MathOverflow

I conjecture the following:
Conjecture: For all $n, m, s \in \mathbb{N}$ with $s \leq n$:
$$\mathfrak{S}_{s,n}^{m} := \sum_{k=s}^{n} \sum_{\boldsymbol{\lambda} \in \wp_k^{k+m-1}} H_{s,n}^{\boldsymbol{\lambda}} = \frac{n - (s - 1)}{s^m},$$

Let $A_1,A_2,B_1,B_2,C_1,C_2$ be six points in Euclidean affine $3$-space, in general position. Put
$$a=A_1A_2,\qquad b=B_1B_2,\qquad c=C_1C_2.$$
For each $(i,j,k)\in\{1,2\}^3$, let

Definitions:
The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set.
Given two planar convex regions $C_1$ and $C_2$ both with unit perimeter, we define the difference between $C_1$ and $C_2$ as the least value of Hausdorff distance between $C_1$ and $C_2$ can have when the regions are placed above one another and transformed...

Consider the Syracuse map's natural Markov "transfer" chain on the $3$-adic integers $\mathbb{Z}_3$: for $x \in \mathbb{Z}_3$, let$$P(x, \cdot) = \sum_{a \ge 1} 2^{-a}\, \delta_{\varphi_a(x)}, \qquad \varphi_a(x) = (3x+1)2^{-a},$$i.e. the transfer model...