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\title{\textbf{Addendum P154}\\[4pt]
Laplace-Type Transform of the G\textsubscript{2} Root Spectral Measure\\[2pt]
and Gap G1}
\author{L\'eon Fernando Vlegels}
\date{May 2026}

\begin{document}
\maketitle

\begin{abstract}
Gap G1~$=\,(432 - \pi\alpha^{-1})/(\pi\alpha^{-1}) \approx 0.3456\%$ remains open after
P151--P153.  The integer $432 = J_{\mathrm{short}}\times J_{\mathrm{long}}$ emerges from
G\textsubscript{2} Killing norms, while the breath period $\Omega = \pi\alpha^{-1}$ arises
from a continuous S\textsuperscript{3} integral.  We ask whether a Laplace-type smearing of
the discrete G\textsubscript{2} root spectral measure $\mu = \sum_\alpha \delta(t -
\langle\alpha,H_s\rangle)$ converts the integer second moment $\sum_\alpha \langle\alpha,
H_s\rangle^2 = 48$ into the transcendental target $\Omega/9 \approx 47.835$.  Six
candidates (A--F) are computed numerically.  All candidates that formally hit the target do
so either by tautological construction or because G1 $< 1\%$ makes the discrete value
already within the tolerance threshold.  No candidate provides geometric input that explains
the transcendental correction.  G1 remains open.
\end{abstract}

%%─────────────────────────────────────────────────────────────────
\section{Setup: G\textsubscript{2} root spectral measure}
%%─────────────────────────────────────────────────────────────────

Fix the TOE constants
\begin{equation}
  \alpha^{-1} = 4\pi^3+\pi^2+\pi,\qquad
  \Omega = \pi\alpha^{-1} \approx 430.512 \qquad (\text{breath period}).
\end{equation}
G\textsubscript{2} has 12 roots.  The projections of all roots onto the coroot $H_s$ of the
short simple root $\alpha_1$ are
\begin{equation}
  \bigl\{\langle\alpha,H_s\rangle\bigr\} \;=\;
  \{-3,-3,-2,-1,-1,0,0,1,1,2,3,3\}.
\end{equation}
The \emph{discrete root spectral measure} is
\begin{equation}
  \mu_{\mathrm{disc}} \;=\; \sum_{\alpha\in\Delta} \delta\!\bigl(t - \langle\alpha,H_s\rangle\bigr).
\end{equation}
Its second moment is the Killing form restricted to $H_s$:
\begin{equation}
  M_2^{\mathrm{disc}} \;=\; \int t^2\,\mathrm{d}\mu_{\mathrm{disc}}(t)
  \;=\; \sum_\alpha \langle\alpha,H_s\rangle^2 \;=\; 48.
\end{equation}
The product of short and long Killing norms gives $J_s\times J_l = 12\times 36 = 432$,
recovering the integer approximant of $\Omega$.  The residual gap is
\begin{equation}
  \mathrm{G1} \;=\; \frac{432-\Omega}{\Omega}
  \;=\; \frac{432-\pi\alpha^{-1}}{\pi\alpha^{-1}}
  \;\approx\; 3.456\times10^{-3}.
\end{equation}
The hypothesis under investigation: some natural smearing of $\mu_{\mathrm{disc}}$ at the
breath scale yields a continuous second moment equal to $\Omega/9 \approx 47.835$, closing
G1.  (The factor 9 appears because $M_2^{\mathrm{disc}} = 48$ and $\Omega/9 \approx 48$, so
$\Omega/9$ is the ``corrected'' first-moment-squared-over-roots comparison.)

%%─────────────────────────────────────────────────────────────────
\section{Candidates A--F: definitions and numerical results}
%%─────────────────────────────────────────────────────────────────

All computations are performed by \texttt{verify\_P154.py} (see \S\ref{sec:script}).  A
candidate \textbf{passes} if its result agrees with its declared target to within $1\%$; a
pass due to tautological construction or to G1 $< 1\%$ is flagged explicitly.

\paragraph{Candidate A — Gaussian smearing at $\sigma^2 = \pi$.}
Convolving $\mu_{\mathrm{disc}}$ with a Gaussian of variance $\sigma^2 = \pi$ shifts the
second moment by $\sigma^2$ per root:
\begin{equation}
  M_2^A \;=\; \sum_\alpha \bigl[\langle\alpha,H_s\rangle^2 + \pi\bigr]
  \;=\; 48 + 12\pi \;\approx\; 85.70.
\end{equation}
This far exceeds $\Omega/9 \approx 47.84$.  \textbf{FAIL.}

\paragraph{Candidate B — Laplace MGF, mean-square tilt.}
Define the one-sided Laplace functional
\begin{equation}
  L(s) \;=\; \sum_\alpha e^{-s\langle\alpha,H_s\rangle^2}
  \;=\; 4e^{-9s}+2e^{-4s}+4e^{-s}+2, \quad s\ge 0.
\end{equation}
$L$ has range $(2,12]$; neither $\alpha^{-1}\approx 137$ nor $\alpha\approx 0.0073$ lies in
this range, so there is no $s^*$ with $L(s^*)=\alpha^{\pm1}$.  The \emph{weighted
mean-square} is $W(s) = -L'(s)/L(s)$, with $W(0) = 48/12 = 4$ and $W(\infty)\to 0$.
We seek $s^*$ such that $N\cdot W(s^*) = \Omega/9$, i.e.\
\begin{equation}
  -\frac{L'(s^*)}{L(s^*)} \;=\; \frac{\Omega}{108} \;\approx\; 3.9863.
\end{equation}
Bisection gives $s^* \approx 9.847\times10^{-4}$.  The leading-order approximation uses
\begin{equation}
  \frac{d}{ds}\!\left[-\frac{L'}{L}\right]\bigg|_{s=0}
  = \frac{-\sum p^4 \cdot N + \bigl(\sum p^2\bigr)^2}{N^2}
  = \frac{-360\times12 + 2304}{144} = -14,
\end{equation}
so $s^* \approx (4 - \Omega/108)/14 = (432-\Omega)/1512$.  The factor $1512 = 14\times 108$
carries no recognisable G\textsubscript{2} invariant; it encodes the fourth-moment
$\sum p^4 = 360$ and the number of roots.  Concretely,
$s^*\times\Omega \approx 0.4239$ and $s^*\times 432 \approx 0.4254$, neither a
simple constant.  Candidate B \textbf{passes numerically} but with $s^* \propto \mathrm{G1}$: the
parameter that solves the equation \emph{is} the gap in disguise.

\paragraph{Candidate C — Bilateral MGF, ratio test.}
\begin{equation}
  M(s) = 4\cosh 3s + 2\cosh 2s + 4\cosh s + 2,\quad
  M''(s) = 36\cosh 3s + 8\cosh 2s + 4\cosh s.
\end{equation}
$M''(s)/M(s)$ is even and increases monotonically from $4$ (at $s=0$) to $9$ (as
$|s|\to\infty$, dominated by $e^{3|s|}$).  The target $\Omega/432 \approx 0.997 < 4$, so no
real $s^*$ satisfies $M''(s^*)/M(s^*)=\Omega/432$.  At $s=0$, the ratio $M''(0)/M(0) =
4.000$, which is within $1\%$ of $\Omega/9/N \approx 3.986$ only because G1 $< 1\%$.
\textbf{PASS at $s=0$, trivially; no proper $s^*$ exists.}

\paragraph{Candidate D — Heat-kernel smearing on $\mathbb{R}$.}
The heat-kernel second moment shifts as $M_2(t) = 48+2t$.  Solving $48+2t^*=\Omega/9$:
\begin{equation}
  t^* = \tfrac{1}{2}\!\left(\tfrac{\Omega}{9}-48\right)
      = -\frac{432-\Omega}{18}
      \;\approx\; -0.08265.
\end{equation}
Since $\Omega < 432$, $t^* < 0$.  Diffusion always \emph{adds} variance; a negative
diffusion time is unphysical.  \textbf{PASS by construction, but unphysical.}

\paragraph{Candidate E — Uniform measure on a circle of radius $R$.}
Replacing the 12 point masses with a continuous uniform measure on a circle of radius $R$
gives a per-root second moment $R^2/2$.  The condition $N\cdot R^2/2 = \Omega$ yields
$R^2 = \Omega/6 = \pi\alpha^{-1}/6$, a transcendental number with no simple form.
The variant requiring $N\cdot R^2/2 = \Omega/9$ gives $R^2 = \Omega/54$, equally
transcendental and by construction.  \textbf{PASS by construction; no predictive content.}

\paragraph{Candidate F — Weyl character at imaginary argument.}
The G\textsubscript{2} adjoint Weyl character ($\dim = 14$) at group element $e^{isH_s}$ is
\begin{equation}
  \chi_{\mathrm{adj}}(s) = 4\cos 3s + 2\cos 2s + 4\cos s + 4.
\end{equation}
$\chi_{\mathrm{adj}}(0)=14$, $\chi_{\mathrm{adj}}(\pi)=-2$; the unique zero in $[0,\pi]$ is
at $s_0 \approx 0.8812\pi$.  Values at $s=\pi/n$ for $n=1,\ldots,12$ are listed in
Table~\ref{tab:weyl}.  The character crosses $\alpha = \alpha^{-1,-1}\approx 7.297\times10^{-3}$
at $s^*\approx 0.8809\pi$, near but not equal to the zero.  No ratio $s^*/\pi$ matches a
simple fraction to four significant figures.  \textbf{PASS in the sense that a crossing
exists, but the target ($\alpha$) is not the second-moment target $\Omega/9$;
the candidates are incommensurable.}

\begin{table}[h]
\centering
\begin{tabular}{rll}
\hline
$n$ & $s=\pi/n$ & $\chi_{\mathrm{adj}}(\pi/n)$ \\
\hline
1 & $\pi$      & $-2$ \\
2 & $\pi/2$    & $+2$ \\
3 & $\pi/3$    & $+1$ \\
4 & $\pi/4$    & $+4$ \\
5 & $\pi/5$    & $+6.618$ \\
6 & $\pi/6$    & $+8.464$ \\
\hline
\end{tabular}
\quad
\begin{tabular}{rll}
\hline
$n$ & $s=\pi/n$ & $\chi_{\mathrm{adj}}(\pi/n)$ \\
\hline
7  & $\pi/7$  & $+9.741$ \\
8  & $\pi/8$  & $+10.640$ \\
9  & $\pi/9$  & $+11.291$ \\
10 & $\pi/10$ & $+11.773$ \\
11 & $\pi/11$ & $+12.140$ \\
12 & $\pi/12$ & $+12.424$ \\
\hline
\end{tabular}
\caption{Weyl character $\chi_{\mathrm{adj}}(\pi/n)$ for G\textsubscript{2}.
Note $\chi_{\mathrm{adj}}(\pi/5)=4+\sqrt{5}\cdot\frac{6+\sqrt{5}}{4}\approx 4+2.618$;
the golden ratio enters but does not produce $\alpha^{-1}$.}
\label{tab:weyl}
\end{table}

%%─────────────────────────────────────────────────────────────────
\section{Script output (verbatim)}
\label{sec:script}
%%─────────────────────────────────────────────────────────────────

\begin{verbatim}
=================================================================
P154 SPECTRAL LAPLACE VERIFICATION
=================================================================
ALPHA_INV        = 137.0363037759
BREATH_PERIOD    = 430.5122452174
BREATH_PERIOD/9  = 47.8346939130   <- transcendental target
J_discrete       = 48          <- Killing form (integer)
G1 = (432-BP)/BP = 3.45577809e-03  (0.345578%)

CANDIDATE A — Gaussian smearing sigma^2 = pi
  M2_A = 48 + 12*pi = 85.69911184
Candidate A: result=85.69911184, target=47.83469391, ratio=1.79156810, FAIL

CANDIDATE B — Laplace MGF  L(s) = Sum_alpha exp(-s*<alpha,H_s>^2)
  L(0)=12.0000,  lim_inf L(s)=2.0000
  -L'(0)/L(0) = 4.000000 = J_discrete/N = 48/12 = 4
  Seek s* where -L'(s*)/L(s*) = BP/108 = 3.9862244928
  s* = 0.0009847302
  s* / (1/BREATH_PERIOD)  = 0.423938
  s* / (pi/432)           = 0.135410
  d(-L'/L)/ds|_{s=0}      = -14.000000
  Linear estimate s* ~ (4 - BP/108)/14 = (432-BP)/1512
  s* * BREATH_PERIOD = 0.42393842
  s* * 432           = 0.42540346
Candidate B: result=47.83469391, target=47.83469391, ratio=1.00000000, PASS
  [s* = 9.847e-04, proportional to G1; no independent geometric form]

CANDIDATE C — Bilateral MGF  M(s)=4cosh(3s)+2cosh(2s)+4cosh(s)+2
  M(0)=12, M''(0)=48, M''(s)/M(s) in [4, 9] (monotone increasing)
  Target BP/432=0.99656 < 4: no real s* satisfies M''/M = BP/432
  Alt target TARGET/N = 3.98622 < 4: no s* for M''/M either
Candidate C: result=4.00000000, target=3.98622449, ratio=1.00345578, PASS
  [at s=0 only; passes because G1 < 1%]

CANDIDATE D — Heat-kernel smearing  M2_D(t) = 48 + 2t
  t* = (BP/9 - 48)/2 = -0.0826530435
  t* = -(432-BP)/18  = -0.0826530435
  t* < 0: unphysical (diffusion adds variance)
Candidate D: result=47.83469391, target=47.83469391, ratio=1.00000000, PASS
  [by construction; unphysical t* < 0]

CANDIDATE E — Uniform circular measure; N*R^2/2 = BREATH_PERIOD
  R^2 = BP/6 = 71.7520408696
  R   = sqrt(pi*alpha_inv/6) = 8.4706576409
  R^2/pi = ALPHA_INV/6 = 22.83938396  (transcendental)
Candidate E: result=430.51224522, target=430.51224522, ratio=1.00000000, PASS
  [by construction; target is BP, not BP/9]
  For BP/9 version: R^2 = BP/54 = 7.9724..., also by construction.

CANDIDATE F — Weyl character  chi(s)=4cos(3s)+2cos(2s)+4cos(s)+4
  chi(0)=14 (=dim G_2),  chi(pi)=-2
  Zero in [0,pi]: s_zero = 2.7683344491  (s/pi = 0.88119)
  Crossing chi(s*) = alpha = 7.297e-3 at s* = 2.7675701265 (s*/pi = 0.88094)
Candidate F: result=0.00729734, target=0.00729734, ratio=1.00000000, PASS
  [crossing exists; but target is alpha, not BP/9]

=================================================================
SUMMARY
=================================================================
G1               = (432-BP)/BP = 3.45577809e-03
BREATH_PERIOD/9  = 47.8346939130
J_discrete       = 48   (integer)
48 - BP/9        = 0.1653060870  (size of gap in moment space)

Candidate A: FAIL  [48+12pi = 85.7  vs  target 47.83]
Candidate B: PASS  [s* proportional to G1; no new geometric content]
Candidate C: PASS  [at s=0; no s* for exact match; G1 < 1% gives free ride]
Candidate D: PASS  [unphysical t* < 0; trivial by construction]
Candidate E: PASS  [by construction; R^2 = BP/54]
Candidate F: PASS  [crossing of alpha; different target from BP/9]

CONCLUSION: No candidate naturally closes G1.
G1 = (432-pi*alpha_inv)/(pi*alpha_inv) requires transcendental input
beyond G_2 geometry. The gap 0.345578% remains open after P154.
\end{verbatim}

%%─────────────────────────────────────────────────────────────────
\section{Findings and implication for G1}
%%─────────────────────────────────────────────────────────────────

\subsection*{What passes and why}

Of the six candidates, only Candidate~A genuinely \emph{tests} a specific physical
hypothesis ($\sigma^2=\pi$) and fails decisively.  The remaining passes are of three
qualitatively distinct types:

\begin{description}
\item[Tautological (D, E)] The smearing parameter is fixed by the equation
  $\text{result}=\text{target}$ itself, so the pass is definitional.

\item[G1-free-ride (C)] The bilateral MGF ratio $M''(0)/M(0)=4$ deviates from the target
  $\Omega/108\approx 3.9862$ by exactly $1+\mathrm{G1}$, so any check with tolerance $> 0.35\%$
  trivially passes.  No smearing brings the ratio below $4$.

\item[Proportional to G1 (B)] The MGF root $s^*$ exists numerically and equals
  $(432-\Omega)/1512$ to leading order.  The numerator is G1 times $\Omega$.  Far from
  providing geometric input, $s^*$ \emph{encodes} the gap --- it is not an independent
  derivation of G1.

\item[Different target (F)] The Weyl character crossing at $s^*\approx 0.881\pi$ locates
  the fine-structure constant $\alpha$ in the character spectrum, which is a genuinely
  interesting fact about G\textsubscript{2}, but its target ($\alpha$, not $\Omega/9$) is
  incommensurable with the second-moment question.
\end{description}

\subsection*{Why no Laplace smearing closes G1}

The root of the difficulty is algebraic.  The discrete second moment is the integer $48$.
The target $\Omega/9 = \pi(4\pi^3+\pi^2+\pi)/9$ is a specific transcendental combination
of $\pi$.  The difference
\begin{equation}
  48 - \frac{\Omega}{9} \;=\; 48 - \frac{\pi(4\pi^3+\pi^2+\pi)}{9}
  \;=\; 48 - \frac{4\pi^4+\pi^3+\pi^2}{9}
  \;\approx\; 0.1653
\end{equation}
is itself transcendental.  A Laplace smearing of $\mu_{\mathrm{disc}}$ modifies the second
moment by an amount that depends on the smearing parameter $s$ and the root positions, both
of which are algebraic (the root projections are integers).  No algebraic function of
integer data can produce the transcendental correction $(432-\Omega)/18$ without
\emph{additional transcendental input}.

The additional input required is precisely the identification $\alpha^{-1} = 4\pi^3+\pi^2+\pi$:
once that identity is granted, $\Omega/9$ is determined, and the ``gap'' is zero by
definition.  What is missing is not a smearing prescription but a \emph{derivation of why}
the coroot second moment $48$ and the continuous breath period $\Omega$ are related by the
factor $9$ up to the residual $\approx 0.345\%$.  Such a derivation would need to originate
from a continuous geometric structure that connects the G\textsubscript{2} root system to
S\textsuperscript{3} holonomy at the TOE scale --- a layer of geometry not present in the
discrete Killing form alone.

\subsection*{What would be needed}

Closing G1 via spectral methods would require one of the following:

\begin{enumerate}
\item A continuous measure on the G\textsubscript{2} root system (e.g.\ the Plancherel
  measure of the associated Lie group) whose second moment evaluates to $\Omega/9$
  \emph{from first principles}, not by setting a free parameter.

\item A spectral identity of the form $\int_{\mathrm{G_2}} \langle\alpha,H_s\rangle^2
  \,\mathrm{d}\mu_{\mathrm{Haar}} = c\cdot\Omega$ for an integer $c$ derived from
  representation theory.

\item An embedding argument showing that the G\textsubscript{2} coroot projections,
  when lifted to S\textsuperscript{3} via the TOE Hopf lapse, acquire a continuous
  spectral density whose second moment is $\Omega/9$ by the same identities that
  produce $\alpha^{-1}=4\pi^3+\pi^2+\pi$.
\end{enumerate}

None of these is established.  The Weyl character result (Candidate~F) hints that
G\textsubscript{2}'s character spectrum makes contact with $\alpha$ at a specific
imaginary argument $\approx 0.881\pi$, which is suggestive of path (3), but the connection
to the second-moment question is not yet explicit.

\subsection*{Status of G1}

\textbf{G1 remains open.}  The discrete G\textsubscript{2} root Killing form ($=48$) and
the continuous breath-period second moment ($\Omega/9\approx 47.835$) are related by
$\mathrm{G1}\approx 0.3456\%$, a transcendental gap that no Laplace-type smearing of the
discrete measure closes without importing the very transcendental it is supposed to derive.

\bigskip
\hrule
\bigskip
{\small \textcopyright\ L\'eon Fernando Vlegels. MIT License.}

\end{document}
