I'm here for some help to prove or disprove a (possibly trivial) conjecture concerning compositions (i.e. ordered partitions) of the natural number $n$, and the contiguous subsequences that they determine within a finite sequence of length $n$. What is conjectured is that only compositions whose parts fall within a relatively small range have a certain property; the computer suggests that the conjecture holds for all $n<34$ (and counting).
The details: let $p$ and $q$ be members of $\{ 2,3,4,\ldots\} $ with $q>2$, and suppose $\left(a_k\right)_{k=1}^{pq}$ and $\left(b_k\right)_{k=1}^{pq}$ are finite sequences with terms in $\left[-1,1\right]\subset\mathbb{R}$.
Any composition $y$ of $pq$ is of the form
$$\sum_{i=1}^{n_y} y_i.$$
Declare $y_0=1$, and for each $\alpha \in \{0,1,2,...,n_y - 1\}$ let $s_\alpha\in\mathbb{R}$ be such that the following finite sequence has minimal RMS:
$$\Big(a_j - s_{\alpha}b_j\Big)_{j=\sum_{i=0}^{\alpha}y_i}^{\sum_{i=1}^{\alpha+1}y_i}\tag{1}\label{1}$$
For example, if $p=2$, $q=3$, and $y$ is the composition $2+4$ of $6$, then $s_0$ minimizes the RMS of the finite sequence
$$\left(a_1-s_0 b_1,a_2-s_0 b_2\right)$$
while $s_1$ does the same for
$$\left(a_3-s_1 b_3,a_4-s_1 b_4,a_5-s_1 b_5,a_6-s_1 b_6\right).$$
Let $C_{\mu}^{\nu}\left(pq\right)$ denote the set of compositions of $pq$ consisting only of terms no less than $\mu$ and no greater than $\nu$ (so that e.g. $C_{3}^{5}\left(2\cdot 4\right)$ contains only the compositions $4+4$, $5+3$, and $3+5$ of $8$).
The issue at the heart of the conjecture: for which compositions $y$ in $C_{p}^{pq}\left(pq\right)$ does the following quantity attain a minimal value (for arbitrary finite sequences of length $pq$ in $\left[-1,1\right]$):
$$\text{rms}\bigg(\Big(a_j - s_{0}b_j\Big)_{j=1}^{y_1}\frown\Big(a_j - s_{1}b_j\Big)_{j=1+y_1}^{y_1 + y_2}\frown\cdots\frown\Big(a_j - s_{\left(n_y - 1\right)}b_j\Big)_{j=\sum_{i=0}^{n_y - 1}y_i}^{\sum_{i=1}^{n_y}y_i}\bigg)$$
(I've used notation suggested here to denote the concatenation of finite sequences. The finite sequences being concatenated are the values of $\eqref{1}$ as $\alpha$ ranges over $ \{0,1,2,...,n_y - 1\}$).
The conjecture is then (finally) that only compositions lying in $C_p^{2p-1}\left(pq\right)$ have the desired property.
I get the sense from the computer that parts of size greater than $2p-1$ 'decompose' into two or more parts with sizes in $\{p,p+1,\ldots,2p-1\}$, but am unsure how this might be formalised (if it indeed actually occurs). Would also be very grateful to learn of any obvious counterexample disproving the conjecture.
