...while this is basically a rules question, posting this here as in Missions all improvements must be done during downtime using "Purchased Hits".
As I understand even for Adepts one needs Arcana to initiate. Now while I feel for a dedicated physad, it is otherwise a rather useless skill since the specialisations only cover writing formulas which the character would never make use of. Nonetheless it apparently is a requirement for all awakened characters
So my character is looking at initiating again. She has the skill at 1 and as I understand, for the purpose of initiation, the test. is Arcana + Intuition, not Logic. It is also an extended test with an interval of 29 days (basically 1 month) so grade 1 would be 1 month grade 2 would take 2 months, and so on. As the character has an intuition of 5, that would give her a pool of 6, so could she initiate three times before having to increase the skill?