TFPT scientific source excerpt
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000256 |   phase, not a unitary matrix entry.
000257 | \item \textbf{Moduli result.} The $D_4$-symmetric $SU(3)$ monodromy (each puncture in the cusp class
000258 |   $\{1,\omega,\omega^2\}$, product $=\mathbf1$) is explicitly constructible ($\|M_1M_2M_3M_4-\mathbf1\|<10^{-12}$),
000259 |   but the $D_4$-fixed locus is \emph{positive-dimensional} (the conjugation invariant $|\mathrm{tr}\,M_1M_2|$
000260 |   varies continuously). Hence $\nabla_F^\star$ --- and with it the exact $\Lambda_{f,j}$ --- is \emph{not}
000261 |   pinned by $D_4$-symmetry $+$ cusp class $+$ product alone.
000262 | \item \textbf{Why it is a wall (sharpening).} The cusp weights $\{0,\tfrac13,\tfrac23\}$ (sum $1$) at the
000263 |   four punctures give total parabolic weight $\sum_iw_i=4$, so strict stability of the split
000264 |   $\mathcal O(-2)\oplus\mathcal O(-1)^2$ would need $w_1{<}2$ (the $\mathcal O(-2)$) and $w_2,w_3{<}1$ (the
000265 |   $\mathcal O(-1)$'s) --- but $w_1=4-w_2-w_3>2$. The bounds are jointly \emph{unsatisfiable}: the
000266 |   configuration sits on the parabolic \emph{stability wall}, so $\nabla_F^\star$ is the unique
000267 |   \emph{polystable} representative there. This is exactly why the $D_4$-fixed locus is positive-dimensional,
000268 |   and it pins $(U)$ down to a wall-selection. \veri{v26\_flavor\_frontier\_unification.py}
000269 | \end{enumerate}
000270 | \noindent\textbf{Update --- the ratios no longer reduce to this gate.} Since \vref{v49}/\vref{v69}/\vref{v70}/\vref{v71} the geometric
000271 | origin of $Q$ (\veri{v69\_d4\_q\_geometry.py}), the lattice selector $\det R=8$ (\veri{v70\_q\_integer\_lift.py})
000272 | and the rigidity of $c_u/c_d=\tfrac{55}{117}$ on the derived stratum (\veri{v49\_readout\_rigidity.py}) are