Extreme Kaleidocycle conjecture

Let $N\ge 6$. Recall that $\mathcal{M}_N$ is the configuration space of $N$-Kaleidocycles, $\pi_c:\mathcal{M}_N\to\mathbb{R}$ is the projection onto the $c$-axis, where $c=\cos\nu$, and $\mathcal{M}_N^+$ and $\mathcal{M}_N^-$ denote the two components used in the description of the configuration space. Let $c_N^*\in[0,1]$ be the boundary value identified numerically, and let $\gamma^{b,\varepsilon}$ denote the strip associated with $b$. Extreme Kaleidocycle conjecture. The following hold: (1) $\pi_c^{-1}(c_N^*)\cap\mathcal{M}_N^-$ is a circle, with the stated involution-induced isomorphisms at $-c_N^*$; (2) the semi-discrete sine-Gordon flow has orbit $\pi_c^{-1}(c_N^*)\cap\mathcal{M}_N^-\simeq S^1$ through every element; (3) when $N$ is odd, the corresponding semi-discrete mKdV and sine-Gordon flows at $-c_N^*$ both coincide with the relevant fibre up to rigid transformations; (4) every strip associated with $b\in\pi_c^{-1}(c_N^*)\cap\mathcal{M}_N^-$ is a 3-half twisted Möbius strip, and no Kaleidocycle has one or two half twisting; and (5) as $N\to\infty$, $N\arccos(c_N^*)$ converges to a constant, while every sequence of corresponding curves has a unique limit curve up to congruence, with constant torsion up to sign. These assertions summarize numerical findings about the boundary fibres of the configuration space and their integrable deformations. The preceding numerical experiments were conducted for $6\le N\le 50$, and the limiting and uniqueness assertions are likewise presented as conclusions of the numerical study; the source does not establish them analytically.