%=============================================================================
% APPENDIX P: CLOCK COUPLING AND MAJORANA SCALE — THEOREM-GRADE DERIVATIONS
% v40: Full theorem chain for k_α = α²/(2π) and M_R = M_P α³
%=============================================================================

\section{Clock Coupling and Majorana Scale}\label{app:P_kalpha_MR}

\subsection{Scope and Convention Lock}

This appendix upgrades two relations used in the microsector framework to theorem-grade status:
\begin{equation}
\label{eq:P_kalpha_target}
\fiteq{k_\alpha = \frac{\alpha^2}{2\pi},}
\end{equation}
\begin{equation}
\label{eq:P_MR_target}
\fiteq{M_R = M_P\,\alpha^3.}
\end{equation}

The derivations follow the same ``no hidden knobs'' methodology used in Appendix~\ref{app:O_alpha57_clean} 
(the $\alpha^{57}$ hierarchy): all dimensionless outputs must be built from
(i)~the unique dimensionless coupling $\alpha$ (already derived from the Chern-Simons microsector at $k_{\max}=60$) and
(ii)~topological integers established in the paper (notably the discrete input $N_{\rm gen}=3$; App.~F generation-count remark).

%=============================================================================
\subsection{Theorem P.1: Schwinger Coefficient $a_e = \alpha/(2\pi)$}\label{sec:P_schwinger}
%=============================================================================

\begin{theorem}[Schwinger one-loop anomalous magnetic moment]\label{thm:P_schwinger}
In QED with one charged Dirac fermion of charge $e$ and mass $m$, the one-loop correction to the on-shell vertex yields
\begin{equation}
\fiteq{a_e:= \frac{g_e-2}{2} = F_2(0) = \frac{\alpha}{2\pi} + \mathcal{O}(\alpha^2),}
\end{equation}
where $\alpha=e^2/(4\pi)$ in $\hbar=c=1$ units and $F_2(q^2)$ is the Pauli form factor.
\end{theorem}

\begin{proof}
Write the renormalized on-shell vertex as
\begin{equation}
\fiteq{\bar u(p')\Gamma^\mu(p',p)u(p)=\bar u(p')\!\left[\gamma^\mu F_1(q^2) + \frac{i\sigma^{\mu\nu}q_\nu}{2m}F_2(q^2)\right]\!u(p),}
\end{equation}
with $q=p'-p$ and $F_1(0)=1$ by charge renormalization. The one-loop vertex graph gives (in Feynman gauge)
\begin{equation}
\fiteq{\begin{aligned}
\Gamma^\mu_{(1)} = (-ie)^3\!\int\!\frac{d^4k}{(2\pi)^4}\,
\gamma_\alpha\frac{(\not\!p'-\not\!k)+m}{(p'-k)^2-m^2} \\
\times\gamma^\mu
\frac{(\not\!p-\not\!k)+m}{(p-k)^2-m^2}\gamma^\alpha\frac{1}{k^2}.
\end{aligned}}
\end{equation}
Projecting onto the Pauli structure and taking $q^2\to 0$ on-shell, standard Feynman-parameter reduction yields
\begin{equation}
\fiteq{F_2(0) = \frac{\alpha}{2\pi}\int_0^1\! dx\,2x(1-x)=\frac{\alpha}{2\pi}.}
\end{equation}
(Any UV divergence resides in $F_1$ and cancels after renormalization; $F_2(0)$ is finite.)
\end{proof}

%=============================================================================
\subsection{Theorem P.2: Clock Coupling $k_\alpha=\alpha^2/(2\pi)$}\label{sec:P_kalpha}
%=============================================================================

\paragraph{Microsector axiom (already used in the paper).}
The ``clock coupling'' is defined operationally by the fractional shift of a purely electromagnetic 
atomic transition under a small static DFD potential $\psi$:
\begin{equation}\label{eq:P_def_kalpha}
\fiteq{\frac{\delta \nu}{\nu} = k_\alpha\,\psi + \mathcal{O}(\psi^2).}
\end{equation}

\paragraph{Key microsector input.}
In the DFD microsector, $\alpha$ is \emph{topologically fixed} (Appendix~\ref{app:microsector}) and therefore 
does \emph{not} vary with $\psi$ at tree level.
Hence the leading nontrivial $\psi$-dependence of EM transition frequencies must arise from the 
\emph{first} quantum correction that links:
\begin{equation}
\fiteq{\psi \longrightarrow \text{(EM vacuum)} \longrightarrow \text{(atomic frequency)}.}
\end{equation}

\begin{theorem}[Clock coupling constant]\label{thm:P_kalpha}
Assume the microsector ``no hidden knobs'' principle: in the weak-field regime, the leading EM-sensitive 
$\psi$ insertion is a single gauge vertex and therefore carries one factor of $\alpha$.
Then the coefficient $k_\alpha$ in \eqref{eq:P_def_kalpha} is forced to be
\begin{equation}
\fiteq{\boxed{k_\alpha = \alpha\,a_e = \frac{\alpha^2}{2\pi}}}
\end{equation}
\end{theorem}

\begin{proof}
By hypothesis, the leading $\psi$ insertion into the EM sector is a single gauge vertex, hence contributes a factor $\alpha$.
The \emph{only} universal, gauge-invariant, dimensionless one-loop EM correction that couples to atomic spin/magnetic 
structure and is independent of atomic details is the Pauli form factor at zero momentum, $F_2(0)=a_e$ (Theorem~\ref{thm:P_schwinger}).
Therefore the leading dimensionless coefficient multiplying $\psi$ in the EM sector is the product $\alpha\,a_e$.
Using Theorem~\ref{thm:P_schwinger} gives $k_\alpha=\alpha^2/(2\pi)$.
\end{proof}

\paragraph{Remark (what is and is not a new assumption).}
The only nontrivial input beyond QED is the microsector rule that the leading $\psi\!\to\!$EM insertion is a 
single gauge vertex (``one $\alpha$''), rather than an arbitrary analytic function of $\alpha$.
This is exactly the same kind of admissible ``no hidden knobs'' restriction used in Appendix~\ref{app:O_alpha57_clean} 
to turn the $\alpha^{57}$ hierarchy into a theorem.

%-----------------------------------------------------------------------------
\subsubsection{Observational Test: Fine-Structure Constant Variation}\label{sec:P_espresso}
%-----------------------------------------------------------------------------

The clock coupling $k_\alpha = \alpha^2/(2\pi)$ predicts that the fine-structure constant varies with 
cosmological gravitational potential:
\begin{equation}
\fiteq{\frac{\Delta\alpha}{\alpha}(z) = k_\alpha \times \Delta\psi(z).}
\end{equation}

Using the $\psi$-screen reconstruction from Section~\ref{sec:psi-tomography} ($\Delta\psi(z=1) \approx 0.27$):
\begin{equation}
\fiteq{\frac{\Delta\alpha}{\alpha}\bigg|_{z=1} = \frac{\alpha^2}{2\pi} \times 0.27 = +2.3 \times 10^{-6}.}
\end{equation}

\paragraph{ESPRESSO comparison.}
The ESPRESSO spectrograph at the VLT has measured $\Delta\alpha/\alpha$ in quasar absorption systems. 
The 2022 ESPRESSO collaboration analysis reports:
\begin{equation}
\fiteq{\frac{\Delta\alpha}{\alpha}\bigg|_{z \sim 1} = (+1.3 \pm 1.3) \times 10^{-6}.}
\end{equation}

\noindent
\begin{tcolorbox}[colback=green!5!white, colframe=green!60!black, title=$\alpha(z)$ Prediction vs.\ ESPRESSO]
\textbf{DFD prediction:} $\Delta\alpha/\alpha = +2.3 \times 10^{-6}$ at $z = 1$

\textbf{ESPRESSO (2022):} $(+1.3 \pm 1.3) \times 10^{-6}$

\textbf{Agreement:} $0.8\sigma$ --- sign and magnitude both consistent
\end{tcolorbox}

\paragraph{Key features.}
\begin{enumerate}
\item \textbf{Positive sign:} DFD predicts $\alpha$ \textit{increases} at higher redshift (larger $\psi$). 
 ESPRESSO data prefer positive $\Delta\alpha/\alpha$.
\item \textbf{Magnitude:} The predicted $\sim 10^{-6}$ level matches current sensitivity.
\item \textbf{$z$-dependence:} $\Delta\alpha/\alpha \propto \Delta\psi(z)$ gives specific predictions for different redshifts.
\end{enumerate}

\paragraph{Predictions for ELT.}
The Extremely Large Telescope will improve sensitivity to $\sim 10^{-7}$. DFD predictions:

\begin{center}
\fittab{%
\begin{tabular}{ccc}
\toprule
$z$ & $\Delta\psi(z)$ & $\Delta\alpha/\alpha$ ($\times 10^{-6}$) \\
\midrule
0.5 & 0.15 & $+1.3$ \\
1.0 & 0.27 & $+2.3$ \\
1.5 & 0.35 & $+3.0$ \\
2.0 & 0.42 & $+3.6$ \\
3.0 & 0.55 & $+4.7$ \\
\bottomrule
\end{tabular}%
}
\end{center}

%=============================================================================
\subsection{Theorem P.3: Majorana Scale $M_R = M_P\alpha^3$}\label{sec:P_MR}
%=============================================================================

\paragraph{Setup.}
The right-handed neutrinos are gauge singlets (see Appendix~\ref{app:higgs-yukawa}).
Let $\mathcal{H}_{\nu_R}$ denote the internal Hilbert subspace supporting the $\nu_R$ degrees of freedom.

\begin{lemma}[Generation multiplicity]\label{lem:P_Ngen}
The number of generations is a topological invariant:
\begin{equation}
\fiteq{\dim(\mathcal{H}_{\nu_R}) = N_{\rm gen}=3,}
\end{equation}
fixed by the index theorem on the internal manifold $\mathbb{C}P^2\times S^3$ with the chosen twist bundle.
\end{lemma}

This is the \textbf{same} Atiyah-Singer index that gives $k_{\max} = 60$ (Appendix~\ref{app:microsector}).
The integer 3 is as topologically protected as 60.

\paragraph{Toeplitz scaling input (same mechanism as Appendix~O).}
Let $\mathcal{K}_{\nu_R}$ be the positive operator controlling the singlet-sector quadratic form in the 
Toeplitz-quantized microsector.
The microsector coupling parameter is $g=\alpha^{-1}$, and constant-symbol scaling acts by 
$\mathcal{K}_{\nu_R}\mapsto g\,\mathcal{K}_{\nu_R}$.

\begin{theorem}[Majorana scale from determinant scaling]\label{thm:P_MR}
Assume (i)~the singlet-sector quadratic form is non-extensive and Toeplitz-quantized on $\mathcal{H}_{\nu_R}$, 
(ii)~the only dimensionless knob is $g=\alpha^{-1}$, and (iii)~$\dim\mathcal{H}_{\nu_R}=N_{\rm gen}$ (Lemma~\ref{lem:P_Ngen}).
Then the unique dimensionless singlet-sector suppression factor is
\begin{equation}
\fiteq{\varepsilon_{\nu_R}(g):=\frac{\det(\mathcal{K}_{\nu_R})}{\det(g\mathcal{K}_{\nu_R})}=g^{-N_{\rm gen}}=\alpha^{N_{\rm gen}}=\alpha^3,}
\end{equation}
and the corresponding Majorana mass scale is forced to be
\begin{equation}
\fiteq{\boxed{M_R = M_P\,\varepsilon_{\nu_R}(\alpha^{-1}) = M_P\,\alpha^3}}
\end{equation}
\end{theorem}

\begin{proof}
Because $\mathcal{H}_{\nu_R}$ is finite-dimensional (non-extensive microsector) with $\dim\mathcal{H}_{\nu_R}=N_{\rm gen}$, 
constant scaling multiplies every eigenvalue by $g$ and therefore multiplies the determinant by $g^{N_{\rm gen}}$:
\begin{equation}
\fiteq{\det(g\mathcal{K}_{\nu_R})=g^{N_{\rm gen}}\det(\mathcal{K}_{\nu_R}).}
\end{equation}
Hence $\varepsilon_{\nu_R}(g)=g^{-N_{\rm gen}}$.
By the ``no hidden knobs'' principle, the Majorana scale can only be the unique fundamental mass $M_P$ multiplied 
by a dimensionless singlet-sector factor built from $g$ and $N_{\rm gen}$; the determinant ratio above is the unique 
such factor with the correct scaling behavior.
Substituting $g=\alpha^{-1}$ and $N_{\rm gen}=3$ gives $M_R=M_P\alpha^3$.
\end{proof}

%-----------------------------------------------------------------------------
\subsubsection{Parallel Structure with Appendix O}\label{sec:P_parallel}
%-----------------------------------------------------------------------------

The $M_R = M_P \alpha^3$ derivation parallels Appendix~O exactly:

\begin{center}
\small
\fittab{%
\begin{tabular}{lll}
\toprule
& \textbf{Appendix O ($\alpha^{57}$)} & \textbf{Appendix P ($\alpha^3$)} \\
\midrule
State space & $\mathcal{H}_{\rm UV}$, dim $= k_{\max} = 60$ & $\mathcal{H}_{\nu_R}$, dim $= N_{\rm gen} = 3$ \\
Operator & Kinetic $\mathcal{K}$, dim ker $= 3$ & Majorana $\mathcal{M}$, no kernel \\
Exponent & $k_{\max} - N_{\rm gen} = 57$ & $N_{\rm gen} = 3$ \\
Dictionary & $\rho_{\rm vac}/\rho_{\rm Pl}:= \varepsilon(\alpha^{-1})$ & $M_R/M_P:= \varepsilon_{\nu_R}(\alpha^{-1})$ \\
Result & $\rho_{\rm vac}/\rho_{\rm Pl} = \alpha^{57}$ & $M_R/M_P = \alpha^3$ \\
\bottomrule
\end{tabular}%
}
\end{center}

Both use the same ``no hidden knobs'' principle: the exponents are topologically forced integers.

%-----------------------------------------------------------------------------
\subsubsection{Neutrino Mass Predictions}\label{sec:P_neutrino}
%-----------------------------------------------------------------------------

With $v = M_P\alpha^8\sqrt{2\pi} = 246.09$ GeV (derived in Section~\ref{sec:quantum}) and the see-saw formula
$m_\nu \sim m_D^2/M_R$:

\paragraph{Numerical result.}
\begin{equation}
\fiteq{\begin{aligned}
M_R &= M_P \times \alpha^3 \\
&= 1.22 \times 10^{19}\,\text{GeV} \times (137)^{-3} = 4.74 \times 10^{12}\,\text{GeV}.
\end{aligned}}
\end{equation}

\paragraph{Mass hierarchy.}
The ratio of neutrino masses follows the generation structure:
\begin{equation}
\fiteq{\frac{m_{\nu,i}}{m_{\nu,j}} = \alpha^{-(j-i)/N_{\rm gen}} = \alpha^{-(j-i)/3}.}
\end{equation}

\begin{center}
\fittab{%
\begin{tabular}{lcc}
\toprule
\textbf{Quantity} & \textbf{Prediction} & \textbf{Observed} \\
\midrule
$m_3/m_2$ & $\alpha^{-1/3} = 5.16$ & $50.8/8.6 = 5.9$ \\
Agreement & \multicolumn{2}{c}{$13\%$} \\
\midrule
$\Sigma m_\nu$ & $\approx 60$ meV & $< 120$ meV (Planck+BAO) \\
Status & \multicolumn{2}{c}{Consistent, testable by DESI + CMB-S4} \\
\bottomrule
\end{tabular}%
}
\end{center}

\paragraph{Absolute scale.}
The Dirac Yukawa is \emph{not} a free placeholder: it is fixed by the
seesaw-consistent closure of Proposition~\ref{thm:new-yD} (Appendix~AT). The
Type-I seesaw $m_{\nu_3}=m_D^2/M_R$ with the locked scales
$m_3=\tfrac{14}{13}\pi M_P\alpha^{14}$ and $M_R=M_P\alpha^3$ forces
\begin{equation}
\fiteq{y_D^2 = \frac{2\,m_3 M_R}{v^2} = \frac{14}{13}\,\alpha,
\qquad
m_D = \sqrt{m_3 M_R} = \frac{v}{\sqrt2}\sqrt{\tfrac{14}{13}\,\alpha} = 15.43~\text{GeV},}
\end{equation}
where the $\pi$ of $m_3$ cancels against the $\sqrt{2\pi}$ in $v=M_P\alpha^8\sqrt{2\pi}$,
and $\tfrac{14}{13}=1+\tfrac{1}{N_{\rm gen}}\sin^2\theta_W$ with $\sin^2\theta_W=3/13$,
$N_{\rm gen}=3$ (Prop.~\ref{thm:new-yD}). This returns the absolute scale
\begin{equation}
\fiteq{m_{\nu_3} = \frac{m_D^2}{M_R} = \frac{14}{13}\pi M_P\alpha^{14} = 50.16~\text{meV}}
\end{equation}
\emph{exactly}. (The earlier placeholder $y_D\sim\alpha^{1/2}$ gave
$m_{\nu_3}\approx93$~meV $=\tfrac{13}{7}\times50.16$~meV; the exact effective exponent
is $y_D\sim\alpha^{0.4925}$, \emph{fixed} by the $14/13$ closure above rather than
tuned to the observed value.)

%=============================================================================
\subsection{Summary}\label{sec:P_summary}
%=============================================================================

\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black, title=Appendix P: Theorem Status]
\textbf{$k_\alpha = \alpha^2/(2\pi)$:} Theorem-grade (given ``one gauge vertex'' axiom).
\begin{itemize}
\item Theorem P.1: $a_e = \alpha/(2\pi)$ (Schwinger, QED --- fully proven)
\item Theorem P.2: $k_\alpha = \alpha \times a_e$ (no hidden knobs axiom)
\item Observational test: ESPRESSO $0.8\sigma$ consistent
\end{itemize}

\vspace{0.3em}
\textbf{$M_R = M_P\alpha^3$:} Theorem-grade (same rigor as $\alpha^{57}$).
\begin{itemize}
\item Input: $N_{\rm gen} = 3$ (discrete input; numerically $\chi_{\rm top}(\mathbb{C}P^2)$ --- App.~F)
\item Theorem P.3: $\det(g\mathcal{M}) = g^{N_{\rm gen}}\det(\mathcal{M})$ (pure linear algebra)
\item Dictionary: $M_R/M_P:= \varepsilon_{\nu_R}(\alpha^{-1})$ (explicit identification)
\item Predictions: $m_3/m_2 = 5.2$ (obs: 5.9, $13\%$); $\Sigma m_\nu \approx 60$ meV
\end{itemize}
\end{tcolorbox}

Both derivations follow the Appendix~O protocol: theorem-grade mathematics plus explicit ``no hidden knobs'' 
axiom or dictionary identification. The exponents (2 for $k_\alpha$, 3 for $M_R$) are not fitted---they emerge 
from the same topological structure that gives $\alpha^{57}$ for the cosmological constant.
