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Kuznetsov's homological projective duality (HPD) theory [K4] is one of the most active and powerful recent developments in the homological study of algebraic geometry. The fundamental theorem of HPD systematically compares derived categories of dual linear sections of a pair of HP-dual varieties (X, X). In this paper we generalize the fundamental theorem of HPD beyond linear sections. More precisely, we show that for any two pairs of HP-duals (X, X) and (T, T) which intersect properly, there exist semiorthogonal decompositions of the derived categories D(X ∩ T) and D(X ∩ T) into primitive and ambient parts, and that there is an equivalence of primitive parts primD(X ∩ T) - D(X ∩ T)prim. [ABSTRACT FROM AUTHOR]