## The Grothendieck group construction

After "snowzilla" disrupted my priorities for about a week I am falling behind in my duties and today I only have time to present a small but essential topic. So let's talk about the Grothendick group construction which appears front and center in the categorical solution to the measurement problem.

 Alexander Grothendieck

There is no easier way to show this then to see it at work in the case of the integer numbers. Basically we introduce a Cartesian pairs of natural numbers and we call the left element positive integers, and the right element negative integers like this:

$$7\equiv (7,0)$$

$$-3\equiv (0,3)$$

Then we can add and subtract the numbers in the most natural way: $$7-3=7+(-3) = (7,0)+(0,3) = (7,3)$$

But this is not what we want: we want 7-3 to be 4 and not (7,3). How to do that? 4 is (4,0) and there is no problem if (7,3) is identical with (4,0). In other words, we need to introduce an equivalence relationship where distinct pairs represent the same thing.

How can we justify (7,3)=(4,0)? One way is to observe that if we subtract the right pair from the left one we get identical answers in both slots: (7-4, 3-0) = (3,3) but we are not allowed to use subtraction because we need to rely only on the operation available from the original commutative monoid. The answer is that we need to add the elements like this:

7+0 = 3+4

In other words, the equivalence relation we seek is as follows:

The pairs a-b and c-d are equivalent $$(a,b)\sim (c,d)$$

if $$a+d = b+c$$. Please notice the outer-inner pattern.

But is this an equivalence relation? To prove that we need to show three properties:

• relexivity
• symmetry
• transitivity
Is this reflexive: $$(a,b)\sim (a,b)$$? Indeed it is because: a+b (from outer) is the same as b+a from inner.

Is this symmetric: if $$(a,b)\sim (c,d)$$ is $$(c,d)\sim (a,b)$$? Trivial.

Is this transitive: if $$(a,b)\sim (c,d)$$ and $$(c,d)\sim (e,f)$$ do we have: $$(a,b)\sim (e,f)$$?

Let's see. We have: a+d=b+c and c+f=d+e. Can we prove a+f=b+e?

If we add the first two we get: a+d+c+f = b+c+d+e or a+f + k = b+e+k where k=d+c and we need to make a tiny generalization:

$$(a,b)\sim (c,d)$$ if and only if there is a k such that a+d+k = b+c+k.

So what does this have to do with quantum mechanics? It will turn out that those a,b,c,d numbers will be the dimensions of the Hilbert spaces involved in the measurement problem. Also if in the case of integers we have the Cartesian pair: (positive number, negative number) in the case of quantum mechanics we have the Cartesian pair: (Quantum system, Observer).

No more hokey pokey endless philosophical debates about the role of the observer in quantum mechanics which devolve into arguments about consciousness, but a sharp (and unique, and natural) mathematical construction which will allow us to bring about rigorous proofs. Please stay tuned.

## What is the relationship between unitary evolution and collapse in quantum mechanics?

Last time we started discussing the measurement problem. In quantum mechanics textbooks, a quantum system is presented as undergoing two distinct time evolution:
• a unitary time evolution before measurement
• a sudden change known as the collapse of the wavefunction
The epistemic explanation of it is very simple: the collapse corresponds to information update. QBism has self consistent explanations to all quantum puzzles, so why bother seek a different solution to the measurement problem? I mean besides trying to erase from your memory the creepy picture Chris Fuchs is using in his talks about qbism:

For some reason this picture of an one-eye man with measurement dials instead of hands is like a bad song you cannot get out your brain.

The answer is that any non-unitary time evolution is fatal to quantum mechanics. Here is why:

Unitarity in the state space formulation of quantum mechanics, just like the Leibniz identity in the algebraic formalism are a consequence of the invariance of the laws of nature under time evolution and moreover each one can be derived from the otherI have showed in prior posts how to reconstruct quantum mechanics using Leibniz identity. What reconstruction of quantum mechanics shows is that breaking the Leibniz identity makes the whole quantum formalism inconsistent: no more Hilbert spaces or Hermitean operators before or after measurement.

So for pure mathematical consistency arguments, can we describe the (non-unitary) update of information using only unitary time evolution? Yes we can, and the answer is that the non-unitary collapse is nothing but a change in the GNS representation, but to show it rigorously I need to build up the required machinery.

Last time I made a strong claim: the transformation

$$(\lambda |\psi_A \rangle + \mu |\psi_B \rangle )\otimes |M_0 \rangle\ \rightarrow \lambda |\psi_A \rangle \otimes |M_A \rangle + \mu |\psi_B \rangle \otimes |M_B \rangle$$

is not correct. In foundations of quantum mechanics the justification for the transformation above stems in part from the following argument: quantum mechanics is universal, and the measurement process should be described quantum mechanically in a Hilbert space. Asher Peres had a rebuttal to this but that was only handwaving inspired by Godel incompletness theorem. This rigorous  reason the argument is faulty is because it turns out that there are many Hilbert spaces involved. I will show that $$|\psi_A \rangle \otimes |M_0 \rangle\ \rightarrow |\psi_A \rangle \otimes |M_A \rangle$$ and $$|\psi_B \rangle \otimes |M_0 \rangle\ \rightarrow |\psi_B \rangle \otimes |M_B \rangle$$ should not be understood as unitary evolution, but as a change in representation. Since changes in representation do not happen in a Hilbert space, there is no superposition and they cannot be combined.

Interestingly enough, the relationship between unitary evolution and collapse is deeply related with the relationship between the addition and subtraction in the case of integer numbers. There is a category construction called the Grothedieck group construction which is at play in both cases. The Grothendieck group construction is an universal property, meaning it is both unique and natural.

For now I only want to point a problem in the easy integer numbers setting. When you learn arithmetic in elementary school , first you learn to add and subtract. Then you learn about multiplication and division. Fast forward to say your high school or early undergraduate years, you learn to formalize those operations in the concepts of groups, rings, and fields. The natural numbers for example are an abelian monoid. Going from monoids to groups one needs to add the inverse elements. Natural numbers N become the integers Z. And in the integers abelian group Z you have two operations: $$+$$ and $$-$$. But wait a minute, a group has only one operation! So what does the subtraction operation mean? Do we have two independent operations? In a field there are two independent operations, addition and multiplication, but not in a group.

Starting from this trivial observation that subtraction must be not independent from addition, the problem is how to express one in terms of the other. The first try is to say: no problem at all, subtraction is addition by negative elements. I bet you already heard that in school. But this is mathematically sloppy and unacceptable answer because the definition of an operation should be decoupled from the nature of the elements it works with. It turns out that there is a unique way one can turn an abelian monoid into an abelian group, and this was figured out by Mr. Grothendieck.

So what does this have to do with quantum mechanics? Do you know a natural abelian monoid there? When we compose quantum systems we use the tensor product: $$\otimes$$. This is associative: it does not matter the order in which we compose Hilbert spaces. It has a unit element: compose with nothing. It is also commutative. We will see that the inverse operation $${\otimes}^{-1}$$ is deeply related with the collapse postulate, but it is not a straightforward relationship. Please stay tuned.

## What is the measurement problem?

After discussing the category theory approach of quantum mechanics reconstruction I want to start a series of posts discussing the so-called measurement problem. Is the wavefunction collapse real? The views on this topics varies wildly and touches a lot of raw nerves because it goes to the heart of quantum mechanics interpretation

Outside the foundations community, one prevalent attitude is that I know very well how to use quantum mechanics in my day to day computations which keep me very busy, and if I only have one free afternoon to think about it I will surely solve it. But this is actually a hard problem, and moreover I will attempt to show that all known solutions are incorrect/incomplete in one form or another. Because the category theory approach was successful in deriving quantum mechanics from physical principles, it is natural to expect that it also offers hints on solving the measurement problem. And the solution turns out to be completely unexpected, an entirely new paradigm.

To set the stage, I can argue that the best solution so far to the measurement problem is offered by QBism. This is not without issues however (but not what people usually use against the epistemic interpretation), and I will attempt to make the epistemic interpretation mathematically rigorous. As an analogy consider the usage of $$ict$$ in special relativity. Time is not an imaginary distance and the proper way to understand relativity is by using the metric tensor. Similarly in quantum mechanics if we are sloppy and ignore a mathematical construct which naturally appears in category theory, we can talk about collapse.

The basic argument used to present the measurement problem is as follows:

Suppose I have a wavefunction $$|\psi \rangle$$ and a measurement device $$|M\rangle$$. To simplify the argument we can consider that there are  only two measurement outcomes: $$|\psi_A \rangle$$ and $$|\psi_B \rangle$$.  If the ready state of the measurement device is $$|M_0 \rangle$$, the pointer state for the $$A$$ outcome is $$|M_A \rangle$$, and the pointer state for the $$B$$ outcome is $$|M_B \rangle$$, from the fact that a repeated experiment confirms the prior value we have:

$$|\psi_A \rangle \otimes |M_0 \rangle\ \rightarrow |\psi_A \rangle \otimes |M_A \rangle$$
$$|\psi_B \rangle \otimes |M_0 \rangle\ \rightarrow |\psi_B \rangle \otimes |M_B \rangle$$

Now by superposition:

$$(\lambda |\psi_A \rangle + \mu |\psi_B \rangle )\otimes |M_0 \rangle\ \rightarrow \lambda |\psi_A \rangle \otimes |M_A \rangle + \mu |\psi_B \rangle \otimes |M_B \rangle$$

This is interpreted differently by various quantum mechanics interpretations.

• In many worlds, the world splits in two branches, and in each branch we have an outcome.
• In Bohmian, we add a hidden variable, the position, and different initial conditions lead to the A or B outcomes.
• In GRW the wavefunction spontaneously collapses to the A or B outcomes
• In Copenhagen, the wavefunction only predicts potential outcomes and the collapse is only an information update. One criticism people level on this is the Wigner's friend problem. (QBism has a good answer to this criticism).
Sometimes the measurement problem is presented as a trilema: any 2 of the following 3 statements contradicts the other one:

S1: Quantum mechanics is complete
S2: Quantum mechanics predicts one outcome
S3: Quantum mechanics evolves linearly according to Schrodinger's equation

Bohmian violates S1 and respects S2 and S3.
MWI violates S2 and respects S1 and S3
GRW violates S3 and respects S1 and S2

What I will attempt to show in the next posts is that the trilema is false: quantum mechanics obeys all 3 properties: S1, S2, S3. The argument:

$$(\lambda |\psi_A \rangle + \mu |\psi_B \rangle )\otimes |M_0 \rangle\ \rightarrow \lambda |\psi_A \rangle \otimes |M_A \rangle + \mu |\psi_B \rangle \otimes |M_B \rangle$$

will turn out to be bogus.

There is only one outcome from any experiment which occurs when the quantum system interacts with the measurement device, the Wigner's friend and quantum eraser has natural explanations, and we can talk about collapse when we sloppily ignore a mathematical structure. The epistemic information update will be rigurously described in a precise mathematical way. Please stay tuned.

## Musings over algebra and topology

One open problem in quantum mechanics reconstruction is the complete classifications of the realization of the algebraic properties. We know quantum mechanics can be formulated over the reals, complex, or quaternionic numbers, but is this all there can be? The problem is solved in the finite dimensional case, but is open in the infinite case. But why is this a hard problem? I think a recent unrelated cute recreational math video shows the heart of the mater. The video attempts to prove that:

1+2+4+8+16+... = -1

This is not the same as 1+2+3+...= -1/12 The rigorous treatment of the Riemann zeta function cannot be covered in only one post (you have to go past the usual Ramanujan tricks), but the current case is much simpler.

Formally, the algebraic manipulations are trivial:

if $$S = 1+q+q^2 + \cdots$$ then $$Sq = q+q^2 + q^3 + \cdots = S-1$$ and so $$S = 1/(1-q)$$ which for q=2 results in -1.

But does it make sense to have this kind of algebraic manipulation? We learn in school that this is allowed only for convergent series which means that $$q\in (-1, 1)$$ and 2 is outside the radius of convergence. Case closed, right? Wrong!

We have do dig deeper into what it means that an algebraic manipulation makes sense. The easiest thing to do is to consider the existence of a metric, which is a positive function which assign a number between any two points subject to the usual properties. The most important property of a metric is the triangle inequality and once we have it we can have the usual epsilon-delta arguments.

If you study functional analysis most of the concepts come from considering a metric. Take for example the notions of continuity or that of compactness. A function is continuous if when we approach a point from both ends the value of the function converges. Also a set is compact if it is bounded and closed. One difficulty in learning topology is in generalizing those common sense ideas and using only open sets. For example a function is continuous if and only if it returns open sets into open sets, and a set is compact if from any covering with open sets we can extract a finite covering.

But once we freed ourselves from the usual metric intuition we can see and appreciate things in a different light. For the problem above the key idea is to reorder the numbers to create a different topology and a different metric where the sum does converge. The trivial algebraic manipulation suggests how to do it and one arrives at the p-adic numbers.

So it looks that there is flexibility in messing with ordering and neighborhoods to satisfy algebraic identities. But how much flexibility is allowed by nature in the case of quantum mechanics?

First, is there a p-adic quantum mechanics? Some publications claim there is, but they are all nonsense. p-adic numbers violate the so-called Archimedean property. While it is conceivable to mathematically imagine universes where probability predictions violate the Archimedean property, a non-Archimedean quantum mechanics must violate the Archimedean property for the Jordan algebra as well and this is where you get in trouble from the physical point of view.

p-adic quantum mechanics is not physical, but can we twist the order of the real numbers in a different way which respects the Archimedean property and yet we get a district topology? To me this looks highly unlikely but I do not have a proof for this impossibility. I did not even began to scratch the surface of topology and algebra in this post, but I hope I succeeded in highlighting the main issue: topology is not as rigid as naive metric epsilon-delta functional analysis proofs from college would made us believe. Categorical arguments nail the algebraic structure of quantum mechanics, but they have nothing to offer on the topological side.

## Happy New Year!

Wishing everyone a Happy New Year! Let 2016 bring you hope, happiness, and prosperity.

No physics post today, only a brainteaser in honor of the new Star Wars: find the panda in the picture below.

## Can quantum mechanics coexist with classical physics?

Continuing the discussion about quantum mechanics interpretations, today I want to look in depth on what it means to have a composite quantum-classical system. In standard Copenhagen interpretation, the measurement apparatus is considered to be described by classical physics. In physics there are only two theories of nature known and possible: quantum and classical mechanics and usually classical mechanics is described as the limit of quantum mechanics when $$\hbar \rightarrow 0$$. How can we introduce a fundamental theory of nature by using its limit case which is of a different character (obeys local realism)? This is one of the usual criticism of Copenhagen and a motivation for people to look for local realistic models of quantum mechanics. More important I think is to decide if there can be any consistent quantum-classical description of a physical system. But are there real world examples of such composite systems? I saw once this example given at a physics conference: a transistor. We do not see transistors in a superposition state and its inner workings are definite quantum mechanical.

So now that the stage is set, we can provide some answers. The framework is yet again the categorical approach to quantum mechanics, and part of what I will state today was discovered by one of Emile Grgin's colleague at Yeshiva University, Debendranath Sahoo: http://arxiv.org/pdf/quant-ph/0301044v3.pdf

So let's start at the beginning: is quantum mechanics defined by using one of its limits? The answer is no, but this is only a recent development with the complete derivation in the finite dimensional case of quantum mechanics from physical principles. Quantum mechanics stands on its own without classical mechanics help.

Is classical mechanics defined by the limit $$\hbar \rightarrow 0$$? Surprisingly, no again! This limit is mathematically sloppy, the proper limit is for $$\hbar$$ to become a nilpotent element: $$\hbar^2 = 0$$. If you remember the map $$J$$ between observables and generators, its dimension  is actually $$\hbar$$ and while in quantum mechanics $$J^2 = -1$$, in classical physics $$J^2 = 0$$, meaning $$\hbar^2 = 0$$ in classical physics. Another way to see this is by looking at deformation quantization approaches and convince yourself this is the proper limit. But if we are not sticklers for math and we adopt a physical point of view, $$\hbar \rightarrow 0$$ is good enough.

Can we combine consistently quantum and classical mechanics? No again because quantum and classical mechanics belong in disjoint composability classes. But what would happen if we try? This question was answered by Debendranath Sahoo in the paper above. Let's recall the fundamental composition relationships (in either quantum or classical mechanics):

$$\alpha_{12} = \alpha_1 \otimes \sigma_2 + \sigma_1 \otimes \alpha_2$$
$$\sigma_{12} = \sigma_1 \otimes \sigma_2 + J^2 \alpha_1 \otimes \alpha_2$$

If $$\alpha_1$$ is the commutator, $$\alpha_2$$ is the Poisson bracket, $$\sigma_1$$ is the Jordan product, and $$\sigma_2$$ is the regular function multiplication, what would $$\alpha_{12}$$ and $$\sigma_{12}$$ be? In quantum mechanics one can have superselection rules and there is nothing which prevents us to combine the 4 ingredients in a marriage of convenience. The penalty however is that we get something which lacks invariance under tensor composition! As such there cannot be any possible generalization of the commutator in the quantum-classical case. And people did try to invent such things but no such proposal withstood scrutiny. But this is not the only penalty!

What Sahoo found (working the problem Grgin-Petersen style) is that there is a lack of backreaction from the quantum to classical system! I double-checked the proof, it is correct, and because it involves a lot of Latex typing I will not repeat it here but you can read it in the paper. The result has two main implications:
• gravity has to be quantized: you cannot get a self-consistent theory of gravity in a mixed quantum-classical setting.
• The measurement devices should not be treated classically because they will not be able to measure anything: there is no information transfer from the quantum to the classical system.
If we were to solve the measurement problem we must do it solely in the quantum world. MWI is not the answer and it is no longer the only pure unitary quantum game in town. The categorical approach shows the way and provides a brand new solution. Despite its classical appearance, the transistor is still only a quantum object. The main problem is not interpreting quantum mechanics to appease our classical intuition but to explain the emergence of classical behavior. Decoherence only provides a partial answer. Please stay tuned.

PS: today is December 24

## Can anyone defend the Many Worlds Interpretation?

Quantum mechanics has many interpretations or classes of interpretations with internal splits: Copenhagen, Bohmian, spontaneous collapse, many worlds, transactional, etc. Because my take on the matter falls within the neo-Copenhagen family, I do not follow very closely interpretations which fall outside my interest. But although I disagree with non-Copenhagen interpretations, I do understand the approaches they take with only one exception: the many worlds interpretation (MWI). Not for the lack of trying but as far as I dug into it, MWI did not make any sense whatsoever to me (except Zurek's approach which technically is not MWI). So here is my challenge: can anyone defend MWI in a way that will answer the issue I will raise below?

The ground rule of any interpretation is first and foremost to recover the standard quantum mechanics predictions, otherwise it cannot call itself a "quantum mechanics interpretation". Quantum mechanics has this novel feature called the Born rule. Let me digress for a bit and expand on why this does not occur in classical physics. If you recall from prior posts, in configuration space in classical mechanics one encounters the Hamilton-Jacobi equation, while in quantum mechanics one has the Schrodinger equation. In classical physics in phase space we need both the position and momenta of a particle to specify the trajectory, and therefore it should come as no surprise that in configuration space where we only have positions there can be crossing trajectories in the Hamilton-Jacobi case. Therefore the information content attached to a configuration point is ambiguous in classical physics: no Born rule in classical physics. However in the quantum case in configuration space we can attach an information interpretation to the Schrodinger wavefunction known as the Born rule. Born rule shows that quantum mechanics is probabilistic and initial conditions are not required. (In the Bohmian case you can add initial conditions only in a contextual (parochial) way respecting an additional constrain called quantum equilibrium otherwise you violate Born rule).

Is MWI compatible with Born rule?

But what is MWI, and why is it considered? MWI supposes to solve the measurement problem without resorting to the collapse of the wavefunction.

Suppose we have two outcomes, say spin up and down. Once spin is measured up, a quick subsequent measurement confirms the result, and the same for down. Since the wavefunction respects the superposition principle we can derive a superposition of up and down with the measurement device pointing up for the spin up, and pointing down for spin down. In other words, we arrive at the famous half dead half alive Schrodinger cat which does not occur in nature. Everett noticed that there is a correlation between how the measurement device points and the spin value and he proposed that the world splits between different outcomes: in each outcome the observer is only aware of his own unique measurement result. One proponent of this narrative was Sydney Coleman!!! (I have a big respect for the late Sidney Coleman, but in this instance I think he was shooting from the hip.) I grant that MWI is an appealing idea, but does it stand up to close scrutiny?

People naturally objected to the idea of split personality or "the I problem" to which supporters can fire back with "you do not take quantum mechanics seriously enough to trust what it shows". Also there is a "preferred basis problem" because the split can happen on an infinite number of basis. But to me the most important problem is the treatment of probabilities and agreement with Born rule. I think it is safe to say that anyone agrees that the original Everett argument of why MWI obeys Born rule is not satisfactory. If Everett's derivation were correct, then there will not be that many new "derivations" of Born rule in the MWI framework. However I found no satisfactory derivation to date of Born rule in MWI. Moreover, the only thing that makes sense to me is branch counting and this is definitely violating Born rule - also no disagreement here.

But why I am not convinced by the proposals of deriving Born rule? A common criticism is that those derivations are circular. I assert something stronger: when not circular, Born rule derivations in MWI are mathematically incorrect. Let me show why.

I have discovered long ago that the simplest problems are the hardest, and I will use this here. Instead of muddling the water with convoluted arguments and examples, let's streamline the basic system to the max. So consider a source of electrons which fire only one particle say once a minute. Pass this through a Stern Gerlach device and select only the spin up branch. In other words, we prepare a source of single electrons with a known vertical spin. Then we pass our electron through a second Stern Gerlach device and we measure spin on say the x axis. Half the time we will get the positive spin x and half the negative spin x. In MWI both outcomes occur and I am split into two "me" each observing one definite outcome. So far so good, but now rotate one of the two devices by some angle theta. The statistics changes!!! but what does MWI predict? The world still splits in two and in one world I detect up and in the other one down. In other words, no changes and this is the root cause of why MWI makes no sense.

Now supporters of MWI are well aware of this fact and attempt to derive Born's rule regardless starting from more or less natural assumptions. Let's dig deeper into their claims.

First we need to collect more data to make up a meaningful statistics. For the first electron we have two branches: one up and one down: u d. For the second electron we have 4 branches: uu ud du dd, for the 3rd electron we have 8: uuu ... Now let's count in those branches how many spins are up and how many are down regardless of the actual order of the events:

1st run:                                           1u    1d
2nd run:                                     1uu  2ud    1dd
3rd run:                          1uuu    3uud     3udd       1ddd
4th run:                1uuuu    4uuud   6uudd    4uddd      1dddd

We get Pascal's triangle and the binomial coefficients. This is nothing like Born rule, and the frequentist approach in statistics is rejected by the MWI supporters. Instead they adopt the Bayesian approach. For simple problems like this, the frequentist and Bayesian approaches predict the same things so something else must be thrown in the mix: "the rational observer".  A "rational observer" would have expectations of probabilities before the actual experimental outcome is obtained, and MWI supporters contend that to a rational observer making rational decisions, while branching is incompatible with Born rule, the sane way for such a person to behave is as Born rule appears to be true. Something like: the Earth moves around the Sun, but to us it appears that the Sun moves around the Earth. This line of reasoning was introduced by Deutsch and continued by Wallace.

Several natural sounding principles were proposed to justify this apparent emergence of Born rule in the MWI world. Now Born rule deals with the complex coefficients in front of the ket basis, and those coefficients are simply ignored by  branching because this is what it means to to have a relative correlation between the wavefunction and the measurement device. To derive Born rule you must deal with those coefficients and moreover you must do it in an indirect way. The only indirect way possible is for your "natural sounding principle" to say something nontrivial about a superposition. And in the best case scenario what you actually say about the superposition is nothing but the Born rule in disguise and you have a circular argument

But it gets worse if you claim you broke the circularity: you become mathematically inconsistent. Here is why:

1. To prove Born rule in MWI you need to reject branch counting.
Why? Because Born rule's prediction changes with changing complex coefficients, but branch counting does not.

2. Branch counting arises as a particular case of Born rule. When? In the particular case when the complex coefficients are equal.

So the very act of proving even an apparent Born rule inherently contains a contradiction. All mathematically consistent proposals of deriving Born rule in MWI I am aware of are circular arguments and all their "natural sounding principle" respects branch counting as well.

In summary, coming back to my physical example with the electron source and the two S-G devices, because branching happens the same way regardless of the orientation of the two devices, there is a one to (uncountable infinite) many degeneracy problem which MWI cannot hope to solve by relative arguments alone. In the frequentist approach it is impossible to derive Born rule which acts as a removal of this degeneracy, and MWI supporters pin their hopes on derivations of an apparent Born rule by using some "natural principles". However all the derivations I studied so far are circular, and I know one by Tippler which is mathematically incorrect-maybe I should write a rebuttal to that one, it was published last year. Moreover they cannot reject branch counting because this follows from Born rule when all the scalar coefficients are equal. If you claim you reject  branch counting you are killing your "apparent" Born rule too.

I am challenging MWI supporters to present a valid non-circular derivation of Born rule (either real or apparent). I don't have the time to closely follow MWI developments and maybe there is a recent proposal I missed which can stand up to scrutiny. However I contend it can't be done for the reasons outlined above.