Treasured tools, charms, rune stones (cave keys), summon cultivation, purification, refines, quick rune (eight gates), myoboku grasp and oil, assist links, reputation hall, battle armor, clothing. Home bond system?(I can't verify because my server is currently under maintenance)

but sudden increase in 200 initiative with full battle armor rank lvl 4, how?

and can you tell me which of the above you told should give more initiative with less time?

Could be that he just recently obtained a purple quick rune. Or if not, cave keys+charms combo did the sudden 200 increase.

Charms basically amplifies the stats obtained from magatamas and rune stones. Purification for refines. Basically your charms and purification bonuses are dependent on the base magatama+rune stones and refines.

It's also possible that he may have increased his treasured tool's traits.

Or he could have maxed out one of his assist link cards.

As for the fastest, personally I think it's the treasured tools - but the community may say otherwise. It would be better if we wait for their input as well.

Well i checked it and found his rune stones are unchanged, and charms also, but dont know about treasured tools or assist link cards, but can this much initiative be acquired from 2 lvl 8 refines and 3 lvl 5 refines while i have more higher refines?

With treasured tools and mount myoboku, yes.