Visuomotor phase-locked loop reproduces elliptic hand trajectories across different rhythms [preprint]

[EetuP]

Rick, I have not said (and seen others saying) that the pure mathematical analysis – the playing with equations – of yours and Maoz et al were different so that Maoz et al were right and you wrong. There are same basic problem in both, but the logic around them is different. Maoz et al do not make such problematic inferences from this analysis as you do. The basic problem in both is the concept of affine velocity (A): it is not explained and understood what that concept is and where it comes from. I have tried to show that it is not a similar independent variable like velocity (V) and curvature (C) and that is why the analysis of Maoz et al is quite questionable and one should make very cautious inferences from it – like Maoz et al themselves do but you don’t. (I would like to know what other discussion their analysis has caused in Power Law research, do Adam know?)

What Maoz et al seem to claim based on that analysis is this: If there is low or no correlation between log A and log C then the power coefficient between log V and log C is (respectively) near or exactly -1/3. There is no correlation between log A and log C when A is constant. This is the known result which comes straight from the way how Pollick and Sapiro defined affine velocity. A new result which Maoz et al produced was that noise can lessen the correlation between log A and log C and thus draw the power efficient near that -1/3.

What they did not infer from that analysis is that there is a necessary mathematical relationship between C and V so that -1/3 is a “true” power coefficient between them in every possible trajectory. Here you generalize from a certain special case to all possible cases (as Martin has stated).

Affine velocity, how ever it is defined, cannot create any mathematical relationship between velocity and curvature but only between itself and both velocity and curvature taken together. Just similarly as the area of a rectangular does not create mathematical relationship between the width and height of that rectangular. Perhaps I should try to show this with simple diagrams but I am sorry that I have no time for that now.

[rsmarken]

Hi Eetu

I forgot to reply to this part of you earlier post.

@EetuP

RM: OK, it looks like that’s your story and your stickin’ to it. But I would like to know where you think the power law fits into PCT. You know where I think it fits in: It’s an example of an irrelevant side effect of controlling. Where do you think it fits in?

In this kind of discussion group we should not be so much interested about what is whose story but what story is true and valid when compared together.

I don’t think this discussion group should be about determining whose story is true; I think it should be about determining whose story (model) fits the data best. Going for “truth” smacks a bit too much of religion for my taste.

The most tragicomic feature in your story (and consequently my criticism of it) is that it has nothing at all to do with PCT except that the fame of PCT as a serious scientific truth seeking enterprise is put under danger.

I agree that my “story” – that the power law is an irrelevant side effect of the control of the perceptions involved in producing curved movement and that you see this particular side effect because of the mathematical relationship between curvature and velocity and the statistical methods used to determine whether a movement conforms to the power law – may have nothing to do with what you think of as PCT but it has everything to do with Bill Powers’ model of the behavior of living organisms.

And I am not interested in PCT being famous; I am interested in it being understood and developed through scientific investigation. And perhaps you are right that my work on the power law puts the “fame” of PCT in danger but that would be ok with me because I’m pretty sure that what you call “PCT” is not the same as the theory I spent 34 years working on with Bill Powers.

Yes I know that you claim that power law is a side effect of controlling. And we know that so think all the other discussers here.

If that’s true, when why are all these other discussers so down on me? In Bill’s model of behavior, side effects of control are “red herrings” that lead researchers down a blind alley. So if your PCT is really Bill’s model of behavior then you would be celebrating the exposure of the power law as a “red herring” and maybe even proposing where research on human movement might go now that the power law is no longer the focus of attention.

But it has nothing to do with your mathematical reasoning which I and others have criticized. And as I have said, side effects are often interesting and important research topics.

Then what does it have to do with?

[EetuP]

Rick, it is nice to discuss all the side topics but I’ll wait that you will answer to the criticism about your use of affine velocity. Your conclusion that power law is a side effect of control may be and quite surely is pertinent, but because the mathematical justification you use for it does not bear closer scrutiny, no one has to take you seriously. That is the price you have to pay if you abandon the truth.

(I have forgotten to reference to Adam’s message which shows that angular speed is very similar construct as affine velocity.)

[rsmarken]

@EetuP

Rick, I have not said (and seen others saying) that the pure mathematical analysis – the playing with equations – of yours and Maoz et al were different so that Maoz et al were right and you wrong.

Take a look at Martin Taylor’s comment in reply to Marken & Shaffer (2017) where he “proves” that we got the math all wrong. Maoz et al (2005) weren’t known to me at the time but since the Marken & Shaffer analysis is exactly equivalent to that of Maoz et al, Martin was implicitly dissing the math of Maoz et al as well. Since Martin now seems to think that the mathematical analysis of Maoz et al is just fine, there is at least one person (and, I suspect, quite a few more than one) who seems to think that our math was wrong and that of Maoz et al is right.

The basic problem in both [Marken & Shaffer and Maoz et al] is the concept of affine velocity (A): it is not explained and understood what that concept is and where it comes from.

The affine velocity variable (called D in Marken & Shaffer and called alpha(t) in Maoz et al) comes from the derivation of the equation relating curvature to velocity. Here’s how Maoz et al derive it:

And here’s how Marken & Shaffer derived it:

I have tried to show that it is not a similar independent variable like velocity (V) and curvature (C) and that is why the analysis of Maoz et al is quite questionable and one should make very cautious inferences from it – like Maoz et al themselves do but you don’t.

So now the mathematics of both Maoz et al and Marken & Shaffer are not wrong? They are just “questionable”? So could you tell me what is the questionable aspect of the math and how the questionability of the math relates to our inferences?

What Maoz et al seem to claim based on that analysis is this: If there is low or no correlation between log A and log C then the power coefficient between log V and log C is (respectively) near or exactly -1/3.

Yes, and that’s just a mathematical fact. So the degree to which a movement is found to conform to the power law depends only on the mathematical/statistical relationships between V, C and A in a particular movement.

My conclusion from this is that the -1/3 power law is a mathematical statistical artefact that has nothing to do with how the movement was produced. Maoz et al come to a different conclusion;

Here’s what I think they are saying:
Don’t worry! Despite what we have demonstrated up to this point, the power law is real, not a mathematical/statistical artifact (ie. not a bogus phenomenon). So you can keep doing power law research (and get the grant money to support it) but just be careful to filter out any random noise, which can reduce the correlation between curvature and affine velocity and artificially “drive the results to the power law.” That is, you have to massage the data properly to make sure that you are seeing the real power law rather than the one determined by the mathematics.

The problem with this is: You can’t fool the math. The math (as shown by Maoz et al and by Marken & Shaffer) says that when you regress curvature on velocity you will find a power coefficient close to -1/3 when there is a low correlation between curvature and the variable omitted from the regression, affine velocity. And such a correlation can occur with or without filtering. There is simply no way to tell whether an observed -1/3 power coefficient relating curvature to velocity is more “real” than another.

I prefer the PCT explanation of the power law; that it is an irrelevant side effect of controlling the variables involved in producing movement and is usually found to have a value close to -1/3 because of the mathematical/statistical relationship between curvature, velocity and affine velocity.

There is no correlation between log A and log C when A is constant. This is the known result which comes straight from the way how Pollick and Sapiro defined affine velocity.

There can also be no correlation between log A and log C when log A is not constant. Indeed, this is the most common case.

A new result which Maoz et al produced was that noise can lessen the correlation between log A and log C and thus draw the power efficient near that -1/3.

Which happens because you can get a low (or zero) correlation between log A and log C when the variance of log A is greater than zero.

What they did not infer from that analysis is that there is a necessary mathematical relationship between C and V so that -1/3 is a “true” power coefficient between them in every possible trajectory.

They didn’t have to infer it; they could see it right in front of their eyes. And you can see it for yourself in Moaz et al 's equation 5:

That -1/3 in front of log (kappa(t)) is the true mathematical coefficient of the power relationship between curvature (kappa(t)) and velocity (v(t)).

[rsmarken]

@EetuP

Rick, it is nice to discuss all the side topics but I’ll wait that you will answer to the criticism about your use of affine velocity.

And I answered in my previous post. I “use” affine velocity for the same reason Maov et al did; it’s all in the derivation of the mathematical relationship between curvature and velocity.

Your conclusion that power law is a side effect of control may be and quite surely is pertinent, but because the mathematical justification you use for it does not bear closer scrutiny, no one has to take you seriously.

The mathematics don’t justify calling the power law a side effect of control. That comes from understanding the nature of control. The mathematics explains why you typically get a power “law” coefficient of close to -1/3.

That is the price you have to pay if you abandon the truth.

And what do you think is the price you pay for thinking you know more than you know?

[rsmarken]

@EetuP

That is the price you have to pay if you abandon the truth.

Is it something like the price paid by Galileo for not knowing the truth?

[EetuP]

RM: Is it something like the price paid by Galileo for not knowing the truth?

No, rather just another way round. Afterwards most scientists take Galileo and his arguments seriously but not so many those of Inquisition. And if you think that you play here the role Galileo, I must say that it is also in another way round.

Because you continuously answer the criticisms just by repeating your starting points, I think that you cannot read or understand them. So I tried once again to collect the most central criticism and express it as understandably as I can in the attached writing. Hope you will read and answer it.

Eetu

power law and pct.pdf (224 KB)

[rsmarken]

@EetuP

Because you continuously answer the criticisms just by repeating your starting points, I think that you cannot read or understand them. So I tried once again to collect the most central criticism and express it as understandably as I can in the attached writing. Hope you will read and answer it.

Well, I appreciate your (and Martin and Adam’s) efforts to explain why I (and Maoz et al) got it all wrong so I will try to give a detailed answer to the attached essay. But in the meantime, could you tell me what you think the power law is. Why is it considered an important phenomenon; why has so much research been (and, apparently, will continue to be) dedicated to the study of the power law?And, finally, what do you think the power law tells you about movement production?

I believe the power law IS an irrelevant side effect of the controlling involved in movement production. I think it’s considered an important phenomenon by power law researchers because it seems to show organisms “slow down through curves” so it is a side effect that is relevant to their intuition that this is the way curved movement , such as the movement of race cars around a track, works. So a lot of research has been dedicated to trying to find out why this happens. I think the power law tells us nothing about movement production but quite a lot about the researchers studying it.

What do you think? Feel free to consult with Adam and Martin and anyone else about it.

[EetuP]

RM: could you tell me what you think the power law is

That is an interesting question but not very important to me. I cannot know the answer and I do not want to hang myself to any speculations about it. It seems very probable that it is a side effect of control, but how relevant or irrelevant it is and in what sense, is a totally different question. I expect that PCT informed researchers like Adam and others will tell something more about it in the future.

I have taken part in to this discussion because I control with high gain for a couple of principles in scientific discussion. One is the requirement of the validity of argumentation: all claims should be grounded by sound and holding arguments. Another one is the Peircean maxim “Do Not Block the Way of Inquiry”.

BTW. your concept of irrelevance is ambiguous. It can mean for example that the irrelevant thing is not caused by someone, or it does not affect someone, or it does not, or should not, interest someone. If you stress that power law does not tell any thing about how the movement is produced you probably mean the first (and the last case). But when you define: “An irrelevant side effect is a side effect of control that is irrelevant to the controlling done by a control system” you seem to meant the second variant, that it does not affect the controlling. However, if Power Law (or rather should be said: power relationship between curvature and speed) is a side effect of controlling, then it caused by the controlling. And it is a general way to explain phenomena by searching the factors which cause it. Also power law clearly affects back to the controlling of the movement by determining its speed in relation to the curvature. So, in what sense it is irrelevant?

[rsmarken]

@EetuP

Because you continuously answer the criticisms just by repeating your starting points, I think that you cannot read or understand them. So I tried once again to collect the most central criticism and express it as understandably as I can in the attached writing. Hope you will read and answer it.

Well, I have read your PDF titled “About the “true” power coefficient between the curvature and velocity – and what does it have to do with PCT” with your 5 theses explaining why I am wrong about the power law being an irrelevant side effect of control and I am now completely convinced that the power law is whatever you (and your friends Adam and Martin) say it is, whatever that is.

Actually, I started to reply but realized I had already answered all your points – to my satisfaction anyway – and that there was absolutely nothing I could say that would convince you that you are chasing an illusion. I was trying to help you understand what I think is one of the most important insights that comes from PCT – the distinction between relevant and irrelevant side effects of control. But, as Tim Carey said in a recent blog post there is a fine line between helping and interfering. I will interfere with you on longer (on this subject anyway).
(Tim, by the way, is one of the few people who agrees with me about just about everything PCT – because he understands PCT – but unfortunately he does not participate on iapct.discouse. I wonder why;-))

I will just say that all 5 of your criticisms of my analysis of the power law – Marken & Shaffer (2017) – apply also to the analysis done by Maoz et al (2005). That’s because the two analyses are exactly the same. The only difference between Marken & Shaffer and Maoz et al are the conclusions they come to based on that analysis.

And I realized that the reason for the different conclusions is that Marken & Shaffer understood movement in terms of the PCT model of behavior and Maoz et al didn’t. Without PCT, Maoz et al were unable to conceive of the power law as a possible side effect of the processes that produce the movement. The only conclusions available to them were that it was either a bogus phenomenon or a true “invariant” that could falsely appear if there were high frequency noise components in the movement. So, of course, they went with the true invariant idea.

[EetuP]

Rick, I must say I am disappointed, even though I understand it is hard to try to defend ones deepest beliefs when everyone and everything seems to be against them. I don’t think I have seen yet your answers to those themes I took up in my essay. Could you try to point to them or recap shortly:

[EetuP]

Oops Discourse cut my email message. Let’s try again:

Rick, I must say I am disappointed, even though I understand it is hard to try to defend ones deepest beliefs when everyone and everything seems to be against them. I don’t think I have seen yet your answers to those themes I took up in my essay. Could you try to point to them or recap shortly:

1. Do you really think that in planar movement we do not have two degrees of freedom, but less, so that if something moves in x dimension, it must necessarily take a certain move also in y dimension? This must be the case if there prevails a mathematical relationship between curvature and velocity.

2. Note, that I do not say there is any miscalculation in your derivation of

V = R1∕3 * D1∕3

But tell me, why does it set a different relationship between V, R, and A than

Width * Height = Area or

Perimeter = Width + Height

sets between Width, Height, and Area/Perimeter? Or does it?

3. In Marken & Schaffer 2018 you say: “regression analysis does not assume that the criterion variable (speed) ‘depends on two independent variables’. The regression analysis takes into account any correlation that might exist between the predictor variables…” which is of course true. The question is not that independent variables must independent among them, but they must be independent from the criterion variable. You have not said anything about this.

4. Fig 1 in Marken & Schaffer 2018 is quite nice. Left pane depicts an elliptical trajectory of the cursor in the screen. This trajectory is intentionally produced by the subject and it follows power law. Let’s call it power law movement. Right pane depicts the movement of the mouse – non power law movement. This movement is not elliptical because the computer caused disturbances to the cursor and the mouse movements had to cancel them in addition to produce the ellipse. Which one of these two movements is (more) a side effect? Both are causally produced at the same by the subject and her muscles and limbs. What there is irrelevant and what is relevant, and why?

Best

Eetu

[rsmarken]

@EetuP

Rick, I must say I am disappointed

Well, as the Dread Pirate Roberts said to Inigo Montoya, “Get used to disappointment”

I understand it is hard to try to defend ones deepest beliefs when everyone and everything seems to be against them.

It is hard but I enjoy doing it because I have a deep commitment to carrying on the work that Bill Powers started.

I don’t think I have seen yet your answers to those themes I took up in my essay. Could you try to point to them or recap shortly:

OK, I’ll give it a try. It’s probably useless but I have some time to waste and I’ll keep my answers short.

1. Do you really think that in planar movement we do not have two degrees of freedom…

No, I know that there are 2 df in planar movement.

2. Note, that I do not say there is any miscalculation in your derivation of

V = R1∕3 * D1∕3

But tell me, why does it set a different relationship between V, R, and A than

Width * Height = Area or

Perimeter = Width + Height

It doesn’t. All these equations are equivalent. They would all be linear if you take the log of both sides in the equations for V and Area.

3. In Marken & Schaffer 2018 you say: “regression analysis does not assume that the criterion variable (speed) ‘depends on two independent variables’…The question is not that independent variables must independent among them, but they must be independent from the criterion variable. You have not said anything about this.

What is the “this” that I said nothing about? I never said that the independent (more accurately called predictor) variables in a multiple regression must be independent of the criterion variable. That would have been ridiculous.

4. Fig 1 in Marken & Schaffer 2018 is quite nice… Which one of these two movements is (more) a side effect?..

Here’s Figure 1 from that paper:

Glad you like it.

Cursor movement is not a side effect; it is a controlled result of mouse movements. The elliptically varying position of the cursor is a controlled variable.

Mouse movement is a side effect of controlling cursor position but it is a relevant side effect because it is part of the process of controlling the cursor (it’s the output variable in the loop that controls cursor position relative to a variable reference).

Mouse movement is also a controlled variable since slight variations in the amount and direction of force applied to the mouse are needed to move the mouse appropriately to counter the effects of disturbances to mouse position, such as slight variations in the resistance of the surface on which the mouse is moved.

The .3 power relationship between velocity (V) and curvature (R) that is observed for the cursor movement in Figure 1 is an irrelevant side effect of the control process that produced that movement; the .05 power relationship between V and R that is observed for the mouse movement is also an irrelevant side effect of the control process that produced the movement.

[EetuP]

Thank you Rick for replying! Now I didn’t need to get used to disappointment .

I start from the end and refer at same time to your and Martin’s discussion. I gladly agree that the power law phenomenon is a side effect of control. And especially if the subject is asked only to draw an ellipse – with no special speed – then it is clearly an irrelevant side effect. (Still there is some ambiguity in the meaning of “irrelevant” in different occasions, but that is another topic.) So that is not a disturbance to me and I think it is neither for Martin – and possibly it would not be a very big disturbance to (all) power law researchers neither because it does not nullify their research topic: it is still a problem why there appears this kind of (interesting side effectual) relationship in some kinds of movements but not it other kinds. This problem requires physical explanation.

Glad that you agree that we still have 2 df in planar movement. That means that if an object happens to be moving with a certain speed straight to x direction and then it starts to turn to y direction, this turning (= adding an y component to the movement) has no necessary (=mathematically predetermined) effect to the x component of the movement. Independently from the change in y component the x component can remain the same or increase or decrease. Turning to left or right has no mathematically predetermined effect on the total speed of the movement because, when the y component accelerates or decelerates, the x component can freely remain constant or accelerate or decelerate. Generally there can be no mathematical relationship between the speed and curvature of the movement. And so all relationships between them must be causally / physically caused. Isn’t that clear?

About the equations [V = R1∕3 * D1∕3]; [Width * Height = Area]; [Perimeter = Width + Height]; and also that of the angular speed could be added here [A = C * V], you said: “All these equations are equivalent. They would all be linear if you take the log of both sides in the equations for V and Area.” The first sentence is simply true depending on what you mean here with equivalence of different equations, but they are all similar so that the left side and right side are equivalent: There is always the same value both side when the variables are instantiated and calculations made. So the relationships between the sides are linear even without taking the logs, aren’t they? (The linear relationship between logs means that there is some power relationship – in this case the power coefficient is 1.)

But you seem the think that the equation [V = R1∕3 * D1∕3] sets (or unveils) a mathematical relationship between V and R so that there is a necessary power relationship between speed and curvature with the “true” coefficient 1/3? And if the equations above are equivalent then there must also prevail a necessary mathematical relationship between the height and with of all possible rectangles. Do you think so? If you think so, then what is the power coefficient of that relationship?

RM: “I never said that the independent (more accurately called predictor) variables in a multiple regression must be independent of the criterion variable. That would have been ridiculous.” Here I think you mean with the word “independent” something different from mathematical / conceptual independence. Perhaps you mean that independence means that there is no correlation? It is natural that we try to use as predictors such variables which correlate with the criterion variable. But if this analysis should have any scientific sense then there must be a possibility that we will not find any significant correlation. That is often an important result. If the criterion is conceptually depending on even some of the predictors, then this result is impossible. The conceptual / mathematical dependence determines the result beforehand. This will happen if you analyze the correlation of a variable with itself, or with some multiplication of itself. And this is exactly what you do in your OVB. You create arbitrary and artificial statistical results.

[MartinT]

You keep asking me to look at Maoz et al., but have refused to provide a link to help me do so. Having now managed to find the relevant paper, I see why you wanted to make it difficult for me to find. Maoz et al. (2005) start by assuming the ⅓ power law, and do some analysis and Monte-Carlo studies to see what might be the consequences of assuming that the ⅓ power law holds. They do not analyze from the other end, which would be to consider (from a PCT standpoint, preferably) all possible functions that might plausibly result in the observed data, and conclude, as you do, that the ⅓ power law is necessarily true.

If you think that your mathematics is the same as theirs, you are saying that you assume that the ⅓ power law is necessarily true, and then prove that under this assumption, the ⅓ power law is necessarily true.

[rsmarken]

@EetuP

Thank you Rick for replying! Now I didn’t need to get used to disappointment .

Yes, now only I do;-)

…it is still a problem why there appears this kind of (interesting side effectual) relationship in some kinds of movements but not it other kinds. This problem requires physical explanation.

Yes, I think it might be worth it to figure out why this particular side effect occurs. Given the level of misunderstanding of PCT on this site I think it might be worth spending some time on it.

Glad that you agree that we still have 2 df in planar movement. That means that if an object happens to be moving with a certain speed straight to x direction and then it starts to turn to y direction, this turning (= adding an y component to the movement) has no necessary (=mathematically predetermined) effect to the x component of the movement.

Actually, that’s not true. While x and y are 2 df of planar movement, velocity (V) and curvature (C) are not. That’s because V and C (or R) are functions of both x and y (see equations 2 and 3 in Marken & Shaffer, 2017).

About the equations [V = R1∕3 * D1∕3]; [Width * Height = Area]; [Perimeter = Width + Height]; and also that of the angular speed could be added here [A = C * V], you said: “All these equations are equivalent. They would all be linear if you take the log of both sides in the equations for V and Area.” The first sentence is simply true depending on what you mean here with equivalence of different equations, but they are all similar so that the left side and right side are equivalent: There is always the same value both side when the variables are instantiated and calculations made. So the relationships between the sides are linear even without taking the logs, aren’t they?..

No. Only the equation for P = H + W is linear.

But you seem the think that the equation [V = R1∕3 * D1∕3] sets (or unveils) a mathematical relationship between V and R so that there is a necessary power relationship between speed and curvature with the “true” coefficient 1/3?

Yes, I do.

And if the equations above are equivalent then there must also prevail a necessary mathematical relationship between the height and with of all possible rectangles. Do you think so? If you think so, then what is the power coefficient of that relationship?

In A = H * W the power coefficient of H is 1 and the power coefficient of W is 1. The equation can be written A = H^1*W^1. The same is true for H and W in P = H + W and for C and V in A = C * V.

RM: “I never said that the independent (more accurately called predictor) variables in a multiple regression must be independent of the criterion variable. That would have been ridiculous.”

Here I think you mean with the word “independent” something different from mathematical / conceptual independence. Perhaps you mean that independence means that there is no correlation? It is natural that we try to use as predictors such variables which correlate with the criterion variable. But if this analysis should have any scientific sense then there must be a possibility that we will not find any significant correlation.

That is simply not true. There is no law against using regression analysis to confirm a mathematical relationship between variables. And in our case, that use of regression had enormous scientific value because it showed that when you do a regression on only a subset of the variables (V and R) involved in a mathematical relationship that involves those variables and one other (D) your result will deviate from the true relationship between the included two by an amount that is proportional to the covariance between the included and omitted predictors. This is a scientifically important finding because it confirms the PCT explanation of the observed behavior.

The conceptual / mathematical dependence determines the result beforehand. This will happen if you analyze the correlation of a variable with itself, or with some multiplication of itself. And this is exactly what you do in your OVB. You create arbitrary and artificial statistical results.

Actually, I didn’t create the arbitrary and artificial results obtained in power law research. The power law researchers who did regression analyses of C on V, omitting D, have been creating them for decades.

[rsmarken]

@MartinT

If you think that your mathematics is the same as theirs, you are saying that you assume that the ⅓ power law is necessarily true, and then prove that under this assumption, the ⅓ power law is necessarily true.

No Martin, I am not saying anything like that. What I’m saying is that our mathematical analysis is identical to theirs. If you think not then show me how the math differs. Write down the relevant equations from Marken & Shaffer (2017) and Maoz et al (2005) and show us how our equations differ from theirs.

I dare you. No, I double dare you;-)

[MartinT]

I didn’t suggest that your math differs from theirs. Since they start by assuming the ⅓ power law is true, I am saying that YOU are saying that you start with the assumption that the ⅓ power law is true — and then use that assumption to prove that it is true.

[rsmarken]

Still wrong Martin. Neither Maoz et al nor I assumed anything about the power law.

But, wait! I thought your beef with Marken & Shaffer (2017) was only about our math. But now you are saying there was nothing wrong with our math, it was our “assumptions”. Truly, you have a dizzying intellect.

[MartinT]

It’s very hard to have a discussion with someone who simply talks nonsense. Maoz et al., and according to yourself therefore you, did explicitly start by assuming that the ⅓ power law holds true. They didn’t go through their math and then conclude that therefore the power law holds true, but you did append that conclusion. Anyway, I didn’t complain about your math beyond pointing out the logical error in that if you find it is true that your variable V agrees with V/V=1, and you also know X/X =1, then V=X.