TFPT scientific source excerpt
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000353 | the actual branch Yukawa tensor; only afterwards is $\Oadm=48=4\cdot12$ read as a consistency check.}
000354 | \label{fig:carrier-closure-cut}
000355 | \end{figure}
000356 | 
000357 | \begin{corollary}[Hard holonomy form of the Yukawa matrices]
000358 | On the admissible branch, each fermion sector obeys the canonical factorization
000359 | \[
000360 | Y_f=D_{L,f}U_f^\star D_{R,f},
000361 | \qquad
000362 | U_f^\star\in SU(3)_F,
000363 | \]
000364 | and, in the family basis fixed by \TFPTxref{thm:exact-transport-closure},
000365 | \[
000366 | (Y_f)_{ij}
000367 | =
000368 | \lambdaY^{L^L_{f,i}+L^R_{f,j}}
000369 | \left\langle e_i,\mathrm{Hol}_{\Gamma_{ij}^{\min}}(\nabla_F^\star)e_j\right\rangle.
000370 | \]
000371 | Any factorization by $U_f^\star$ is therefore a consequence of the hard holonomy closure, not an
000372 | independent flavor input. The diagonal path-length rows \TFPTxref{eq:path-length-yukawa} are the
000373 | singular-value reduction of this matrix statement.
000374 | \end{corollary}
000375 | 
000376 | \begin{proof}[Proof sketch]
000377 | The finite $C_6$ transport geometry fixes the graph-distance weights, while the rigid
000378 | $D_4$-equivariant family connection fixes the holonomy factors on the canonical path frame. The
000379 | diagonal ladder formulas arise once left and right admissible words are paired on the same
000380 | singular-value channel, so the matrix factorization is recovered as a corollary of one closed
000381 | holonomy statement rather than as a separate ansatz.
000382 | \end{proof}
000383 | 
000384 | \begin{definition}[Center-neutral admissible algebra]