Regular-graph numerical contradiction threshold

By [F5], choose a common rotation $R_c$ making the first order parameter real and nonnegative; when the original ${\rho }_1=0$, take $c=0$. Addition of a fixed phase is continuous in the torus coordinates and has the continuous inverse $R_{-c}$, so $R_c$ is a homeomorphism. If $U$ is a neighborhood witnessing local minimality at the original state, then $R_{c}U$ is a neighborhood of the rotated state. For every $\varphi \in R_{c}U$, energy invariance in [F6] gives ${\mathcal{E}}_G(\varphi )={\mathcal{E}}_G(R_{-c}\varphi )\ge {\mathcal{E}}_G(\theta )={\mathcal{E}}_G(R_c\theta )$. Thus the rotated state is stable. Choose its principal representatives using [F5] and denote this normalized stable state again by $\theta$. Consequently the hypotheses of [F1] and [F2], including countable choice, apply to this state.

By [F1], $X\ge 1-2\alpha =0.8368>0$. If $d>0$, [F2] gives $M\le {\alpha }^2(1-X)\mathrm{/}X\le {\alpha }^2(2\alpha )\mathrm{/}(1-2\alpha )<{\alpha }^2\mathrm{/}4$, because $4(2\alpha )=0.6528<0.8368=1-2\alpha$. If $d=0$, [F2] gives $M=0<{\alpha }^2\mathrm{/}4$. Thus in either case $M<{\alpha }^2\mathrm{/}4=0.00166464<0.00832$.

For each $x\in {\mathcal{C}}_{\xi }$, [F4] gives $\xi \le \mathrm{\mid }{\theta }_x\mathrm{\mid }\le \pi$. By [F3] and the definition of $H$, $H({\theta }_x)=H(\mathrm{\mid }{\theta }_x\mathrm{\mid })\ge H(\xi )={\sin }^2\xi$; all other summands are nonnegative. Hence $nM\ge \mathrm{\mid }{\mathcal{C}}_{\xi }\mathrm{\mid }{\sin }^2\xi$. If the asserted threshold held, this would give $M\ge 0.00832$, contradicting step 2.1. The condition is therefore impossible.