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So, normally, I'd take this to the Rules forum, but we're dealing with a unique to PFS item.
The question is, how does Dragon Ferocity (Style Feat) interact with the Braid's activated ability?
Does this mean that when the Braid is active, I'm in Dragon Style w/ Dragon Ferocity, and I flurry, my first attack is at 2.5x STR Bonus and my remaining attacks at 2x STR Bonus? Or do these abilities not stack in any way? The language specifically calls out an increase of "1/2" versus? Or does the "to a total of" override that and cap things at 2x & 1.5 respectively?

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The damage section of the PRD should help to answer this question.
Note: When you multiply damage more than once, each multiplier works off the original, unmultiplied damage. So if you are asked to double the damage twice, the end result is three times the normal damage.
EDIT
More on point, p. 12 of the CRB says this:
Multiplying: When you are asked to apply more than one multiplier to a roll, the multipliers are not multiplied by one another. Instead, you combine them into a single multiplier, with each extra multiple adding 1 less than its value to the first multiple. For example, if you are asked to apply a ×2 multiplier twice, the result would be ×3, not ×4.

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I think your calculation is correct based on the multiplier rules cited by Michael Eshleman. The "to a total of" is just a clarification that Dragon Ferocity adds to the 1.5x Strength modifier on your first unarmed strike received from Dragon Style (demonstrating that the normal stacking rules apply to the advanced style feat). When you activate the Braid, it just adds another 0.5x Strength on top of whatever bonus you already receive. If, for example, you were using a staff two handed and were not in Dragon Style, you would normally receive a 1.5x Strength bonus to damage and if you activate the Braid, it would become 2x Strength. I don't see anything in the rules for the style and the magic item to suggest that these bonuses would not stack. That said, it appears that not everyone agrees, so I would suggest you discuss it with your GM in advance to be certain that you're on the same page.