The primitive codimension-growth conjecture for Jacobi spectral curves

Let the connected reducibility locus be the locus in parameter space where $b_1\cdots b_{n-1}\ne0$ and the spectral polynomial is reducible. Let $\mathcal R_n^{\rm prim}$ denote the union of its irreducible components that are not induced from proper contiguous subchains and are not obtained from scalar-diagonal degenerations.

Primitive codimension-growth conjecture. One has

$$\operatorname{codim}\mathcal R_n^{\rm prim}\longrightarrow\infty\qquad\text{as }n\to\infty.$$

More precisely, outside special coincidence strata among the diagonal entries, one expects

$$\operatorname{codim}\mathcal R_n^{\rm prim}\ge\left\lfloor {n\over2}\right\rfloor.$$

This organizes the known nontrivial connected reducibility mechanisms by their expected codimension: palindromic and scalar blocks of length $m$ have codimension $m-1$, while generic constant branches have expected codimension $\lfloor m/2\rfloor$. The statement remains an expectation, with special coincidence strata excluded from the more precise estimate.