Hmm this is really confusing me. I understand why you say "two mistakes instead of one!" is specious, but your interpretation of this comment is different from mine. I was thinking that the value of the mistakes of incorrectly folding to 34 was the same as the value of the mistakes of incorrectly folding to AJ, so if mistake "A" for 34=mistake "A' " for AJ, then any additional mistakes (incorrectly calling) for AJ would necessarily create more EV. Under this condition, it's not specious to say two mistakes are better than one. Now I see that 34 can create more mistakes as a bluff than AJ, so the EV for the play as a bluff is not equal. so the value of that portion of the play for AJ, 34 is not equal.

However...you can't possibly lose more money by going all-in with AJ here than by going all-in with 34. UNLESS you forego value that could be obtained with a smaller value bet. But let's think about this. How could value betting be profitable? Certainly we cannot bet another 250$ and expect this to be called by hands worse than AJ that would not call a push. In fact, we are more likely to get a push called here by a worse hand than a 250$-300$ value bet. If we go much lower, calling becomes a more attractive option to the opponent holding eights, tens, or KJ. BUT, raising as a bluff becomes even more attractive-and Lukie would not possibly be able to call this bluff the way the hand played out. It cannot mathematically be proven because we don't know villains exact propensities on the river facing any given bet, but its pretty certain to me at least that there is no amount between 0 and all in that could be more profitable than both.

But you'll say that it's still a mathematical possibility for 34 to be creating more mistakes than AJ, such as when his calling range is QQ or better. That is true, sir. But I hope you see that this is not correct:
I would rather do this with a hand that has virtually no potential to win money postflop and / or be the best hand. At least that way I'm sure what I'm trying to accomplish.

Talk about simplistic and specious! Just because your bluff with this hand may POSSIBLY be creating more mistakes does not mean you should rather have it here. You NEED to create more mistakes to approach the same EV because your hand is of a lesser value.

The way the hand played out, and assuming opp isnt bad, Lukie's best river bet must be either 0 (accompanied by a fold to a significant bet) or all in. This is also true of 34. These facts by themselves make it blatantly obvious that you should rather have AJ than 34 here.