= Solution
For complex solutions of the <Klein-Gordon equation>, define the current
$$
j^a=i(\overline\alpha\nabla^a\beta-\beta\nabla^a\overline\alpha).
$$
Its divergence is
$$
\nabla_aj^a=i\left(\overline\alpha\Box\beta-\beta\Box\overline\alpha\right)=0,
$$
because $\Box\beta=m^2\beta$ and $\Box\overline\alpha=m^2\overline\alpha$ for real $m^2$. The <Klein-Gordon inner product> is the flux $\int_{\Sigma}n_aj^a\,d\Sigma$. Applying <Stokes theorem> to a slab between two <Cauchy hypersurfaces> gives equality of the fluxes, with the future normal on each surface and opposite boundary orientations.
For noncompact surfaces this requires the integrals to converge and no net flux through spatial infinity, for example spacelike compact support or suitable decay. <Global hyperbolicity> alone does not ensure convergence for arbitrary solutions. Under these conditions,
$$
\boxed{(\alpha,\beta)_{\Sigma_1}=(\alpha,\beta)_{\Sigma_2}.}
$$
Complex conjugation also gives
$$
\overline{(\beta,\alpha)}=-i\int_\Sigma d\Sigma\,n^a(\beta\nabla_a\overline\alpha-\overline\alpha\nabla_a\beta)=(\alpha,\beta).
$$
Thus it is a <Hermitian form>, antilinear in its first argument and linear in its second. It is indefinite on the full complex solution space; positivity will arise only on a chosen <positive-frequency solution> subspace.
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