Two-barrier Ore-degree conjecture for direct-sum matchings

Let $V$ be an $n$-dimensional vector space over the field with $q$ elements, and let $\genfrac{[}{]}{0pt}{}{V}{k}$ denote the set of $k$-dimensional subspaces of $V$. For a family $\mathcal F\subsetneq\genfrac{[}{]}{0pt}{}{V}{k}$, let $\sigma_{k,q}(\mathcal F)$ be its projective Ore-degree and let $\nu_{\oplus}(\mathcal F)$ be its maximum direct-sum matching size. For integers $k\ge2$, $s\ge2$, and $n\ge ks$, define $D_q(n,k,s-1)$ as in the paper.

Two-barrier Ore-degree conjecture. If

$$\sigma_{k,q}(\mathcal F)>\max\left\{\genfrac{[}{]}{0pt}{}{k}{1}_{q}D_q(n,k,s-1),\;\genfrac{[}{]}{0pt}{}{\max\{0,(s+1)k-n-1\}}{1}_{q}\genfrac{[}{]}{0pt}{}{sk-2}{k-1}_{q}\right\},$$

then $\nu_{\oplus}(\mathcal F)\ge s$.

This corrects the single-threshold formulation, which fails in the full range because both the linear-stability obstruction and the subspace-clique obstruction can control the Ore-degree threshold.