Guessing models imply the singular cardinal hypothesis

In this article we prove three main theorems: (1) guessing models are internally unbounded, (2) for any regular cardinal \kappa \ge \omega _2, \mathsf {ISP}(\kappa ) implies that \mathsf {SCH} holds above \kappa , and (3) forcing posets which have the \omega _1-approximation property also have the countable covering property. These results solve open problems of Viale [Ann. Pure Appl. Logic 163 (2012), no. 11, 1660-1678] and Hachtman and Sinapova [J. Symb. Log. 84 (2019), no. 2, 713-725].