TFPT scientific source excerpt
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000140 | \[
000141 | \rho_F:\pi_1(X_f^\circ)\to SU(3)_F
000142 | \]
000143 | uniquely up to global $SU(3)_F$ conjugation.
000144 | 
000145 | Construct the flat Hermitian bundle explicitly as
000146 | \[
000147 | L_F:=(\widetilde X_f^\circ\times\mathbb C^3)/\pi_1(X_f^\circ),
000148 | \]
000149 | where $\pi_1(X_f^\circ)$ acts on $\mathbb C^3$ through $\rho_F$. Because the image of $\rho_F$ lies
000150 | in $SU(3)_F$, the standard Hermitian form on $\mathbb C^3$ descends to a Hermitian metric on
000151 | $L_F$, and the trivial connection on $\widetilde X_f^\circ\times\mathbb C^3$ descends to a flat
000152 | Hermitian connection realizing the same monodromy representation.
000153 | 
000154 | Conversely, any admissible flat Hermitian family connection with the same puncture data has the same
000155 | monodromy representation up to global $SU(3)_F$ conjugation. The unitary Riemann--Hilbert
000156 | correspondence for flat Hermitian bundles then identifies it with the descended bundle above.
000157 | Hence there is exactly one admissible flat Hermitian connection class realizing the rigid puncture
000158 | data. We denote this distinguished class by
000159 | \[
000160 | [\nabla_F^\star].
000161 | \]
000162 | \end{proof}
000163 | 
000164 | \begin{theorem}[Canonical transport kernel and single-winding closure]
000165 | \label{thm:exact-transport-closure}
000166 | \label{thm:canonical-transport-kernel}
000167 | The admissible cusp set is
000168 | \[
000169 | \left\{1,\frac23,\frac13\right\},
000170 | \]
000171 | obtained from the singlet-projected hypercharge spectrum. On the base hexagon the residue packets
000172 | are
000173 | \[
000174 | R_u=(0,3,1),
000175 | \qquad
000176 | R_d=(2,5,1),
000177 | \qquad
000178 | R_e=(3,5,2).
000179 | \]
000180 | Let $\mathcal G_{D_4}\subset X_f^\circ$ denote the unique $D_4$-invariant geodesic spine of the
000181 | four-punctured family sphere, and let
000182 | \[
000183 | \Gamma_{ij}^{\min}
000184 | \]
000185 | be its canonical admissible edges. Then the Yukawa matrices are fixed intrinsically by the operator
000186 | formula
000187 | \[
000188 | (Y_f)_{ij}
000189 | :=
000190 | \left\langle e_i,\mathrm{Hol}_{\mathcal G_{D_4}}(\nabla_F^\star)\,K_f^{\mathrm{str}}e_j\right\rangle,
000191 | \]
000192 | and on diagonal reduction this becomes the hard holonomy form
000193 | \[
000194 | (Y_f)_{ij}
000195 | =
000196 | \lambdaY^{L^L_{f,i}+L^R_{f,j}}
000197 | \left\langle e_i,\mathrm{Hol}_{\Gamma_{ij}^{\min}}(\nabla_F^\star)e_j\right\rangle,
000198 | \]
000199 | so the bounded coefficients $\Lambda_{f,j}$ are exact matrix entries of the structural transport
000200 | kernel rather than independent $O(1)$ amplitudes. The resonance pole of the transport kernel is the
000201 | unique admissible root
000202 | \[
000203 | \delta_{\mathrm{ph}}\in\left(\frac13,\frac23\right)
000204 | \]
000205 | of \TFPTxref{thm:algebraic-transport-pole}. The exact sum rule is
000206 | \[
000207 | \sum_{f,j}L_{f,j}=\Oadm\gamma=40.
000208 | \]
000209 | The base residue sum is $22$, so the missing $18=3\cdot 6$ forces exactly one full winding in
000210 | each first-generation direction. Therefore uniquely
000211 | \[
000212 | L_u^{\mathrm{diag}}=(7,3,0),\quad
000213 | L_d^{\mathrm{diag}}=(7,5,2),\quad
000214 | L_e^{\mathrm{diag}}=(8,5,3)
000215 | \]
000216 | on the closed branch.
000217 | \end{theorem}
000218 | 
000219 | \begin{proof}
000220 | By \TFPTxref{thm:rigid-family-local-system}, the flat Hermitian family-connection class
000221 | \[
000222 | [\nabla_F^\star]
000223 | \]
000224 | is uniquely fixed on the rigid four-punctured family sphere. By
000225 | \TFPTxref{def:canonical-cycle-holonomy,thm:d4-monodromy-rigidity}, its holonomy on the unique
000226 | $D_4$-invariant geodesic spine $\mathcal G_{D_4}$ is therefore intrinsic. At the same time,
000227 | \TFPTxref{def:structural-transport-kernel,thm:algebraic-transport-pole} fixes the structural positive
000228 | transport kernel and its unique admissible resonance pole
000229 | \[
000230 | \delta_{\mathrm{ph}}\in\left(\frac13,\frac23\right).
000231 | \]
000232 | Consequently the operator product
000233 | \[
000234 | \mathrm{Hol}_{\mathcal G_{D_4}}(\nabla_F^\star)\,K_f^{\mathrm{str}}
000235 | \]
000236 | is uniquely determined on the canonical family basis, and its matrix entries are theorem-level
000237 | transport data. This proves the operator formula for $Y_f$ and shows that the former coefficients
000238 | $\Lambda_{f,j}$ are not independent amplitudes but exact entries of the same closed transport
000239 | operator.
000240 | 
000241 | On the diagonal branch, the canonical admissible edges $\Gamma_{ij}^{\min}$ recover the word-length
000242 | reduction
000243 | \[
000244 | y_{f,j}^{\mathrm{UV}}=\lambdaY^{L_{f,j}}\Lambda_{f,j}.