BB(3, 4) > Ack(14)

2^60 is very little! Is it known what fraction of them has an insanely large run time?

Not really no. I suggested that you can decouple tenure (1900s) from modern peer review (1970s). Citations aren't an issue when publishing (you can publish anywhere) a result but are maybe more of an issue when you have a collaborative body of work (e.g. closer to open-source software development). But even still you can have citations (e.g. the citation using the Discord channel). For some reason you seemed to take the position that citations are inextractable from peer review. The grant mechanism is definitely a problem because of how it interacts with university funding, but the grant selection mechanism can evolve to be closer to how entrepreneurs work in the business market (which has its own pluses and minuses). What I suggested though is that even if you completely remove the modern peer review system, you'll still be left with the replication crises because. You've seem to have taken issue both with the suggestion that peer review is removable from academia and completely failed to engage with the issues that have nothing to do with peer review.

1. Issues around funding are a higher order problem with peer review only being a symptom at best (if at all). For example, Sabine talks about the issues and focuses on grants and spends 0 time on modern peer review.

2. Fraud didn't come into being because of peer review but grows with funding. The more funding the bigger the problem. Conversely the less funding the more likely proportionally the research is fraudulent or of poor quality because there's a smaller community checking the work. We know that the more funding we spend, the more research activity a field experiences. We don't have good ways to sift out fraud proactively - it takes disproportionately more work to root out fraud and bad science than it is to publish that & reap the rewards. This is true beyond academia - it's easier to spew BS than it is to explain the truth.

3. Not registering for null results has nothing to do with peer review. It's more just "hey I did this work and I'm not going to get rewarded so I'm not going to bother spending the work to publish the null result". That exists independent of the modern peer review system & even publish/perish is ancillary to this - a null result amounts to "failure" emotionally and that can be hard to deal with. That's why there's systems now to mandate pre-registration of the experiment - so that meta analysis can determine whether or not a result has actually been replicated enough to reduce the risk of p-hacking.

4. The replication crises for particle physics is a stark example how peer review is not really contributing as much. There's two schools of thought. The first is that we just follow the math and use data to refine which mathematical model to pick. The second is that we need to come up with better philosophical underpinnings for what the math is telling us. For now the first school is winning in terms of funding dollars (& results), but it's really hard to determine a priori which path is actually the one we should be following. Moreover, the orthodoxy exists independent of the peer review system (& even independent of grant proposals).

In Python:

    def L():
        global index, tape
        if index: index -= 1
        else: tape.insert(0, 0)

    def R():
        global index, tape
        index += 1
        if index >= len(tape): tape.append(0)

    table = {
     ('A', 0): (1, R, 'B'),
     ('A', 1): (3, L, 'B'),
     ('A', 2): (1, R, 'Z'),
     ('A', 3): (2, R, 'A'),
     ('B', 0): (2, L, 'C'),
     ('B', 1): (3, R, 'B'),
     ('B', 2): (1, L, 'C'),
     ('B', 3): (2, R, 'A'),
     ('C', 0): (3, R, 'B'),
     ('C', 1): (1, L, 'B'),
     ('C', 2): (3, L, 'C'),
     ('C', 3): (2, R, 'C'),
     }

    state = 'A'
    tape = [0]
    index = 0
    while state != 'Z':
        tape[index], direction, state = table[state, tape[index]]
        direction()

Can you clarify what you mean by BBλ being "provably optimal"? IIUC BB functions for any Turing-complete computation model should be equivalent up to a constant. Maybe something like: there exists N, c: forall n >= N: BBλ(n) < BB(n + c) and vice-versa. I am not sure what the exact equation will be off-hand, maybe in this case we will need to replace BB() with a version based on binary encoding of TMs.

As the OEIS entry says:

> for any other busy beaver BB based on self-delimiting programs, there is a constant c such that a(c+n) >= BB(n)

This requires an information theoretic measure of program size, such as bits or bytes, whereas the TM-based BB measures in states. I don't believe there are additively tight bounds on how many states are needed to encode an arbitrary string of n bits.