The local-to-global companion-point criterion

Let $y$ be a point of the relevant patched, global, or local Bernstein variety, and let $\varrho$ be its associated generic potentially crystalline Galois representation with distinct Hodge–Tate weights. Let $\mathbf h$ be its decreasing Hodge–Tate weights, let $\lambda$ be the associated dominant weight, and let $\texttt r(\varrho)$ have its generic $P$-filtration with associated Weyl-group element $w_{\mathscr F_{\varrho}}$. For $w\in\mathscr W_{\min,F_{\wp}}^P$, write $x$ for the corresponding point and use the dot action on weights. Companion-point conjecture. The point belongs to the indicated companion-point locus precisely when

$$w w_0\geq w_{\mathscr F_{\varrho}}.$$

This gives the expected companion-point criterion on Bernstein eigenvarieties, patched Bernstein eigenvarieties, and Bernstein paraboline varieties. The source presents it as part of the program attacking the socle conjecture and does not provide a resolution here.