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I'm trying to better understand Harmonious Academic Methodology (a Lore charm).
It can be used to train others, but the rules states that the lawgivers can also train himself.
So can a character with HAM train himself up to 4 dot in the allowed abilities spending 1 week of training for dot without using XP?
Not without using XP
From the same wiki, description of the Training Charm-keyword :
though they may award competence quickly and easily, the character must still pay for what they have gained.
From the HAM page you linked: "She can train them"... "or up to her own own ratings, whichever is lower" (emphasis mine). Any point she might try to gain would always be raising her rating to a level she doesn't already have it up to, and therefore would always be beyond the ability of HAM.
All effects in Exalted with the Training keyword require that the recipient(s) expend the appropriate amount of XP. If the target doesn't have enough XP, they go into XP debt. When a character with XP debt gains XP, half of the gained XP is spent buying off the debt. A character with XP debt cannot be the target of a training effect.
Thus, while you can use training effects to get around some of the more laborious training times, you can't use them to get around the associated XP costs.
