Since people keep thinking about the possibility of phasing the AS72 WFS with a longitudinal drive of SRC or PRC, we think it is thus useful to provide a mathematical proof showing that 

the AS72 (both the SUM and each quadrant individually) will NOT see longitudinal perturbations to the linear order. 

Please see the attached note for the details of the proof. The only assumption we made in the derivation is that the carrier field is exactly on- or anti-resonance in the cavities. This should be a reasonable assumption as by design the carrier field should be resonant in PRC and anti-resonant in SRC. (In reality the alignment imperfections and contrast defect will inevitably detune the recycling cavities by a small amount, yet when a length drive shows up in AS72, it will show up in POP90 which behaves very much similar to AS72 SUM. The signal showing up there is actually a power fluctuation in the cavities quadratic in the length drive.)