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\title{\textbf{Addendum P116: $\tau$ Mass from P$_{13}$ Peirce Geometry}\\
\large Fold-Map Density Modification and the Hopf S$^1$ Fiber in J$_3(\mathbb{O})$}
\author{L\'eon Fernando Vlegels}
\date{2026-05-14}

\begin{document}
\maketitle

\begin{abstract}
P115 closed the charged lepton sector for the electron and muon: $Z_e = 1$ by the T1
spectral anchor, $Z_\mu$ from the P$_{12}$ Peirce bridge EM walk.
The $\tau$ was stated to arise from the $B^4/S^3$ fold-map (P36).
Here we identify the \emph{Peirce sector origin} of that fold-map:
the P$_{13}$ block (J$_1 \leftrightarrow$ J$_3$, the maximal off-diagonal sector)
is precisely the Hopf S$^1$ fiber over which the fold rotation $R_\Theta$ acts.
This identification promotes the P36 mass formula to a first-principles derivation:
the S$^1$-fiber occupation fraction $\lambda_\tau = \mu_1/(\mu_0+\mu_1) = \mathrm{MU}/(1+\mathrm{MU})$
is fixed by detailed balance in the fold-map; it enters the spectral density
$\rho_\lambda(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi\lambda x$
as a scale on the monadic S$^1$-fiber term; the resulting self-lensing energy
$E_\mathrm{self}(\lambda_\tau) = 13.419$ gives
\[
  m_\tau/m_e = \exp\!\bigl(\mathrm{MU}\cdot(E_\mathrm{self}(\lambda_\tau)-\pi)\bigr) = 3474.6
  \quad (-0.075\%\text{ vs PDG }3477.23).
\]
The P$_{13}$ walk amplitude $A_\mathrm{walk}(P_{13}) = \cos^2(3\pi/14)\cos(6\pi/14) = 0.136$
is topological (not an EM correction), confirming that the mechanism is qualitatively
distinct from the P$_{12}$ muon renormalization.
All three charged leptons are now derived from first principles within J$_3(\mathbb{O})$.
\end{abstract}

\tableofcontents

%------------------------------------------------------------
\section{Background: Peirce block hierarchy and lepton assignments}

In J$_3(\mathbb{O})$ the Peirce decomposition is
\begin{equation}
  \mathrm{J}_3(\mathbb{O}) = J_1 \oplus J_2 \oplus J_3
  \oplus P_{12} \oplus P_{13} \oplus P_{23},
\end{equation}
with sector assignments:
\begin{align}
  J_1 &= \text{electron sector} \quad (m_e = \text{spectral anchor}), \\
  J_2 &= \text{muon sector} \quad (m_\mu\text{: spectral ladder, P36/P80}), \\
  J_3 &= \text{tau sector} \quad (m_\tau\text{: fold-map, P36}).
\end{align}
P115 derived the three mass mechanisms:
\begin{center}
\begin{tabular}{llll}
\toprule
Lepton & Sector & Bridge & Mechanism \\
\midrule
$e$ & $J_1$ & none (self) & Spectral anchor; $Z_e=1$ by T1 \\
$\mu$ & $J_2$ & $P_{12}$ (adjacent) & EM wavefunction renorm $Z_\mu$ \\
$\tau$ & $J_3$ & $P_{13}$ (maximal) & Topological fold-map density mod \\
\bottomrule
\end{tabular}
\end{center}
Here we derive the $\tau$ entry from the P$_{13}$ Peirce geometry.

%------------------------------------------------------------
\section{P$_{13}$ as the Hopf S$^1$ fiber}

\subsection{Peirce block distances}

In J$_3(\mathbb{O})$ the three off-diagonal blocks have different ``distances''
in the G$_2$ Weyl chamber, measured by the number of minimal Weyl steps $\theta_C = \pi/14$:
\begin{align}
  P_{12} &: \quad \Delta_{12} = 1 \quad (\text{adjacent sectors}) \\
  P_{23} &: \quad \Delta_{23} = 1 \quad (\text{adjacent sectors}) \\
  P_{13} &: \quad \Delta_{13} = 3 \quad (\text{non-adjacent, maximal diagonal})
\end{align}
The P$_{13}$ block spans the full $J_1 \to J_3$ diagonal.
In the B$^4/S^3$ fold-map (P36), the Hopf fibration $\pi: S^3 \to S^2$ has
fiber $S^1$ whose angular coordinate $\theta$ ranges over $[0, 2\pi)$.
The P$_{13}$ block, as the maximal off-diagonal sector connecting the
first and third diagonal idempotents, is the algebraic representative of this S$^1$ fiber:
\begin{equation}
  P_{13} \cong \mathbb{O} \quad (\dim = 8)
  \qquad \Longleftrightarrow \qquad
  \text{Hopf S}^1\text{ fiber over }S^2 \subset B^4/S^3.
\end{equation}

\subsection{Walk amplitude}

The G$_2$ walk amplitude through P$_{13}$ at three minimal Weyl steps is:
\begin{equation}
  A_\mathrm{walk}(P_{13})
  = \cos\!\tfrac{3\pi}{14}
  \times \cos\!\tfrac{6\pi}{14}
  \times \cos\!\tfrac{3\pi}{14}
  = \cos^2\!\tfrac{3\pi}{14} \cdot \cos\!\tfrac{6\pi}{14}.
\end{equation}
Using the complementary-angle identity $\cos(3\pi/14) = \sin(2\pi/7)$ and
$\cos(6\pi/14) = \sin(\pi/14) = \lambda_\mathrm{Wolfenstein}$:
\begin{equation}
  A_\mathrm{walk}(P_{13}) = \sin^2\!\tfrac{2\pi}{7} \cdot \sin\!\tfrac{\pi}{14} = 0.1358.
\end{equation}

\begin{remark}
This is a \emph{topological} amplitude, not an EM coupling.
For P$_{12}$ the amplitude entered as an EM self-energy correction $Z_\mu = 1 + \alpha A_\mathrm{NLO}$.
For P$_{13}$ the amplitude characterises the S$^1$-fiber density support:
it is the fractional weight of the monadic sector $J_3$ in the full spectral measure.
No factor of $\alpha$ appears because the mechanism is topological (fiber area),
not electromagnetic (loop integral).
\end{remark}

%------------------------------------------------------------
\section{Fold-map detailed balance and $\lambda_\tau$}

\subsection{The fold-map $F_\Theta$}

From P36, the fold-map is $F_\Theta = \Pi_\downarrow \circ R_\Theta \circ \Pi_\uparrow$,
where $R_\Theta$ is rotation by angle $\Theta$ in the P$_{13}$ fiber and
$\Pi_\uparrow$, $\Pi_\downarrow$ are the projection and embedding operators
between the spectral layers $\mu_0$ (bulk) and $\mu_1$ (edge).

The spectral energy moments are:
\begin{align}
  \mu_0 &= \int_0^1 \rho(x)\,dx = 4\pi^3 + \pi^2 + \pi
         \quad (\alpha^{-1}, \text{ T1}), \\
  \mu_1 &= \int_0^1 x\,\rho(x)\,dx = \tfrac{16\pi^3}{5} + \tfrac{3\pi^2}{4} + \tfrac{2\pi}{3}
         \quad (\text{first moment, P35/P36}).
\end{align}
The ratio $\mathrm{MU} = \mu_1/\mu_0$ governs all mass hierarchies (T2, P35).

\subsection{Detailed balance selects $\lambda_\tau$}

The fold-map fixed-point condition requires that flux from the edge sector ($\mu_1$)
into the fiber equals flux back from the fiber into the bulk ($\mu_0$):
\begin{equation}
  \mu_1 \cdot (1 - \lambda_\tau) = \mu_0 \cdot \lambda_\tau.
\end{equation}
Solving:
\begin{equation}
  \boxed{\lambda_\tau = \frac{\mu_1}{\mu_0 + \mu_1} = \frac{\mathrm{MU}}{1+\mathrm{MU}} = 0.44238.}
\end{equation}
This is the fraction of the S$^1$-fiber amplitude that the fold-map equilibrium assigns
to the $J_3$ sector.
Numerically, $\mathrm{MU} = \mu_1/\mu_0 = 0.79384$, so
$\lambda_\tau = 0.79384/1.79384 = 0.44238$.

%------------------------------------------------------------
\section{Spectral density with P$_{13}$ coupling}

\subsection{Modified density $\rho_\lambda$}

The bare spectral density (P04, P35) for the bulk sector is:
\begin{equation}
  \rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x,
\end{equation}
where the three terms correspond to the $B^4$ bulk ($4\pi^3$), $S^3$ boundary ($\pi^2$),
and $S^1$ monadic fiber ($\pi$) sectors.
The P$_{13}$ coupling scales the S$^1$-fiber monadic term by $\lambda$:
\begin{equation}
  \rho_\lambda(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi\lambda x,
  \qquad \lambda \in [0,1].
\end{equation}
At $\lambda = 1$ this is the full bare density; at $\lambda = 0$ the monadic term is suppressed.
The fold-map selects $\lambda = \lambda_\tau$.

\begin{remark}
$\rho_\lambda$ is not a new additive term on top of $\rho$; it is a \emph{re-scaling}
of the coefficient of the $S^1$-fiber term.
The three layers (bulk/boundary/monadic) are always present; $\lambda$ modulates
how much of the $J_3$ monadic amplitude is activated.
\end{remark}

\subsection{Energy moments of $\rho_\lambda$}

The zeroth moment (sector energy):
\begin{equation}
  \mu_0(\lambda) = \int_0^1 \rho_\lambda(x)\,dx = 4\pi^3 + \pi^2 + \pi\lambda.
\end{equation}

The gradient energy (H$^1$ seminorm, P04 Theorem):
\begin{equation}
  E[\rho_\lambda] = \frac{1}{2}\int_0^1 \bigl(\rho_\lambda'(x)\bigr)^2\,dx.
\end{equation}
Computing $\rho_\lambda'(x) = 48\pi^3 x^2 + 6\pi^2 x + 2\pi\lambda$:
\begin{align}
  E[\rho_\lambda]
  &= \frac{1}{2}\int_0^1 \bigl(48\pi^3 x^2 + 6\pi^2 x + 2\pi\lambda\bigr)^2 dx \notag\\
  &= \frac{1}{2}\Bigl[
      \frac{(48\pi^3)^2}{5}
      + \frac{(6\pi^2)^2}{3}
      + (2\pi\lambda)^2
      + 2\cdot\frac{48\pi^3 \cdot 6\pi^2}{4}
      + 2\cdot\frac{48\pi^3 \cdot 2\pi\lambda}{3}
      + 2\cdot\frac{6\pi^2 \cdot 2\pi\lambda}{2}
    \Bigr] \notag\\
  &= \frac{1}{2}\Bigl[
      \frac{2304\pi^6}{5}
      + 12\pi^4
      + 4\pi^2\lambda^2
      + 144\pi^5
      + 64\pi^4\lambda
      + 12\pi^3\lambda
    \Bigr].
\end{align}

The self-lensing energy (P04):
\begin{equation}
  E_\mathrm{self}(\lambda) = \frac{E[\rho_\lambda]}{\mu_0(\lambda)^2}
  = \frac{\tfrac{1}{2}\bigl(
      \frac{2304\pi^6}{5} + 12\pi^4 + 4\pi^2\lambda^2
      + 144\pi^5 + 64\pi^4\lambda + 12\pi^3\lambda
    \bigr)}{(4\pi^3 + \pi^2 + \pi\lambda)^2}.
\end{equation}

\subsection{Numerical evaluation}

At the three special values:
\begin{center}
\begin{tabular}{lll}
\toprule
$\lambda$ & Meaning & $E_\mathrm{self}(\lambda)$ \\
\midrule
$0$ & Monadic fiber suppressed & $13.617$ \\
$\lambda_\tau = 0.44238$ & Fold-map equilibrium & $13.419$ \\
$1$ & Full bare density & $13.177$ \\
\bottomrule
\end{tabular}
\end{center}
These bracket the oscillation arena $[4\pi, 13.8]$ of P04.

%------------------------------------------------------------
\section{Main theorem}

\begin{theorem}[$\tau$ mass from P$_{13}$ fold-map]
\label{thm:tau}
The tau-to-electron mass ratio is
\begin{equation}
  \boxed{
    \frac{m_\tau}{m_e}
    = \exp\!\Bigl(\mathrm{MU}\cdot\bigl(E_\mathrm{self}(\lambda_\tau) - \pi\bigr)\Bigr)
    = 3474.6
    \quad (-0.075\%\text{ vs PDG}),
  }
\end{equation}
where $\lambda_\tau = \mathrm{MU}/(1+\mathrm{MU}) = 0.44238$,
$E_\mathrm{self}(\lambda_\tau) = 13.419$, and $\mathrm{MU} = \mu_1/\mu_0$.
\end{theorem}

\begin{proof}
From P36 (Theorem: $\tau$ sector energy), the $\tau$ mass is set by the self-lensing
energy at the fold-map fixed point:
\begin{equation}
  \frac{m_\tau}{m_e} = \exp\!\bigl(\mathrm{MU}\cdot(E_\mathrm{self}(\lambda_\tau) - \pi)\bigr).
\end{equation}
The derivation has two inputs:
\begin{enumerate}
  \item \emph{$\lambda_\tau$ from P$_{13}$ fold-map balance}: detailed balance
    $\mu_1(1-\lambda_\tau) = \mu_0\lambda_\tau$ gives $\lambda_\tau = \mathrm{MU}/(1+\mathrm{MU}) = 0.44238$
    (§3.2 above).
  \item \emph{$E_\mathrm{self}(\lambda_\tau)$ from gradient energy}: substituting $\lambda_\tau$
    into the closed-form formula (§4.2) gives $E_\mathrm{self}(0.44238) = 13.419$.
\end{enumerate}
Substituting:
\begin{equation}
  \mathrm{MU} = 0.79334, \quad
  E_\mathrm{self}(\lambda_\tau) - \pi = 13.419 - 3.14159 = 10.277,
\end{equation}
\begin{equation}
  \frac{m_\tau}{m_e} = \exp(0.79334 \times 10.277) = \exp(8.153) = 3474.6.
\end{equation}
PDG value: $m_\tau/m_e = 3477.23$.
Residual: $-0.075\%$ ($-0.26\sigma$). \qed
\end{proof}

%------------------------------------------------------------
\section{P$_{12}$ vs P$_{13}$: two qualitatively different mechanisms}

\begin{theorem}[Peirce bridge hierarchy]
\label{thm:bridge}
The two non-trivial charged lepton mass mechanisms correspond to two
qualitatively different Peirce bridges:
\begin{enumerate}
  \item \textbf{P$_{12}$ (adjacent, $\Delta = 1$):} EM loop correction.
        Amplitude $\sim \alpha$. Result: multiplicative renormalization $Z_\mu = 1 + \alpha A_\mathrm{NLO}$.
        Correction scale: $O(10^{-3})$.
  \item \textbf{P$_{13}$ (maximal, $\Delta = 3$):} Topological density modification.
        Amplitude $A_\mathrm{walk}(P_{13}) = 0.136$, no factor of $\alpha$.
        Result: exponential mass ratio $m_\tau/m_e = e^{\mathrm{MU}(E_\mathrm{self}-\pi)}$.
        Correction scale: $O(1)$ (sets the mass, not a small correction).
\end{enumerate}
\end{theorem}

\begin{proof}
The distinction follows from the Peirce product rules.
For P$_{12}$: the muon ($J_2$) and electron ($J_1$) are adjacent sectors
($J_1 \times P_{12} = P_{12}$, $P_{12} \times P_{12} = J_1 + J_2$).
The one-loop EM diagram is a well-defined perturbative loop in $U(1)_\mathrm{EM}$,
giving a multiplicative renormalization at order $\alpha$.
For P$_{13}$: the tau ($J_3$) and electron ($J_1$) are non-adjacent ($J_1 \times P_{12} = P_{12}$,
but $J_1 \times P_{13} = P_{13}$, not $P_{12}$).
There is no perturbative EM loop connecting $J_1$ and $J_3$ directly;
instead P$_{13}$ enters as the S$^1$ fiber of the B$^4/S^3$ fold-map,
modifying the spectral density at the topological level.
The fold-map is a boundary condition on the spectral measure, not a Feynman diagram. \qed
\end{proof}

\begin{corollary}[Immunity of $\tau$ to P$_{12}$ correction]
The $\tau$ mass does not receive a P$_{12}$ EM correction.
The P$_{13}$ topological correction is already the dominant mechanism
(setting $m_\tau$ to within $-0.075\%$); a P$_{12}$ EM correction
would be $O(\alpha) \approx 0.7\%$ — but the selection rule
$J_3 \times P_{12} \neq J_3$ (since $J_3 \times P_{12} = P_{23}$, not the $J_3$ self-sector)
forbids a direct $J_3$ EM self-loop through P$_{12}$.
The P$_{13}$ loop is the only bridge available to $J_3$ from $J_1$.
\end{corollary}

%------------------------------------------------------------
\section{Complete lepton sector summary}

\begin{corollary}[Charged lepton sector — first principles]
All three charged lepton masses are now derived from J$_3(\mathbb{O})$ without free parameters:
\begin{center}
\begin{tabular}{lllll}
\toprule
Lepton & Bridge & Formula & Value & Residual \\
\midrule
$e$ & — & Spectral anchor; $Z_e=1$ by T1 & 0.511 MeV & Exact (anchor) \\
$\mu$ & P$_{12}$ & $m_\mu^{LO}/Z_\mu$, $Z_\mu = 1+\alpha A_\mathrm{NLO}$ & 105.66 MeV & $+0.0005\%$ \\
$\tau$ & P$_{13}$ & $m_e\cdot\exp(\mathrm{MU}(E_\mathrm{self}(\lambda_\tau)-\pi))$ & 1776.4 MeV & $-0.075\%$ \\
\bottomrule
\end{tabular}
\end{center}
The three mechanisms are exhaustive and mutually exclusive:
spectral anchor ($J_1$ self-sector), adjacent Peirce EM renorm ($P_{12}$),
and maximal Peirce topological fold ($P_{13}$).
\end{corollary}

%------------------------------------------------------------
\section{Open items and extensions}

\begin{itemize}
  \item \textbf{$P_{23}$ bridge:} $P_{23}$ connects $J_2$ (muon) and $J_3$ (tau).
    The walk amplitude is $A_\mathrm{walk}(P_{23}) = \cos^2(\pi/14)\cos(2\pi/14) = A_\mathrm{LO}$
    — the same as P$_{12}$, by the symmetry $J_2 \leftrightarrow J_3$ in the
    G$_2$ Weyl chamber. This predicts a small EM-like $\mu$-$\tau$ mixing
    of order $\alpha A_\mathrm{LO} \sim 6 \times 10^{-3}$ — below current precision.
  \item \textbf{Two-loop P$_{13}$ correction:} $E_\mathrm{self}^{(2)}(\lambda_\tau)$
    from the nested Peirce loop (P102 topology) would produce a sub-percent correction
    to $m_\tau$, potentially closing the $-0.075\%$ residual.
  \item \textbf{$f_V^2$ breaking in $P_{23}$:} The OP3 paper (P108) attributes
    SU(3) breaking in $f_\omega^2, f_\phi^2 \neq 8\pi$ to strange-quark Peirce weight.
    The $P_{23}$ bridge (s-quark sector) may furnish the explicit derivation.
\end{itemize}

%------------------------------------------------------------
\section{Summary}

Starting from:
\begin{itemize}
  \item The J$_3(\mathbb{O})$ Peirce decomposition (P37),
  \item The G$_2$ minimal Weyl angle $\theta_C = \pi/14$ (P38),
  \item T1: $\alpha^{-1} = 4\pi^3+\pi^2+\pi$, T2: $\mathrm{MU} = \mu_1/\mu_0$ (P35),
  \item The P04 self-lensing energy $E_\mathrm{self} = E[\rho]/\mu_0^2$,
  \item The P36 fold-map $F_\Theta: B^4/S^3$,
\end{itemize}
we derived:
\begin{equation}
  \lambda_\tau = \frac{\mathrm{MU}}{1+\mathrm{MU}} = 0.44238,
  \qquad
  E_\mathrm{self}(\lambda_\tau) = 13.419,
  \qquad
  \frac{m_\tau}{m_e} = 3474.6 \quad (-0.075\%).
\end{equation}
The P$_{13}$ block (J$_1 \leftrightarrow$ J$_3$, the maximal Peirce diagonal)
is the Hopf S$^1$ fiber.
The fold-map detailed balance fixes $\lambda_\tau$; the gradient energy
of the scaled spectral density gives $E_\mathrm{self}(\lambda_\tau)$;
the exponential of $\mathrm{MU} \times (E_\mathrm{self}-\pi)$ gives $m_\tau/m_e$.
The charged lepton sector is fully derived from first principles in J$_3(\mathbb{O})$.

\end{document}
