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000003 | Let
000004 | \[
000005 | \omega=e^{2\pi i/3},
000006 | \qquad
000007 | U_6=\mathrm{diag}(1,\omega,\omega^2,-1,-\omega,-\omega^2).
000008 | \]
000009 | This makes the phase grammar concrete rather than rhetorical. The six admissible phases are shown as a finite hexagonal transport geometry in \Cref{fig:z6-circle}: the cusp values live on the same real axis as the positive-real phase, the admissible gap is visible directly, and the holonomy phase can be read as a triangle on the same discrete transport base.
000010 | 
000011 | \begin{remark}[Hexagonal spectral geometry]
000012 | The operator $U_6$ is a unitary circulant with eigenphases $e^{in\pi/3}$ for $n=0,\dots,5$ and is unitarily equivalent to the one-step translation operator on a six-site ring. The transport kernel $D_y=y\,\mathbf 1-\delta_{\mathrm{ph}}U_6$ is therefore the resolvent denominator of a finite quantum walk, equivalently a tight-binding transfer problem, on the discrete $C_6$ cycle. The regularized inverse introduced below plays the role of the corresponding Green kernel. CKM and PMNS mixings, mass hierarchies, and the Möbius quotients of \Cref{eq:mobius-quotient} are thus not loose analytic fits but spectral-theory statements on a regular hexagon. The near-pole hierarchy at $y\approx\delta_{\mathrm{ph}}$ becomes the resonance of a hexagonal resolvent rather than tuned parameter choice.
000013 | \end{remark}
000014 | 
000015 | \begin{figure}[t]
000016 | \centering
000017 | \scriptsize
000018 | \begin{tikzpicture}[scale=1.1,>=Latex]
000019 | \coordinate (O) at (0,0);
000020 | \foreach \name/\ang in {p0/0,p1/60,p2/120,p3/180,p4/240,p5/300} {
000021 |     \coordinate (\name) at ({2*cos(\ang)},{2*sin(\ang)});
000022 | }
000023 | \draw[thick, black!70] (p0) -- (p1) -- (p2) -- (p3) -- (p4) -- (p5) -- cycle;
000024 | \draw[gray!25, dashed] (O) -- (p0);
000025 | \draw[gray!25, dashed] (O) -- (p1);
000026 | \draw[gray!25, dashed] (O) -- (p2);
000027 | \draw[gray!25, dashed] (O) -- (p3);
000028 | \draw[gray!25, dashed] (O) -- (p4);
000029 | \draw[gray!25, dashed] (O) -- (p5);
000030 | \foreach \P in {p0,p1,p2,p3,p4,p5} {
000031 |     \filldraw[blue!60!black] (\P) circle (2pt);
000032 | }
000033 | \node[anchor=west, font=\scriptsize] at (2.18,0.12) {$1$};
000034 | \node[anchor=south west, font=\scriptsize] at (1.1,1.85) {$e^{i\pi/3}$};
000035 | \node[anchor=south east, font=\scriptsize] at (-1.1,1.85) {$e^{2i\pi/3}$};
000036 | \node[anchor=east, font=\scriptsize] at (-2.18,0.12) {$-1$};
000037 | \node[anchor=north east, font=\scriptsize] at (-1.1,-1.85) {$e^{-2i\pi/3}$};
000038 | \node[anchor=north west, font=\scriptsize] at (1.1,-1.85) {$e^{-i\pi/3}$};
000039 | 
000040 | \draw[->, thick, red!65!black] (1.15,1.05) arc[start angle=42,end angle=-42,radius=1.2];
000041 | \node[red!65!black, font=\scriptsize, anchor=west] at (2.55,0.25) {CP: $\theta\mapsto-\theta$};
000042 | 
000043 | \filldraw[orange!80!black] (0.67,0) circle (2pt);
000044 | \filldraw[orange!80!black] (1.33,0) circle (2pt);
000045 | \filldraw[green!60!black] (1.1998,0) circle (2.4pt);
000046 | \node[below, font=\scriptsize] at (0.67,-0.08) {$\frac13$};
000047 | \node[below, font=\scriptsize] at (1.33,-0.08) {$\frac23$};
000048 | \node[above, font=\scriptsize] at (1.1998,0.08) {$\delta_{\mathrm{ph}}$};
000049 | \draw[<->, thick, black!65] (1.1998,-0.32) -- (1.33,-0.32);
000050 | \node[below, font=\tiny] at (1.265,-0.34) {$\epsilon_f$};
000051 | \node[align=left, anchor=west, font=\scriptsize, text=black!70] at (2.55,-0.7)
000052 |     {positive-real branch\\near-pole lock at $y\approx\delta_{\mathrm{ph}}$\\gap $\epsilon_f=\frac23-\delta_{\mathrm{ph}}$};
000053 | \end{tikzpicture}
000054 | \caption{Finite transport geometry on the hexagon $C_6$. The six $U_6$ phases form the discrete
000055 | transport base, while the admissible cusps $y\in\{1,\frac13,\frac23\}$ and the branch pole
000056 | $\delta_{\mathrm{ph}}$ lie on the positive-real axis with gap
000057 | $\epsilon_f=\frac23-\delta_{\mathrm{ph}}$.}
000058 | \label{fig:z6-circle}
000059 | \end{figure}
000060 | 
000061 | The family-side holonomy phase is separated in \Cref{fig:holonomy-triangle} so that the transport
000062 | base and the flavor readout remain visually distinct.
000063 | 
000064 | \begin{figure}[t]
000065 | \centering
000066 | \scriptsize
000067 | \begin{tikzpicture}[>=Latex]
000068 |     \node[draw=black!65, fill=gray!10, rounded corners, text width=4.8cm, minimum height=0.85cm, align=center, font=\bfseries] at (0,1.15)
000069 |         {family-side holonomy on the transport grammar};
000070 |     \coordinate (A) at (0,0.1);
000071 |     \coordinate (B) at (-1.45,-1.8);
000072 |     \coordinate (C) at (1.45,-1.8);
000073 |     \draw[thick, orange!75!black] (A) -- (B) -- (C) -- cycle;
000074 |     \filldraw[orange!75!black] (A) circle (2pt) (B) circle (2pt) (C) circle (2pt);
000075 |     \node[above, font=\scriptsize] at (A) {$1$};
000076 |     \node[left, font=\scriptsize] at (B) {$2$};
000077 |     \node[right, font=\scriptsize] at (C) {$3$};
000078 |     \draw[->, thick, blue!60!black] (-0.2,-0.45) arc[start angle=110,end angle=15,radius=0.95];
000079 |     \node[draw=green!50!black, fill=green!8, rounded corners, text width=3.7cm, minimum height=0.95cm, align=center] at (0,-3.15)
000080 |         {$\delta_{\mathrm{CKM}}=\arg\tr\Pi_\triangle(\nabla_F^\star)$\\oriented holonomy triangle};
000081 | \end{tikzpicture}
000082 | \caption{Holonomy triangle on the same discrete transport grammar. The CKM phase is read as the
000083 | oriented three-cycle phase $\delta_{\mathrm{CKM}}=\arg\tr\Pi_\triangle(\nabla_F^\star)$ rather
000084 | than as an additional transport parameter.}
000085 | \label{fig:holonomy-triangle}
000086 | \end{figure}
000087 | 
000088 | \subsection{Regularized Yukawa kernels and path-length masses}
000089 | 
000090 | \[
000091 | P_{\mathrm{str}}:=P_{\Seam,+}P_{\mathrm{sing}}P_\Theta,
000092 | \qquad
000093 | P_{\Seam,+}:=\frac{\mathbf 1+\iC}{2},
000094 | \qquad
000095 | P_\Theta:=\frac12(\mathbf 1+Q_\Theta),
000096 | \qquad
000097 | P_{\mathrm{sing}}:=\frac13(\mathbf 1+Z_c+Z_c^2).
000098 | \]
000099 | 
000100 | \begin{definition}[Structural transport kernel]
000101 | \label{def:structural-transport-kernel}
000102 | For each fermion sector define the structural positive transport kernel by
000103 | \[
000104 | K_f^{\mathrm{str}}:=\bigl(P_{\mathrm{str}} D_f^\dagger D_f P_{\mathrm{str}}\bigr)^{-1/2},
000105 | \]
000106 | whenever the inverse exists on $\mathrm{Ran}(P_{\mathrm{str}})$. The final admissible kernel is
000107 | the restriction of this operator to the gap-stable branch selected below. The transport pole
000108 | $\delta_{\mathrm{ph}}$ is fixed by the structural stationarity of the exact determinant functional
000109 | \[
000110 | \mathcal F_{\mathrm{tr}}(\delta)
000111 | :=
000112 | \sum_{y\in\{1,1/3,2/3\}}
000113 | \Bigl[-\log\det\!\bigl(D_{y,\delta}^\dagger D_{y,\delta}+\varepsilon^2\bigr)\Bigr],
000114 | \]
000115 | with
000116 | \[
000117 | D_{y,\delta}:=y\,\mathbf 1-\delta U_6.
000118 | \]
000119 | On the closed branch one has
000120 | \begin{equation}
000121 | \label{eq:transport-bridge-offset}
000122 | \partial_\delta \mathcal F_{\mathrm{tr}}(\delta_{\mathrm{ph}})=0,
000123 | \qquad
000124 | \delta_{\mathrm{ph}}\in\left(\frac13,\frac23\right).
000125 | \end{equation}