Derived invariance of Hodge numbers for smooth projective complex fivefolds with $K_X = 0$

We prove that derived-equivalent smooth projective complex fivefolds with trivial canonical bundle have the same Hodge numbers. The proof combines the derived invariance of Hochschild homology with a refinement of the Mukai pairing on the individual Hochschild diagonals. After composition with complex conjugation, the pairing gives rise, in odd Hochschild degree, to Hermitian forms preserved by Fourier-Mukai equivalences. Their signatures are computed using the Lefschetz decomposition and the Hodge-Riemann bilinear relations. Together with a Hirzebruch-Riemann-Roch relation specific to fivefolds with trivial first Chern class, this determines all Hodge numbers.