French DJS: to write or not to write the “e caduc”, and the e in “per”

Hello,

So I have these two French DJS theory questions.

-Why does SME (and Sénécal too I believe) write acheter without the so-called unstable e (A-Ch-T-E instead of A-CH-E-T-E) but include it in pèlerinage (P-E-L-E-R-I-N-J), for example (I have found many others)?

-Why write performance according to the established rule stating that -per- should be written P-R, yet write péremption with the first E? I first thought maybe SME distinguished per and pér, but then why peroxyde with P-E-R?

Another example: perdre, perdu, perdition all have the E, but perle doesn’t. Perron has it, persil doesn’t. I’m really clueless about the underlying pattern – I’m hoping there is one?

Thanks!