The question is how common or rare are advantageous functional sequences among the possible combinations of the DNA code? The answer is 1x10^77. That’s a 10 with 77 zeros. Those odds are so big that even billions of years could not produce that many outcomes.
I assume you meant to write 1/10^77, in order that we're actually talking about a probability or a frequency.
It's important to consider what this number actually represents. If this is the proportion of combinations of DNA that give rise to functional proteins; then it's not a probability. It would be a probability if we were taking random stretches of DNA and seeing if they coded for functional proteins, but where does that happen?
That's not what it's supposed to be, of course. What Axe was trying to estimate was the likelihood of a protein sequence (not DNA) performing a specific function. This is an important difference, because you seem to be trying to use it to imply evolution could never happen because useful proteins are too unlikely to form. But even if Axe's calculation is correct (and 1/10^77 is his lower estimate) it only applies to one specific function - is that specific function necessary for life? If not, then why does this matter?
You mentioned another study that claimed 1/10^33 proteins had a useful function. No idea what study that was, but note that this is a difference of 43 orders of magnitude. It's still a very small number, yes, but does it not make you think there might be different things being measured here? If the same thing is being measured, then I'd question whether we know how to measure it.
Here's another study for you. They created 6,000 million random proteins of 80 amino acids in length. 4 of these proteins bound to ATP. That's 1/10^9 - 68 orders or magnitude different from Axe's estimate. Binding to ATP is, of course, not an essential requirement of a functional protein in any abstract sense - it's just a criterion that makes sense to look for in real organisms.
None of these numbers are much use unless you know what they mean.
So were you planning on following up on anything about the numbers posted in the OP?
According to Karl Popper ToE is a pseudoscience.
Karl Popper said the Theory of Evolution was not only a scientific theory but also "the most successful explanation of the relevant facts"; but what does he have to do with the topic anyway? Poor man's been dead quarter of a century - leave him alone.
I understand you may be busy; but you just managed to write 4 posts and well over a thousand words without mentioning anything from the opening post or any of the responses to it.
But note your numbers for this positive event. Even if we assume (as your example does) that the fixation is 100% guaranteed to spread given the appearance of the 2 beneficial mutations, you're showing mutation rates of 10000% (1x10^2) and 500% (5x10^1). That is nowhere near reality. For bacteria, the mutation rate is around 1x10^-8, and probably an order of magnitude less for eukaryotes. So ya, with a mutation rate 10-100 billion times more than normal, you're virtually guaranteed to see your 2-mutations-to-increased-fertility appear with a very modest population.
These numbers don't make any sense, but what I think you're doing is confusing an estimate of the average mutation rate per nucleotide with that per individual. A mutation rate of 1x10^-8 would mean about 0.5 mutations per individual with a genome the size of the malaria parasite's. You're many orders of magnitude off.
Why are you arbitrarily singling out living things? Applying design principles is agnostic with regards to the materials it works on. You can detect design when looking at certain rock formations. You can detect design when looking at certain magnetized deposits. You can detect design when looking at some formations of wood, flowers, or other organic products. You can detect design among elecromagnetic or light signals. There is no limitation on what you can consider for design. Why would living organisms be an exception?
The techiques are used in other sciences that I already mentioned. Archaelogy, forensics, cryptography. SETI is good example of looking for design signs of something that clearly would not be from humans. Why would one have to differentiate design from one agent (known) or another agent (unknown) or presumably evolution. Design is design.
How does this process work, exactly? I'm not familiar with the techniques for detecting design in cryptography or anything - of the topics you listed the only one I have read something on was archaeology. When archaeologists are trying to determine whether some objects were designed or not, at least in the examples I know the techniques rely fundamentally on knowing who the designers are and how they build things.
For example, the Lomekwi stone tools. These are purported tools dating all the way back to 3.3 million years ago, and whenever someone presents evidence of human-made tools this old, the argument is about how do we know these are actually tools, and not just some similar looking pieces of rock fractured entirely naturally?
To figure this out, archaeologists look at what rocks that have fractured naturally look like, and they look at what rocks that are intentionally broken by humans to make tools look like; and see which one the purported tools look most similar to. This can be a bit more sophisticated than simply looking - it can involve statistical analyses of the distribution of the shapes and size of stones in a particular sample, or of the prevalence of certain types of fracture. But it all depends fundamentally on the pre-existing knowledge of what human-made stone tools look like and what agglomerations of rocks gathered by other processes look like.
You seem to be saying archaeologists have a separate technique for figuring this out that would be independent of such knowledge. What is it, and why don't archaeologists use it to settle the interminable arguments over various proposed 'earliest' tools.
For more details, I went reading some about how SETI is searching for radio signals; and how they know what to look for - what would be a radio signal produced by an alien civilisation
They're searching for very narrow band frequencies. They reckon a signal with a bandwidth of less than about 300 Hz would probably be from an alien civilisation. Why? Because we know of no natural source that produces such narrow bandwidths; but we do it all the time with communication signals.
I don't see any sophisticated technique to detect design here. They're just asking 'what's the difference between a radio signal produced by humans and a radio signal produced by known natural sources?'. Just as the archaeologists are asking 'what's the difference between rocks intentionally shaped by humans and rocks shaped by natural processes?'
I'm not seeing how this is applicable to organisms yet.