Are you defining the function of A + B here? Can we call this C that meets HS 
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The function of A+B is:
Either, merely function A and B together.
Or, function A and B together, with some new function C that arises from those two.
It does not really matter though. If you have A and you have B, by definition of function C, you have C.
I think that works The function A+B=C. So for HS you are defining function C and the bits that make up A and B. How those bits came about is a separate issue.
Waiting time being improved by adding resources is certainly true. I think @gpuccio would agree with you and Neil.
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The safes example avoids those contingencies, and yes there are contingencies, but contingency is also opportunity for change. It the thief gets stuck on one lock, perhaps another will open.
Itâs not a bit like changing combinations. The combinations never change. The amount of money in each safe can change (biological reward), and new sets of safes may appear (or be removed) by contingency, but the sequence to unlock each safe should never change. Changing the sequence would be equivalent to changing the physical laws that govern chemistry - those do not change.
The thief might forget a combination, or forget to pass the secret on to his grandson (loss of function), but the biological function the combination represents in this example cannot change.
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You are taking a polemic approach, not a conservative one.
You have no basis for claiming that this is about a single mutation. Youâre ignoring the number of different clones (and the number of different branches of the same clone) in which this is evolution is happening simultaneously in a typical vertebrate.
If you were being conservative and not polemic, you would have included factors for both of these well-documented realities. We have sequencing data. Why donât you look at some before taking a position?
You are making the false assumption that there is only a single cell division. We know that the average clone (germinal center) is about 10000 cells and the doubling time is about 6 hours.
If you were looking for the reality, you would consider the number of cell divisions, but that would give an FIb far too high for your desired conclusion.
There is no reason to pretend that only one position is involved, as mutations in many different positions have been shown to increase affinity. People have done microsequencing of immunoglobulin genes from microsamples taken from different parts of single germinal centers. Tens of thousands of monoclonal antibodies have been made. The data from those studies show your assumption to be false.
Since youâve ignored many factors that significantly increase the probability, your estimated probability is wrong.
The bottom line is that your estimate of the amount of new FIb produced (you still havenât estimated FIa) should be far higher than your ridiculously low polemic estimate.
Since you offered no reasons for doing so, it appears all but certain that you have revised it in hindsight because you have to lower it to preserve @gpuccioâs claim. As shown above, your estimate has little basis in reality. Youâve ignored multiple factors that increase both the information content and the probability of mutations that increase it.
I also havenât seen an estimate of FIa. How come?
We know a lot about why that is and you are ignoring the majority of what we know. We know enough to know that your estimate is wishful thinking and that @gpuccioâs claim is false.
I donât see how a system shared by virtually all vertebrates that functions constantly throughout life can be called âspecial.â @gpuccio was the one who specified vertebrates, wasnât he?
Your estimate does not address what we know about the process. You just made it up.
Since @gpuccio specified vertebrates and this is a universal vertebrate system that functions constantly, thereâs no reason, other than a need to avoid science and engage in polemics, to call something like that âspecialâ or âquite rare.â
In fact, it should be one of the most obvious and first systems one should study if looking at the biological creation of functional information, quadruply obvious for one trained as a physician.
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Iâm still missing an explanation for why, if a handful of mutations can fix incrementally when antibodies are evolved to target a particular antigen, why could more mutations not fix over even longer timescales and the FI increase further then? I donât understand what this mysterious barrier to reach 500 bits of FI is supposed to be.
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You are confused here. I am not at all making the false assumption that there is only a single cell division. I know perfectly well that there are many activated B cells that undergo many cell divisions. But these factors have nothing to do with FIb; rather, they pertain to the probabilistic resources of the system. And it is these probabilistic resources that explain why the system is able to find the target (an antibody with higher affinity) so easily.
Now, if you are interested, I can give you a more thorough account of how I arrived at this value of 50 bits for FIb.
I suspect itâs the same as the mysterious barrier Creationists claim makes it impossible for micro-evolution changes to accumulate over time into macro-evolution. 
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I donât think so. Your probability calculation assumed a single division and included no factors representing the numbers of cell divisions. That means that the number you calculated, if everything else in it was right (it wasnât), would only represent the probability for a single cell division and therefore useless.
That makes no sense.
That would be fun. Whereâs your estimate for FIa?
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You are perfectly right here, and this is precisely why I said I have used a conservative approach in my first attempt at computing FIb. Because if, as it is indeed the case, different mutations at different positions can increase affinity, or, put differently, if the target space increases, then FIb would decrease. Hence my use of the word « conservative ». There is nothing polemical here!
The reason I have revised my estimation of FIb is precisely to take into account that more than one mutation can increase affinity. Are you going to blame me for trying to be more accurate in my estimation of FIb?
It makes no sense for you because you donât understand the basic concepts used by IDers. Let me try to explain. As @gpuccio defined it, FI is a measure of the improbability of finding the target space in one random event. The important point here is the expression « in one random event ». This is why when computing FIb, you have first to compute the probability that a single B cell will produce a receptor with a higher affinity through a single cell division. I have estimated this probability to be 1/1000, which corresponds to a FI of 10 bits. If, in order to produce this amount of FI, the immune system had to rely on only one B cell division, ie., was only allowed to perform a single try, the somatic hypermutation (SHM) process would not work. But why is it that it works? The answer is that during the SHM process, many different B cells are allowed to undergo many cell divisions, which considerably increase the probability of the system to find the target. IOW, the system works because it allows a large number of try.
Or, put differently, the system works because its probabilistic resources are important.
Let me now give you a non biological example of these concepts.
Imagine you have a random letter generator set to output a sequence of 3 letters.
And imagine you want it to produce the word « cat ».
In order to compute the FI associated to the word « cat » in this situation, you will have to calculate the probability to produce the word « cat » through a SINGLE try. This probability is equal to 1/26^3, that is 1/17576, which corresponds to an FI of 14 bits.
Now, if you are allowed to play that game not once but many times (if the probabilistic resources at hand increase) you will obviously enhance your chance to find the target. IOW, the more the probabilistic resources, the more the chance of finding the target. But the important point to see here is that the probabilistic resources have nothing to do whatsoever with the FI. The FI associated to the word « cat » in this example remains the same (14 bits) whatever the number of try. I hope things are clearer for you now.
So why canât a complex protein evolve?
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Ok. Below is a more complete account of how I have tried to assess FIb.
When a foreign antigen enter the body, the first phase of the humoral response is the recognition of the antigen by some B cells patrolling the human body that happen to express a receptor with, most often, a weak affinity for the antigen. The second phase occurs when these activated B cells migrate into follicles and proliferate to form germinal centers where they undergo V-region somatic hypermutations (SHM) that improve affinity for the antigen. Since several rounds of selection occurs during this SHM phase, the FI produced at each round must be estimated in order to calculate the FI produced by the whole SHM process.
So letâs look first at the new FI produced during the first round of selection. Here, we first have to define a function and then calculate the probability for the organism to find this function. The function for the activated B cells can be defined as follow: « bind the antigen with higher affinity ». Letâs call P the probability to find that function.
I have first taken a conservative approach and assumed that in order to produce the function, a specific mutation (a specific aa change) must occur at a specific position in the V region of a B cell receptor within a single activated B cell. Letâs call L this specific location and R this specific mutation. Now letâs define the following 3 probabilities P1, P2 and P3, with:
P1: the probability that a mutation occurs in the V region of a B cell receptor during a cell division. (Iâve found in my immunology textbook that P1 is close to Âœ)
P2: the probability that the above mutation occurs at position L in the V region (given that the V region of immunoglobulins is 120aa long, we have P2=1/120)
P3: the probability that the above mutation that occurred at position L corresponds to R (given that there are 20 different aa, and assuming for simplicity that each aa change is equiprobable, we have P3=1/20)
Now given that P=P1xP2xP3, we have P=1/4800.
But this is undoubtedly an underestimation of P because in my calculation above, I have wrongly assumed that the function could only be achieved by a single specific mutation occurring at a specific position of the V region. IOW, I have assumed that the target space was equal to 1, which is obviously not the case. I donât know the size of the target space here, but my educated guess would be around 5. In that case, we arrive at P=1/1000. Thus, the FI produced during the first round of selection of the SHM phase is equal to 10 bits.
Now, in order to calculate FIb, that is the total amount of FI produced during the SHM process, we must know what is the number of selection rounds that occurs during that process. Here again, I donât know the answer, but a mean of 5 would be my educated guess. In that case, assuming that the same level of FI is produced at each round, we arrived at FIb=10x5, ie., 50 bits.
I am well aware of at least some of the limitations and shortcuts in my calculation of FIb but I think it is not too far from the truth.
If someone here is willing to offer another better estimation, that would be great.
Explain to us again why this is relevant since no one in science says or thinks proteins arose through one random event?
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This all makes sense to me, the problem is I donât see how this could in any way be taken to imply there is some sort of barrier to evolving any known protein.
What is missing from Gpuccio, or you, or anyone, is some sort of argument that connects this calculation of FI to a known protein sequence, which concludes that therefore this known protein sequence could not have evolved(or is at least extremely implausible to have evolved), so must have been designed instead.
What we need is the demonstration that there are proteins that actually have so many bits of FI that their evolution becomes implausible.
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Letâs start from the position that his calculation may have error due to not measuring all possibilities. It got stuck in some optimal position through mutation and selection or drift. The issue is still real for long proteins as even 50% substitutability for prp8 would yield 2335 bits.
This problem is so large when you take into account all the machines in the eukaryotic cell that require coordinated function the implausibility by mutation and selection is an accurate description.
You can always imagine a way it could happen but if we look at what the evidence is showing us there is almost no way to make evolutionary calculations work.
Show the calculation then.
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50% substitutability for prp8 is 2335 bits since 10 different amino acids work in every position.
Try to find a condition where prp8 can evolve given human and mouse are 99% similar. You will see that preservation is a very relevant observation.
The function to sequence space window for evolution to work is minuscule. If you can find it now for prp8 you need the coordinated binding and function of another 200 proteins.
Why is that a problem?
The problem is that there is a minuscule window for evolution to work and gpuccio preservation data is showing that window is exceedingly unlikely. When the dust settles you will realize the only known generator of 2335 functional linear bits is a mind.
Out for the day. Take care.