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We develop a higher semiadditive version of Grothendieck-Witt theory. We thenapply the theory in the case of a finite field to study the higher semiadditivestructure of the $K(1)$-local sphere at the prime $2$. As a furtherapplication, we compute and clarify certain power operations in the homotopy ofthe $K(1)$-local sphere.

Higher semiadditive Grothendieck-Witt theory and the $K(1)$-local sphere