Okay, maybe we can use a concrete example to see if this makes sense:

You have a beamsplitter. You shoot photons in the bottom, you see them come out the top and right. You shoot photons in the left and see the same thing. You conclude that, from an MWI perspective, the beamsplitter acts in either case to split the total number of branches in the wavefunction evenly.

Now you send in photons to both the bottom and left simultaneously, in an even superposition. You observe that they only exit the top. So now your branch counting procedure would imply that the beamsplitter is fusing the two branches together, instead of splitting them.

So, it would appear that you need some rule to determine whether the beamsplitter is going to divide the branches or join them. If you do more experiments, you’ll find that this rule has to be internal to the branches themselves, instead of the beamsplitter. Furthermore, you’d see that in general when you send in the even superposition you could have the beamsplitter always merge the branches up, or always merge them to the right, or any probability between these.

So now your theory has a number of branches for each outcome, and each branch also some number on it that, it turns out, is most sensibly written as an angle between 0 and 2 pi.

As a second, and perhaps lesser concern, you’d also find that states didn’t necessarily have a unique label. For example, let’s say you are measuring the polarization of some light, and you find it to be 50% (Horizontal) + 50% (Vertical, 0).

I’m using a notation I just made up now that is tailored to an MWI perspective. The percentage is the number of branches of each outcome. (Horizontal) or (Vertical) is the outcome of the measurement itself, and the (,0) represents the angle between the two parts that determines, using some prescribed rules, how the branches should merge or not merge when combined in a beamsplitter-like operation (it is always implied that the first branch has the angle 0).

Anyway, the point is that the photons that are identified as 50% (Horizontal) + 50% (Vertical, 0) would, in another experiment, be measured as 100% (Diagonal).

So I guess the question is whether in your opinion this counts as “naive branch counting leading to the Born rule,” or whether the necessity of this funny angle and the rules associated with it, as well as the multiple possible labels for the same state, spoil it for you.

My immediate reaction is that you are not doing any time evolution that would require you to connect amplitudes to the probabilities. I agree that in principle you could have an S-matrix like program that tried to avoid the wave function entirely, in which case, yes, you would just assign this probabilities to what you are calling "branches." But these aren't the same "branches" I'm talking about, which are superposed wave functions.

So let me step back for a second to explain what I mean. It turns out that we can describe the time evolution of QM states using a wave equation (for simplicity we can just use the x-basis, especially considering the basis issue you raised above). Now the basic problem regarding the Born rule is that in order for this wave equation to work, the amplitudes do not correspond to probabilities. For that you have to use the Born rule. In your above example you are not confronting this central issue,

, because you are not at all using Schrodinger evolution, in which case, indeed, you can describe everything using probabilities. This gets at the central philosophical ontology of the MWI, which is that there is this thing called the wave function that evolves in time via Schrodinger, that underlies all of the probabilities. This wave function has an amplitude that you would like to associate with branch density.

So I think a better example than the one you gave would accept the wave picture as a starting point, take a wave packet and evolve it into 2 or 3 wave packets with unequal amplitude, and show that if you treat ratios of amplitudes as ratios of "worlds" then you get the Born rule. The issue I'm getting at is that detractors say this can't work. If it did, then a 1:2 amplitude would correspond to a 1:2 probability, but it doesn't.