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Fun fact: $Z_{ndc}$ has the form $z'=A+\frac{B}{z}$ and is also linearly interpolated in screen-space for z-buffering:

Fun fact: $Z_{ndc}$ has the form $Z_{ndc}=A+\frac{B}{z}$ (where $z$ is in view space) and is also linearly interpolated in screen-space for z-buffering:

Fun fact: NDC z-coordinates of the form $z'=A+\frac{B}{z}$ are also linearly interpolated in screen-space for z-buffering:

Thus both steps amount to LERP of NDC z.

Fun fact: $Z_{ndc}$ has the form $z'=A+\frac{B}{z}$ and is also linearly interpolated in screen-space for z-buffering:

Thus both steps amount to a lerp of $Z_{ndc}$.

The vertex NDCs are $I_1/Z_1$ and $I_2/Z_2$ and the perspective-correct interpolation equation gives: $$ \left[\frac{I_1}{Z_1^2}+s\left(\frac{I_2}{Z_2^2}-\frac{I_1}{Z_1^2}\right)\right]Z_t $$ which is clearly not $I_t/Z_t$ as defined in $\eqref{*}$. The reason this equation doesn't work in this case is that its derivation is based on the fact that the attribute varies linearly across the triangle in 3D space (view space). But screen-space NDC coords don't vary linearly in 3D space because of the perspective divide, so the premise is broken. (Clip-space coords do vary linearly in view-space because they are the result of a linear transform.) Perspective-correct interpolation of values that have undergone perspective divide doesn't work. They are not in 3D space anymore, rather in 2D projection space and thus require linear interpolation in screen-space to get correct results. In practice it's accomplished using perspective-correct interpolation of clip-space coords, plus perspective divide per-pixel.

The vertex NDCs are $I_1/Z_1$ and $I_2/Z_2$ and the perspective-correct interpolation equation gives: $$ \left[\frac{I_1}{Z_1^2}+s\left(\frac{I_2}{Z_2^2}-\frac{I_1}{Z_1^2}\right)\right]Z_t $$ which is clearly not $I_t/Z_t$ as defined in $\eqref{*}$. The reason this equation doesn't work in this case is that its derivation requires the attribute to vary linearly across the triangle in 3D space (view space). But screen-space NDC coords don't vary linearly in 3D space because of the perspective divide, so the premise is broken. (Clip-space coords do vary linearly in view-space because they are the result of a linear transform.) Perspective-correct interpolation of values that have undergone perspective divide doesn't work. They are not in 3D space anymore, rather in 2D projection space and thus require linear interpolation in screen-space to get correct results. In practice it's accomplished using perspective-correct interpolation of clip-space coords, plus perspective divide per-pixel.