Assuming 100bb stacks in a standard single raised pot situation and only 2 options for sizing (33% and 66%) how much equity should our range have before 66% become the more optimal sizing? 60%+? Please explain the thought process behind it

There's no equity threshold for a certain sizing to become preferable because the equity distribution is more important than the simple equity. In BTN vs BB single raised pot the BTN has around 55-58% equity on almost all boards but is using 67% bet size more than 33% even when it has less equity.

On paired boards with two aces, BTN can almost always cbet their whole range for about 33%, while on an AK2 two-tone flop, for example, it's optimal for BTN to bet infrequently for 150-200% pot, even though it has less equity.

In fact, in may be that higher range vs. range equity is correlated with a lower (but more frequent) bet size %.

could you explain why the optimal sizings are different on different textures even if the range equity is similar? Like on ak2 vs aa2 Id assume the range equity is similar right? So why is the sizing larger on ak2? Is it because you have more nutted hands/can make more nutted hands?

At least for the paired boards, more hands completely miss and have to fold to almost any bet. Then on the AAx flops the card removal effect is very strong and because both players have so many Ax, when you're not holding an Ace, the chances that your opponent is goes way up and vice versa when you do hold one. Weaker hands ("bluffs") have less incentive to bet very large because of how inelastic so much of the defender's range is. It's no-limit so you can always bet way over the pot on later streets with the unbeatable hands; you don't have to right on the flop.

Then on the AK2 flop is the classic example of a big nut advantage. In the simulations I used, BB doesn't even defend A2o and 3bets most AQ & AJ, so BTN can go right for stacks with most of its value range and rarely run into better hands, thus the bigger bet.

These are just my speculations and it's hard to say with total certainty everything that's going on on each board that might inform a solution's result.

thanks for explaining. This next part is pretty elementary so sorry for the noob question but on a ak2 type flop where you still have a significant range advantage if you use a larger sizing you gotta be more polarized right, meaning if you completely miss the flop youre better checking those air hands back and stabbing on later streets? As opposed to just betting range?

When betting 100% pot most pure air with no BDFD will check behind along with the vast majority of Kx, most A8-A3, and high-mid pairs. On brick turns it will delay cbet a lot of the weak Ax, some Kx, and most of the lower air, but will still save some air to bluff the river with.

okay thanks, yea thats pretty similar to what I assumed. Thanks for the info appreciate it.

Will do some more theory study next few days so might come back to further discuss certain textures etc if you dont mind tapping the glass a bit more