[afk]

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and we'll get that heart about one in five times. We have an equity in this situation because we have a 1 in five chance of winning the pot and the bet on the turn only represents 1/12 of the pot. Its the same reason why we could pump the flush draw on the flop. Our equity in the situation hasn't decreased enough relative to the size of the pot.
On the flop we stood to win by the river about 1 in 3 times. On the turn, it's now about 1 in 5. The only difference is there's more money in the pot. If no one calls, we take it down UI. If everyone calls we still have a 1 in 5 chance of winning. Not to mention any bets we collect on the river.

Let me put it another way: Are you folding to a turn bet? Are you folding to a turn bet if you're potentially the only caller? If not, why aren't you betting?

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Ok I think there are a few problems in your logic here, I'll try to sort them out.

First off, I think you're saying that putting a bet in on the turn gives you 11:1 odds while your odds of hitting are 4:1. This is wrong. You use pot odds to determine whether you would call a bet or not. For example, on the turn if someone ahead of you bet you'd be getting 12:1 on a 4:1 draw, so of course you call.

The concept of pumping a flush draw focuses on how many callers you'll get, not the size of the pot.

So on the turn, why would you put in one bet (when you most likely won't take it down UI, and you won't get enough callers to make betting your draw profitable) when you could just check and see the river for free? Especially since you're closing the action.

Folding to a bet on the turn is ridiculously unprofitable, but in most cases (including this one), so is betting, though maybe less so. The correct play here is to check. What it comes down to is the fact that betting the turn here is -EV.