The amended reducibility conjecture for finite Jacobi pencils

Let $J_n(w)$ be a finite Jacobi pencil and let

$$\chi_n(\lambda,w)=\det(\lambda I+J_n(w)).$$

A Jacobi chain is connected if all its couplings are non-zero. A factorization is primitive if it is not obtained from a proper disconnected subchain. For a connected block $I=[r,s]$, write $P_I(\lambda,t)$ for its characteristic polynomial; call $I$ palindromic when

$$a_{r+k}=a_{s-k},\qquad b_{r+k}^2=b_{s-k-1}^2$$

for all admissible $k$, and call it scalar-diagonal when $a_r=\cdots=a_s$. A constant branch is a factor $\lambda+a_j$ for which $P_I(-a_j,t)\equiv0$.

Amended reducibility conjecture. Every reducible factorization of $\chi_n$ is expected to be obtained by iterating the following operations on connected blocks: cutting the chain at a zero coupling $b_i=0$; extracting a constant branch $\lambda+a_j$ from a connected block for which $P_I(-a_j,t)\equiv0$; decomposing a palindromic connected block into its symmetric and anti-symmetric parts under reversal; or decomposing a scalar-diagonal connected block by factoring the characteristic polynomial of its constant off-diagonal part. After all these operations have been exhausted, the remaining factors are expected to be irreducible in $\mathbb C[\lambda,w]$.

This amended formulation includes scalar-diagonal blocks, which are a genuine source of connected reducibility. It is an expected classification of all reducibility mechanisms for finite Jacobi pencils; no resolution is supplied in the source.