The straight chord misses the curved double-well minimum-energy path

Along $C$, the first coordinate is $\cos a>0$, and $\mathrm{\mid }y\mathrm{\mid }\mathrm{/}\cos a\le \tan a<1$, so the polar angle lies in $[-a,a]\subset (-\pi \mathrm{/}4,\pi \mathrm{/}4)$. Its radius is between $\cos a=\sqrt{3}\mathrm{/}2>3\mathrm{/}4$ and $1<5\mathrm{/}4$. Thus the whole chord is in the declared sector.

The chord's tangent is $e_2=(0,1)$. At its midpoint $x_0=(\cos a,0)=X(0,n_0)$ with $n_0=\cos a-1\mathrm{\ne }0$. In the chart $X_s=(1+n)T$ and $X_n=N$, where $T(0)=e_2$ and $N(0)=e_1$. The coordinate derivatives of $V$ there are ${\mathrm{\partial }}_{s}V=W^{\mathrm{\prime }}(0)=0$ and ${\mathrm{\partial }}_{n}V=n_0$, so by [F3] $\mathrm{\nabla }V(x_0)\cdot e_1=n_0\mathrm{\ne }0$.

Since $e_1$ is perpendicular to the chord tangent $e_2$, step 1.2 gives a nonzero normal component of the gradient at the chord midpoint. This violates the MEP condition in [F1], proving the chord is not the circular MEP. The improved tangent rule in [F2] only changes the discrete force direction; it does not make this straight segment satisfy the continuous normal-gradient equation.