First of all this paper leaves a lot to be desired in terms of rigor and it is for good reason not how mathematics is written by most mathematicians. It is very conversational and assumptions and derivations are jumbled together, while you sometimes find these in breakthrough papers from visionary mathematicians it often makes it harder to verify.

My understanding of the central argument in the paper is the following:

Definition: Let N be a positive integer denoting the length of the history, M be a positive integer denoting the number of assets. A market realization is an element of {-1, 1}^(NxM), i.e. M vectors of length n where all entries are either +1 or -1.

Definition: We call a function f: {-1, 1}^(TxM) -> {0, 1}^M satisfying f \circ s = s \circ f for s any element in the symmetric group S_M a technical strategy of lookback T.
The symmetric group condition just states that permuting the vectors of length T among the M assets is the same as permuting the output the strategy. I.e. that the strategy has no inherent preferences among the assets.

The payoff of a technical strategy s on a market realization h is given by payoff(s, h) = \sum_{i=T}^N s(h_{i-T, ..., i-1}) \cdot h_i where the indexation is in the time dimension i.e. h_i denotes a length M vector.
The budget of a technical strategy is budget(s) = max_{v \in {-1, 1}^TxM} s(v) \cdot (1, ..., 1). That is the maximal number of assets it wants to hold in any given state of the world.

Given a market realization h, positive integers B and K we say that h is (B, K) EMH-inconsistent if there exists a technical strategy s such that budget(s) <= B and payoff(s, h) >= K. If a market realization h is not (B, K) EMH-inconsistent we call it (B, K) EMH-consistent.

Claim (presented as a theorem in the paper): The problem of determining whether a market realization is (B, K) EMH-consistent is in P if and only if the knapsack problem is in P.

Claim: The weak efficient market hypothesis is true if and only if EMH-consistency is in P.

In the second part of the paper he indicates a model of an order book where he wants to encode 3-SAT as combinations of market orders. I do not understand how this is intended to work, i.e. if all information is available and incorporated into the market and the information generating process is stopped, and I have bid-offer spreads because of transaction costs, and I irrationally (remember I am not interested in the buying or selling the security, I am just interested in solving a 3-SAT problem, thus my actions should not influence the price generation process of an efficient market) enter an OCO-3 order to buy A, B or C at mid. Why should this result in a transaction? In the case (a or a or a) and (!a or !a or !a) I make one trade with myself in the case (a or a or a) and (b or b or b) and (!a or b or b) I make one trade with myself, but one of the problems is satisfiable the other is not. Now it seems obvious that by inventing new order types we can get order book rules that allow for complex computation to resolve clearing, however this is a problem with the proposed order types not the efficient market hypothesis? A (to me) equivalent avenue of investigation would be to imagine different order types such that to decide the clearing of an order book it would involve solving an undecidable problem - i.e. what are the most reasonable order types and order book rules such that we can encode the halting problem?