One physical split instrument

This is a pointwise identity. It is not an assertion that diagonal filtering preserves every two-mode visibility or every measured fringe contrast.

For the diagonal density in ${\tau }_h(z)$, nonzero maxima exist when $z>1/(8\eta v_{x,h})$. Their separation is

This follows by differentiating $-2(\eta x^2-z{)}^2-x^2/(2v_{x,h})$. The inverse square root of the negative log-density curvature at a peak is

This explicitly defined local RMS-width approximation is not assigned an ambiguous “packet width” convention.

Even at the same apparatus outcome $z$, the conditional peak locations can differ between hypotheses. A window selected through a hypothesis-dependent $d_h(z)$ is not one common acceptance rule. Use fixed $z$-windows or calibrated hypothesis-independent bounds.