Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus

We prove the rational Hodge and generalized Hodge conjectures in every cohomological degree of every self-power of a projective complex K3 surface whose transcendental quadratic space, with its cup-product form, admits a rational isometric embedding in ${\mathbb{U}}_{\mathbb{Q}}^{\oplus 2}\perp ⟨-1{⟩}^4$. This includes every ample P-polarized K3 surface, including all Picard jumps, for $P=\mathbb{U}\oplus D_8(-1)\oplus D_4(-1)$. On this locus we construct algebraic correspondences inducing every prescribed standard even-Clifford Kuga–Satake tensor. More generally, for any projective complex K3 surface, algebraicity of one exact standard even-Clifford Kuga–Satake tensor implies both conjectures for every self-power.