For the markets to be efficient today they must calculate P=NP to achieve the solution. Nobody has done that yet. Markets today are inefficient as nobody has implemented P=NP to do the required calculation. Markets in the future could be efficient, but today's ones are not as we have to gauge market value the traditional way with incomplete information and guesstimation and modelling.

But the scientific consensus so far is heavily biased towards the presumption that P!=NP.

I think I understand the sketch of the proof and pretty familiar with the P=NP debate. Perhaps my point is pedantic, and I've also looked at the paper and seen that it doesn't try to prove an "iff" claim, but:

To prove an "iff" you have to prove that Markets are efficient -> P = NP and P=NP -> Markets are efficient . I buy the first claim entirely, but not the second.

Let's assume P really does = NP, but as you've said "nobody has done that yet." Then, markets today must be inefficient, but that is T -> F which is impossible if P=NP -> Markets are efficient □

In lay words, how can it be true that `markets are efficient if and only if P=NP`, if it is possible that `P=NP but markets aren't efficient`.

I think we're on the same page, just different paragraphs. I think I get the gist of your line of reasoning.

Let me try an analogy for the if True case. Gravity (P=NP) exists, and always has. But we didn't understand it a few hundreds of years ago. Before we understood it, we didn't have interplanetary rockets (Efficient Markets). But we do now, and they were never outright impossible before our understanding of delta-v and other aspects. But before we had that understanding, we just straight up didn't have rockets.

Meanwhile if hypothetically gravity was, perhaps stronger or more pervasive over longer distances (P!=NP), it could have stopped interplanetary travel altogether and we would have never have achieved it.

We are just in an equivalent time period to that time in the 1600s before we understood, one way or another, what was ahead of us. If markets can be efficient in the future, they could always have been now and in the past. We just didn't have the tools to figure it out.

Analogy seems roughly right, if it was a statement like "P=NP -> Markets could possibly be efficient" then I'd buy it.

But I think what's more interesting is that

"Markets (present-day) are efficient -> P=NP" could plausibly be true, suggesting that the contrapositive is true: "P!=NP -> Markets (present-day) aren't efficient." Given current state of thought around plausibility of P=NP, that seems pretty powerful nonetheless.

With the slightly stronger statement "Nobody currently possess a polynomial time algorithm to solve a problem in NP -> Markets (present-day) aren't efficient" it starts to become quite likely markets aren't fully efficient.