Semi-topological Galois cohomology and Weierstrass realizability

Semi-topological Galois theory associates a canonical finite splitting covering to a monic Weierstrass polynomial. The inverse limit of the corresponding deck groups defines the absolute semi-topological Galois group, $\text{PiST}(X,x)$. This paper develops a cohomology theory for $\text{PiST}(X,x)$ with discrete torsion coefficients, establishing its fundamental properties and canonical comparison maps to singular cohomology. A Lyndon-Hochschild-Serre spectral sequence is used to yield an obstruction theory for semi-topological embedding problems. We prove several structural and vanishing results, including ST-fullness for free fundamental groups and triviality for finite fundamental groups. As applications, we provide a criterion for lifting finite projective monodromy to linear monodromy, formulate the ${\pi }_1$-detectable Weierstrass realizability conjecture for divisor classes and show that this conjecture is true for abelian varieties, smooth complex projective curves and ruled surfaces over positive-genus curves.