Composita Stability Theorems for Enhanced Koszul Properties in Galois Cohomology

We investigate how enhanced Koszul properties of Galois cohomology behave under composita of fields. Given fields $K_1$ and $K_2$ containing ${\mu }_p$, with intersection $k$ and compositum $K=K_1{K}_2$, we formulate an abstract composita stability theorem: under a pro-$p$ amalgam decomposition $G_K\cong G_{K_1}{\ast }_{G_k}{G}_{K_2}$ of maximal pro-$p$ Galois groups, and natural Mayer-Vietoris compatibility assumptions on the mod-$p$ cohomology rings $H^{\bullet }(G_{K_1},{\mathbb{F}}_p)$, $H^{\bullet }(G_{K_2},{\mathbb{F}}_p)$, and $H^{\bullet }(G_k,{\mathbb{F}}_p)$, the quadratic presentation of $H^{\bullet }(G_K,{\mathbb{F}}_p)$ arises from a fiber-product construction on degree-$1$ generators and quadratic relations. Assuming stability of universal Koszulity under this quadratic gluing, we obtain that universal Koszulity of $H^{\bullet }(G_{K_1},{\mathbb{F}}_p)$ and $H^{\bullet }(G_{K_2},{\mathbb{F}}_p)$ implies universal Koszulity of $H^{\bullet }(G_K,{\mathbb{F}}_p)$. As a concrete application, we prove a composita stability theorem for certain Pythagorean fields whose maximal pro-$2$ Galois groups decompose as free pro-$2$ products of Demuškin groups and free factors. For suitable composita $K=K_1{K}_2$ of such fields, the mod-$2$ Galois cohomology ring $H^{\bullet }(G_K(2),{\mathbb{F}}_2)$ remains quadratic and universally Koszul. This provides large classes of fields, built from local, global, and Pythagorean base fields by admissible extensions and composita, whose maximal pro-$p$ Galois groups have universally Koszul cohomology, and yields inverse Galois obstructions: any finitely generated pro-$p$ group with nonquadratic or non-universally Koszul mod-$p$ cohomology cannot occur as the maximal pro-$p$ Galois group of a field in these families.