= Flat-direction exception to the adjoint cubic stationarity bound
For a traceless Hermitian adjoint scalar with potential $-m^2\operatorname{tr}\Phi^2+a(\operatorname{tr}\Phi^2)^2+b\operatorname{tr}\Phi^4$, the stationary eigenvalues satisfy $4bx^3+(4aR^2-2m^2)x-\zeta=0$. The usual bound of three distinct eigenvalues requires this polynomial not to vanish identically. With $b=0$, $a,m>0$, every traceless matrix of squared norm $R^2=m^2/(2a)$ is a global minimum, including matrices with all eigenvalues distinct. In <SU(5)>, this permits an unbroken maximal torus $U(1)^4$, outside the generic three-block classification.
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