You would also want to consider whether or not the Big Blind is likely to play a check raising range, right?

what is geometrically betsize?

what is the size of the OPEN ,and 3Bet, and the CBET?

Hard to give a sizing bet to use on the turn in your hand description...

Texture and his pre and flop sizing gotta matter a lot here. 1/4-2/3 pot I'd guess, but it will defend on remaining stacks as well.

well we can bet from 1/3 to 2/3 most of the times, depends if we bet on K332r of 7862s

-assuming you balance only 1 bet sizing ott, what is that sizing?  how much does the sizing you choose depend on texture?

While texture will play a factor in determining your bet sizing, it's imperative to consider how both your RANGES relate to the texture of the board. 

as BB's range depolarizes it makes sense to me that we'd build the pot more geometrically from the turn to the river, lining up for stacks with a polarized range.  vs a polarized range it seems better to bet small ott.  

I understand your logic, but once again it depends more on your ranges. For example, assuming that the Villain has a more depolarized range, but if the vast majority of your range can't bet big without making his range too strong, then you're going to need to size your bets down. In other words, if a very large part of your range is semi-strong hands instead of nutted hands than you might need to make your bet smaller than geometric pot size. But you're correct in the sense that if the BBs range is either very strong hands which will check-raise or very weak hands which will fold to a small bet, then betting small is the best size. 

if we also consider BB's defending fqcy ott, i think that if he's making a habit of defending at 1-a he's doing something wrong with his range otf (this probably applies more to boards where he has a significant range advantage).  it seems like that would incentivize IP to use small bet-sizing, since he's in a situation where his bluffs are printing money.  so he would use small sizing at the expense of maybe losing some value by not betting equal fractions ott/otr?

So I think this is where you start to get confused and your analysis falls short. While it's almost certainly true that the BB shouldn't be defending close to 1-a, this doesn't mean that we should be betting small OR that we're "printing money" with our bluffs. I think your confusion stems from thinking that since all the Button's bluffs will be +EV as a bet that they're higher EV than checking. The reason the BB should check-fold at a frequency less than 1-A is that the equilibrium will be found there because the Buttons bluffs will ALSO be +EV as checks. The Button will NOT be betting ALL his bluffs on the turn, he'll be checking back some of them to bluff on the river OR that will get there on the river, etc... therefore they're higher EV as a check. 

None-the-less, when you bet smaller the Villain's frequency of defending on the turn will INCREASE -- not past 1-a, but just because you bet smaller doesn't suddenly mean that you're getting a cheaper price on your bluffs because the Villain will also almost always defend more. If it's GTO correct for you to use a larger bet size, then, by definition, it ALSO must mean that your BLUFFS are HIGHER EV with the larger bet size as well. This doves tail into your comment, 

"so he would use small sizing at the expense of maybe losing some value by not betting equal fractions ott/otr?....I think it's somewhat relatable to what sauce was saying in one of his video threads about the EV gained from using a small c-bet sizing OTF from the more unpolarized hands in the range making up for the EV lost by the more premium hands that are forced to protect the line."

From a GTO standpoint, ALL hands at ALL points must be played in a way that maximizes their EV. In other words, you'd never play a hand "at the expense of losing value" in order to make another hand more EV. If you should use a smaller bet with your nutted hands to protect your marginal hands, it's only because it's the HIGHEST EV line for your nutted hands because your opponent will be raising your marginal hands. 

what i understand about building the pot more geometrically is that it forces villain to felt a slightly wider range than if you bet really small one street and jam large on the next.  

This is absolutely correct. 

-how much does that ^ apply when we move into a model where BB is folding turn too much?

-is the amount that BB usually CF's ott incentivizing us to bet small, since our bluffs are printing money .. as opposed to building the pot geometrically and getting villain to felt slightly wider range in theory (which would give us a little more value?)

As explained above the BB is NOT folding too much when he's defending less than 1-a. 

But I don't think I answered your question. I'll try and do that in a bit. 

On my phone now, but what you should do is come up with ranges for both players on the turn for 2 different board textures -- ranges on the turn should be different on the turn based on flop play.