% Addendum P204: Loop Coefficient Self-Similarity and the Simplicial f-Vector
% Author: Léon Fernando Vlegels
% © Léon Fernando Vlegels. CC BY 4.0.

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\title{\textbf{Addendum P204}\\[4pt]
Loop Coefficient Self-Similarity and the Simplicial $f$-Vector\\[2pt]
{\normalsize P18-T2 Programme: $G_2$-structured Perturbative Expansion for $m_\mu/m_e$}}
\author{L\'eon Fernando Vlegels}
\date{17 May 2026}

\begin{document}
\maketitle
\tableofcontents
\bigskip

%─────────────────────────────────────────────────────────────────────────────
\section{Introduction}
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The P18-T2 programme has produced a four-loop perturbative formula for the
muon-to-electron mass ratio in terms of TOE constants. In the natural expansion
parameter $\alpha = (4\pi^3+\pi^2+\pi)^{-1}$, this reads (P201)
\begin{equation}\label{eq:R4}
  R_4 \;=\; 207\bigl(1 + a_1\alpha + a_2\alpha^2 + a_3\alpha^3 + a_4\alpha^4\bigr),
\end{equation}
where the loop coefficients are
\begin{align}
  a_1 &= -\tfrac{1}{2\pi}, \label{eq:a1}\\
  a_2 &= f_2 \;=\; \tfrac{1}{\pi^2}+\tfrac{11}{16}, \label{eq:a2}\\
  a_3 &= N \;=\; -\tfrac{11}{4}\,\Omega_0\,\alpha^{-1}, \label{eq:a3}\\
  a_4 &= c_4 \;\approx\; 400305.875316\ldots, \label{eq:a4}
\end{align}
with $\Omega_0 = \pi^3/4$, and $R_4 = 206.7682830$ (CODATA 2018) by construction.

P200 identified $f_2$ as a pure $G_2$ structure constant;
P201 carried out an exhaustive PSLQ search for a closed form for $c_4$,
found none, and noted the near-coincidence
\begin{equation}\label{eq:near}
  c_4 \;\approx\; -N\alpha^{-1} \;=\; \tfrac{11}{4}\,\Omega_0\,\alpha^{-2},
  \qquad c_4 + N\alpha^{-1} = -1.501\ldots
\end{equation}
with a residual of $0.19$ CODATA standard deviations.

The present addendum examines the ratios $a_{n+1}/a_n$ between consecutive
loop coefficients. The main finding is that $a_4/a_3 \approx -\alpha^{-1}$
to within $4\times10^{-6}$ in relative terms (the near-coincidence of
eq.~\eqref{eq:near} in a different guise). We call this the
\emph{self-similarity hypothesis}: from loop order three onward the expansion
coefficients form an approximate geometric sequence with ratio $-\alpha^{-1}$.
We also exhibit a combinatorial structure that assigns exactly one simplex
class of the regular 4-simplex to each loop order; the total face count
$30 = h(E_8)$ (Coxeter number of $E_8$) links the simplicial geometry to
the exceptional Lie algebra.

\textbf{Status: OPEN.} The self-similarity is numerically established to
$3.7\times10^{-6}$ relative precision but has no analytic derivation.
The combinatorial $f$-vector assignment is exact.

%─────────────────────────────────────────────────────────────────────────────
\section{Ratio Analysis}
%─────────────────────────────────────────────────────────────────────────────

\subsection{Three consecutive ratios}

Table~\ref{tab:ratios} records the ratios $r_{n,n+1} = a_{n+1}/a_n$ for
$n = 1,2,3$.

\begin{table}[h]
\centering
\caption{Ratios between consecutive loop coefficients. The anomalous
  behaviour at $r_{12}$ and $r_{23}$ gives way to $r_{34} \approx -\alpha^{-1}$
  at four loops.}
\label{tab:ratios}
\begin{tabular}{@{}llll@{}}
\toprule
Transition & Ratio & Numerical value & Comment \\
\midrule
$r_{12} = a_2/a_1$ & $-2\pi f_2$ & $\approx -4.953$ & Sign change; $O(1)$ \\
$r_{23} = a_3/a_2$ & $N/f_2$ & $\approx -3703$ & Large; driven by $\alpha^{-1}$ in $N$ \\
$r_{34} = a_4/a_3$ & $c_4/N$ & $\approx -137.04$ & $\approx -\alpha^{-1}$ \\
\bottomrule
\end{tabular}
\end{table}

The ratio $r_{12} = -2\pi f_2$ is anomalous: it is $O(1)$ and sign-alternating.
The ratio $r_{23} = N/f_2$ is large because $N$ itself carries an explicit
$\alpha^{-1}$ factor (eq.~\eqref{eq:a3}); numerically $|r_{23}| \approx 3703$.
The ratio $r_{34}$, by contrast, is close to $-\alpha^{-1} \approx -137.036$.

\subsection{The $r_{34}$ near-coincidence}

From P201 eq.~(24): $c_4 + N\alpha^{-1} = -1.501$, so
\begin{equation}
  r_{34} \;=\; \frac{c_4}{N} \;=\; -\alpha^{-1} + \frac{1.501}{|N|}
           \;\approx\; -\alpha^{-1} + 5.14\times10^{-4}.
\end{equation}
The relative deviation from $-\alpha^{-1}$ is
\begin{equation}\label{eq:dev}
  \frac{|r_{34} - (-\alpha^{-1})|}{\alpha^{-1}}
  \;=\; \frac{|c_4 + N\alpha^{-1}|}{|N|\,\alpha^{-1}}
  \;=\; \frac{1.501}{|N|\,\alpha^{-1}}
  \;\approx\; 3.7\times10^{-6}.
\end{equation}
This is $3.7\times10^{-6}$ in relative terms, or equivalently a $0.00037\%$
agreement between $|r_{34}|$ and $\alpha^{-1}$.

\begin{remark}
  Equation~\eqref{eq:dev} makes precise what P201 called the ``near-coincidence'':
  the residual $c_4 + N\alpha^{-1} \approx -1.501$ is $0.19$ CODATA
  standard deviations from zero, and the corresponding relative deviation in
  $r_{34}$ is $3.7\times10^{-6}$.  The two statements are equivalent reformulations.
\end{remark}

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\section{Self-Similarity Hypothesis and Partial Sums}
%─────────────────────────────────────────────────────────────────────────────

\subsection{The hypothesis}

\begin{definition}
  The \emph{self-similarity hypothesis} states: for $n \geq 3$,
  \begin{equation}\label{eq:hyp}
    a_{n+1} \;\approx\; -\alpha^{-1}\, a_n.
  \end{equation}
  In particular, the four-loop hypothesis value is
  \begin{equation}\label{eq:a4hyp}
    a_4^{\rm hyp} \;=\; -\alpha^{-1}\, N \;=\; \tfrac{11}{4}\,\Omega_0\,\alpha^{-2}.
  \end{equation}
\end{definition}

Under this hypothesis the loop series is geometric from order three onward,
with common ratio $-\alpha^{-1} \approx -137$.  The sum of the geometric tail
from order three is
\begin{equation}
  \sum_{n=3}^{\infty} a_n \alpha^n \;=\; \frac{N\alpha^3}{1 + \alpha^{-1}\alpha}
  \;=\; \frac{N\alpha^3}{2},
\end{equation}
which converges since $|\text{ratio}|\cdot\alpha = \alpha^{-1}\cdot\alpha = 1$; this is
exactly the boundary of convergence. The hypothesis therefore does not define a
convergent series; it is a finite-truncation statement at four loops only.

\subsection{An exact algebraic identity}

\begin{proposition}\label{prop:S2=S4hyp}
  Under the self-similarity hypothesis,
  \begin{equation}
    S_4^{\rm hyp} \;=\; S_2,
  \end{equation}
  where $S_n = 207\sum_{k=0}^{n} a_k \alpha^k$ (with $a_0 = 1$) denotes the
  $n$-loop partial sum.
\end{proposition}

\begin{proof}
  $S_4^{\rm hyp} = S_3 + 207\,a_4^{\rm hyp}\,\alpha^4 = S_3 - 207\,N\alpha^{-1}\cdot\alpha^4
  = S_3 - 207\,N\alpha^3$.
  Since $S_3 = S_2 + 207\,N\alpha^3$ it follows that
  $S_4^{\rm hyp} = S_2 + 207\,N\alpha^3 - 207\,N\alpha^3 = S_2$, using
  $\alpha^{-1}\cdot\alpha = 1$ exactly.
\end{proof}

\begin{remark}
  Proposition~\ref{prop:S2=S4hyp} shows that the self-similarity hypothesis
  \emph{exactly undoes} the three-loop correction at one order higher.
  The three-loop term $207\,N\alpha^3 \approx -0.235$ drives $S_3$ far below
  the experimental value; the hypothesis value $a_4^{\rm hyp} = -N\alpha^{-1}$
  restores exactly $+207\,N\alpha^3$, returning the sum to $S_2$.
\end{remark}

\subsection{Straddling of the CODATA value}

From P200: $S_2 = 206.76828388 > R_{\rm exp} = 206.7682830$ (the two-loop sum
overshoots by $\delta_2 = 8.8\times10^{-7}$).
From P201: $S_3 = 206.5333\ldots < R_{\rm exp}$ (the three-loop sum undershoots
by $0.235$).
By Proposition~\ref{prop:S2=S4hyp}: $S_4^{\rm hyp} = S_2 = 206.76828388 > R_{\rm exp}$.
The exact value $S_4 = R_{\rm exp}$ (by construction of $c_4$).

Therefore:
\begin{equation}
  S_3 \;<\; R_{\rm exp} \;=\; S_4^{\rm exact} \;<\; S_4^{\rm hyp} \;=\; S_2,
\end{equation}
and $S_4^{\rm hyp} - R_{\rm exp} = S_2 - R_{\rm exp} = 8.8\times10^{-7}$.

\begin{table}[h]
\centering
\caption{Partial sums of the $G_2$ loop series. The hypothesis value
  $a_4^{\rm hyp} = -\alpha^{-1}N$ places $S_4^{\rm hyp} = S_2$ above
  $R_{\rm exp}$; the exact $c_4$ returns it to $R_{\rm exp}$.}
\label{tab:sums}
\begin{tabular}{@{}lll@{}}
\toprule
Sum & Value & Relation to $R_{\rm exp}$ \\
\midrule
$S_2$ (two-loop, P200)
  & $206.76828388\ldots$
  & $> R_{\rm exp}$ by $8.8\times10^{-7}$ \\
$S_3$ (three-loop, P201)
  & $206.5333\ldots$
  & $< R_{\rm exp}$ by $0.235$ \\
$S_4^{\rm exact}$ (four-loop, P201)
  & $206.7682830$ (CODATA)
  & $= R_{\rm exp}$ by construction \\
$S_4^{\rm hyp}$ (hypothesis)
  & $= S_2 = 206.76828388\ldots$
  & $> R_{\rm exp}$ by $8.8\times10^{-7}$ \\
\bottomrule
\end{tabular}
\end{table}

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\section{The Simplicial $f$-Vector Assignment}
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\subsection{The regular 4-simplex}

The regular 4-simplex $\Delta^4$ has five vertices.  Its $f$-vector
$(f_0, f_1, f_2, f_3)$ — listing the number of faces at each dimension from
vertices to tetrahedra — is
\begin{equation}\label{eq:fvec}
  (f_0, f_1, f_2, f_3) \;=\; (5, 10, 10, 5).
\end{equation}
These are exactly the binomial coefficients $\binom{5}{k}$ for $k = 1, 2, 3, 4$:
\begin{equation}
  f_{k-1} \;=\; \binom{5}{k}, \qquad k = 1,2,3,4.
\end{equation}
The $f$-vector is palindromic: $(f_0, f_1, f_2, f_3) = (f_3, f_2, f_1, f_0)$.

\subsection{Assignment to loop orders}

The Wheel (kernel coherence instrument) has exactly five operators:
Fork, Weld, Plateau, Oscillate, Perturb.
These five operators are identified with the five vertices of $\Delta^4$.
A loop amplitude of order $k$ involves an unordered $k$-tuple of operators;
the number of such tuples is $\binom{5}{k}$.

\begin{definition}[Simplicial loop assignment]
  \label{def:assign}
  Loop order $k$ is assigned to the $(k-1)$-faces of $\Delta^4$. The number of
  $(k-1)$-faces is $f_{k-1} = \binom{5}{k}$. Explicitly:
  \begin{center}
  \begin{tabular}{@{}llll@{}}
  \toprule
  Loop order & Faces of $\Delta^4$ & Count $= \binom{5}{k}$ & Coefficient \\
  \midrule
  $k=1$ (one-loop) & vertices & $\binom{5}{1} = 5$ & $a_1 = -1/(2\pi)$ \\
  $k=2$ (two-loop) & edges & $\binom{5}{2} = 10$ & $a_2 = 1/\pi^2 + 11/16$ \\
  $k=3$ (three-loop) & triangles & $\binom{5}{3} = 10$ & $a_3 = N$ \\
  $k=4$ (four-loop) & tetrahedra & $\binom{5}{4} = 5$ & $a_4 = c_4$ \\
  \bottomrule
  \end{tabular}
  \end{center}
\end{definition}

\begin{remark}
  The palindromic symmetry of the $f$-vector — equal counts at one-loop ($k=1$)
  and four-loop ($k=4$), and at two-loop ($k=2$) and three-loop ($k=3$) — is
  a structural symmetry of $\Delta^4$ (the $A_4$ Weyl symmetry on vertices).
  Whether this symmetry has a physical content in the loop expansion is an
  open question.
\end{remark}

\begin{remark}
  The assignment is purely combinatorial: it counts distinct Wheel-operator
  multi-combinations at each loop order.  It does not constrain the numerical
  values of the coefficients $a_k$.
\end{remark}

%─────────────────────────────────────────────────────────────────────────────
\section{The $E_8$ Connection}
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The total number of proper faces of $\Delta^4$ (all dimensions, excluding the
empty face and $\Delta^4$ itself) is
\begin{equation}\label{eq:total}
  \sum_{k=1}^{4} \binom{5}{k} \;=\; 5 + 10 + 10 + 5 \;=\; 30.
\end{equation}
The Coxeter number of $E_8$ is $h(E_8) = 30$ (a classical result; $E_8$ has
rank 8 and dimension 248, and its Coxeter number appears in the exponents
$1, 7, 11, 13, 17, 19, 23, 29$ through $h = 1 + e_{\rm max} = 30$).

\begin{proposition}\label{prop:E8}
  The total proper face count of the regular $4$-simplex equals the Coxeter
  number of $E_8$:
  \begin{equation}
    \sum_{k=0}^{3} \binom{5}{k+1} \;=\; 30 \;=\; h(E_8).
  \end{equation}
\end{proposition}

\begin{proof}
  Direct computation: $5 + 10 + 10 + 5 = 30$, $h(E_8) = 30$. \qed
\end{proof}

\begin{remark}
  The numerical coincidence $\sum f_k = 30 = h(E_8)$ is exact.  Whether
  it reflects a geometric relation between the $A_4$ root system (underlying
  $\Delta^4$) and the $E_8$ lattice — such as through the embedding
  $A_4 \hookrightarrow E_8$ or through the McKay correspondence — is an
  open question (see \S\ref{sec:open}).
\end{remark}

\begin{remark}
  Additional Lie-algebraic reading: $\sum f_k = 30 = \dim A_4 + h(A_4) = 24 + 5$,
  or $= 5 \cdot 6 = f_0 \cdot (f_0 + 1)$.  These factorisations are
  consistent with the $G_2$ decomposition $207 = 14 \cdot 15 - 3$ established
  in P196, where the same pattern $n(n+1)$ appears with $n = 14 = \dim G_2$.
\end{remark}

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\section{Open Problems and Status}\label{sec:open}
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\subsection{Status}

\textbf{OPEN.}  Two results in this addendum are exact:
\begin{enumerate}
  \item Proposition~\ref{prop:S2=S4hyp}: the identity $S_4^{\rm hyp} = S_2$
    (Proposition~\ref{prop:S2=S4hyp}).
  \item Proposition~\ref{prop:E8}: $\sum f_k = 30 = h(E_8)$.
\end{enumerate}
One result is numerical:
\begin{enumerate}
  \setcounter{enumi}{2}
  \item The self-similarity $r_{34} \approx -\alpha^{-1}$ (equivalently,
    eq.~\eqref{eq:near}) is established to $3.7\times10^{-6}$ relative
    precision from CODATA 2018, but has no analytic derivation.
\end{enumerate}
Open Problem OP-C4 from P201 remains in force: $c_4$ has no known closed-form
expression in the TOE constants.

\subsection{Open problems}

\begin{openprob}[Self-similarity at five loops]
  Does $a_5/a_4 \approx -\alpha^{-1}$?  If so, the five-loop coefficient
  would be $a_5 \approx -\alpha^{-1} c_4 \approx -(11/4)\Omega_0\,\alpha^{-3}
  \approx -5.49\times10^7$.  An improved measurement of $m_\mu/m_e$ to
  $\lesssim 10^{-8}$ would test this prediction.
\end{openprob}

\begin{openprob}[Analytic derivation of self-similarity]
  Derive $a_{n+1} = -\alpha^{-1} a_n$ from the $G_2$ loop integral structure.
  The three-loop coefficient $a_3 = N$ contains an explicit $\alpha^{-1}$
  from the TOE geometry (P199); does the four-loop amplitude inherit a second
  factor of $\alpha^{-1}$ from the same source?
\end{openprob}

\begin{openprob}[$A_4$--$E_8$ geometric link]
  The $A_4$ root system underlies the 4-simplex; $E_8$ contains $A_4$ as a
  sub-root-system.  Does the equality $\sum f_k(\Delta^4) = h(E_8)$ arise from
  the McKay correspondence for the binary icosahedral group $I_{120}$, whose
  McKay graph is the $E_8$ Dynkin diagram?
\end{openprob}

\begin{openprob}[Palindromic symmetry]
  The palindrome $(f_0, f_1, f_2, f_3) = (5, 10, 10, 5)$ implies that
  one-loop and four-loop amplitudes share the same combinatorial weight
  ($=5$), as do two-loop and three-loop ($=10$).  Does this symmetry
  constrain the loop-coefficient ratios, or produce a relation between
  $a_1 a_4$ and $a_2 a_3$?
\end{openprob}

\end{document}
