equality.

\paragraph{Route 2 (reinterpret $c$).}
If $c$ can be shown to equal some TOE-native expression
$f(\pi,\muu,\oh)$ — for instance via a deeper analysis of the
$\hat{R}$-residual — then the conjecture follows from the closed-form
identity~\eqref{eq:closed-form}.

\subsection{Geometric open question}

The two coincidences $F = h^\vee(E_8) = 30$ and $F^2/h^2 = 100$ are
intriguing.  A structural question for Post-P220 work:
\begin{quote}
  \emph{Is the factor 100 in the near-miss OP-rSinf-100 derivable from the
  f-vector of the 4-simplex (or equivalently, the Coxeter number of $E_8$)
  via the TOE operator algebra?}
\end{quote}
Specifically: the 4-simplex arises naturally as the $A_4$ root polytope
(or the boundary of the 5-cell), and $E_8 \supset A_4$ is a well-known
embedding.  Whether the f-sum $F=30$ appears as a scaling factor in any
of the TOE's 31-paper derivations has not yet been checked.

\subsection{OP-rSinf-100: updated status}

\begin{center}
\begin{tabular}{ll}
\toprule
Attribute & Value \\
\midrule
Open problem ID & OP-rSinf-100 \\
First appearance & P217 \\
PSLQ status (P216) & Null in 20-element basis \\
PSLQ status (P220) & Null in 3 bases, $\leq 20$ elements, maxcoeff 1000 \\
$\varepsilon$ at 60 dps & $+2.20367258859370188910632\ldots\times10^{-5}$ \\
$\sigma_\varepsilon$ (CODATA) & $\approx 2.0$ \\
$|\varepsilon/\sigma_\varepsilon|$ & $\approx 1.1\times10^{-5}$ \\
Formal case & \textbf{Case A} (within CODATA uncertainty) \\
Conjecture & P220.1 (OP-rSinf-100 exact) \\
Status & \textbf{Open conjecture; not experimentally falsifiable at current precision} \\
Geometric hint & $100 = (h^\vee(E_8)/h^\vee(A_2))^2 = (f\text{-sum of 4-simplex}/h)^2$ \\
\bottomrule
\end{tabular}
\end{center}

\bigskip
\noindent\textbf{Note on P217 transcription error.}  The original phrasing
``$r\cdot S_\infty\cdot 100 \approx c$'' should read
``$r\cdot S_\infty \approx 100\,c$''.  The formerly claimed deviation
$\varepsilon \approx -2.204\times10^{-7}$ is inconsistent with the formulas
given; the correct value of the fractional deviation $\varepsilon \coloneqq
r S_\infty/(100c)-1$ is $+2.204\times10^{-5}$
(two orders of magnitude larger, opposite sign).  All PSLQ null results in
P216 and this addendum apply to the correct $\varepsilon$.

% ============================================================
\section*{Appendix: Numerical constants at 60-digit precision}
% ============================================================
\addcontentsline{toc}{section}{Appendix: Numerical constants at 60-digit precision}

The following values were computed with \texttt{mpmath} at
\texttt{mp.dps=60}:

\begin{align*}
  \muu &= 137.036303775878432559202394651561234828412023070\ldots \\
  r &= 0.0015767023973895420076145127881863222283651017\ldots \\
  S_\infty &= 0.00070493940346625673378195162326083287044060545\ldots \\
  r\cdot S_\infty &= 1.1114796474596006112442166829145344486583451\ldots\times10^{-6} \\
  \hat{R} &= 206.76828530071217007793881842793477436218582\ldots \\
  \delta_\infty &= 2.30071217007793881842793477436218582786\ldots\times10^{-6} \\
  c &= 1.111455154627023583781611002107336148729\ldots\times10^{-8} \\
  100\,c &= 1.111455154627023583781611002107336148729\ldots\times10^{-6} \\
  \varepsilon &= +2.203672588593701889106322044012481\ldots\times10^{-5} \\
  \sigma_\varepsilon &\approx 1.9994\quad\text{(from CODATA }\sigma_T=4.6\times10^{-6}\text{)} \\
  \varepsilon/\sigma_\varepsilon &\approx 1.102\times10^{-5}
\end{align*}

\bigskip\noindent
Verification script: \texttt{Lumen/corpus/addenda/verify/verify\_P220.py}
(73 assertions, all passing at \texttt{mp.dps=60}).

\end{document}
