Conjecture on the eigenvalue multiplicities of the Stern recurrence matrices

For each positive integer $r$, let $A_r$ be the matrix governing the recurrence for $u_r(n)$. Write $[a_0,\dots,a_{q-1}]_q$ for the periodic function $f\colon\mathbb{N}\to\mathbb{R}$ satisfying $f(n)=a_i$ when $n\equiv i\pmod q$, and let $e_r(\theta)$ denote the number of eigenvalues of $A_r$ equal to $\theta$. An eigenvalue is semisimple when the minimum polynomial is not divisible by $(x-\theta)^2$, equivalently when all its Jordan blocks have size one.

Eigenvalue multiplicity conjecture. For positive integers $s$, the following hold:

(a)

$$e_{2s-1}(0)=\frac{1}{3}s+[0,-\frac{1}{3},\frac{1}{3}]_3.$$

The eigenvalue $0$ is semisimple; there are no other multiple eigenvalues, and $1$ is not an eigenvalue.

(b)

$$e_{2s}(1)=\frac{1}{6}s+[-1,-\frac{1}{6},-\frac{1}{3},-\frac{1}{2},-\frac{2}{3},\frac{1}{6}]_6,$$ $$e_{2s}(-1)=e_{2s+6}(1).$$

The eigenvalues $1$ and $-1$ are semisimple, and there are no other multiple eigenvalues.

These assertions would sharpen the naive recurrence-order bounds by identifying repeated eigenvalues and their Jordan structure. They are presented as conjectures based on the paper's computations; no proof or resolution is supplied.