It's likely that str adjusts your damage curve, while attack power is the potential range on that curve.
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We know what control does (lessens sparking and instability) and craftsmanship advances the progress bar, but what magic craftsmanship does is beyond me. Everything I hear/ read seems to be speculation so yes clarification would be really nice, especially since so much of the newer gear has more Mag. Craft than Craft. Increased HQ chance? Increased Crit chance?
OK, so there's no direct correlation between STR and Attack Power.. but what is the difference? What benefit does STR give that Attack Power doesn't?
Had actually already added that in (about at the same time you posted funnily enough); it's asterisked just before the recap (just now made it easier to see). Going to try and find some gear for my 30THM that offers a +4 or possibly 4 +1 pieces to INT, PIE or MND to see if it is in fact +4 for every 1 no matter the level or perhaps starting after a certain point. I can confirm like you that at 50 +4 adds 1. I have another theory I'm working out but I'll wait to post anything on that for now, until I've tested it.
Thanks for the research and links, I'll add them to OP for further reading!
Edit1: As for the information regarding STR and Attack Power, I think there may be a slight misunderstanding between the phrasing of Attack Power stat and your actual attack.
My personal thought is that adding STR will not raise the Attack Power stat. The equation likely is based on the target's VIT. Maybe adding STR to surpass the targets VIT will increase your minimum damage and Attack Power will increase your damage further after reaching that. Just a theory, will need some actual testing (feel free to test anyone, I may not be able to get to it right away). It would also make sense in regard to the reports coming in that adding STR outparses adding Attack Power. The closer you get to the targets VIT the more damage you'd do, once surpased it would likely plateau in which case adding attack power would be beneficial.
Edit2: As for crafting, my theory is that tools that use STR/VIT/DEX use Craftsmanship to determine success rate and tools that use INT/MND/PIE use Magic Craftmanship to determine success rate. It's just a theory based on an educated guess with gear and optimal/required classes in relation to their tools but a dev response or even hint would be great.
There are two ways to interpret Bayohne's post.
1) The damage formula is like this: Damage Dealt = STR + [rest unknown]. If this is the case, each point of STR adds a flat 1 point of damage.
2) The other (and I consider this more likely) possibility is that he was simply trying to explain that STR is not related to ATK the way it was in FFXI (2 STR would, in addition to its own effect, add 1 ATK). In this case we are no closer to understanding the damage formula and we should merely stop trying to spot changes in ATK when we add STR.