 Ambiguity of harmonic wedge-pairing representatives

On a compact oriented Riemannian manifold without boundary, a smooth $(n-p)$-form $\chi$ annihilates all harmonic $p$-forms under wedge integration exactly when its harmonic projection vanishes. Indeed ${\int }_M\phi \wedge \chi =⟨\ast \phi ,\chi {⟩}_{L^2}$, and the Hodge star maps ${\mathcal{H}}^p$ isomorphically onto ${\mathcal{H}}^{n-p}$. By the Hodge decomposition theorem, this annihilator isThus both exact and coexact additions are allowed; a harmonic representative is unique. The potentials of these additions need not be unique, and out-of-range degree spaces are zero. For a concrete coexact ambiguity, take the torus $(\mathbb{R}/2\pi \mathbb{Z}{)}^2$ with metric $d{x}^2+d{y}^2$. The form $\chi =\delta (\sin ydx\wedge dy)=\cos ydx$ annihilates every harmonic one-form by orthogonality, yet $d\chi =\sin ydx\wedge dy$ is nonzero. Thus an annihilating representative need not even be closed, let alone exact.