# New article shows a fatal math error in SR

Improved Andrew Banks said:
Let a point-like detector exist, stationary in coordinate system k, somewhere to the left of the $$\eta$$-axis and only capable of detecting light to its right (including light originating at the origin of coordinate system k).

Assuming everything stationary in system k moves to the right with velocity v (in the x-direction) in system K, assume the origins of system k and K correspond at their respective zero times. Thus
$$t = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( \tau + \frac{v}{c^2} \xi \right) \\ x = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( \xi + v \tau \right) \\ y = \eta$$
Is it possible that for $$0 < v < c$$ the mirror could have such a large $$\eta$$ value that in the K frame a light flash from the time the origins were at the same position arrives at the detector from the left, preventing detection in one description of reality but not the other, supposedly equivalent one?

Andrew Banks correctly decides that the pulse from $$(\tau, \xi, \eta) = (0,0,0)$$ to $$(\tau_0, -\xi_0, +\eta_0)$$ would be seen in system K as a pulse from $$(t, x, y) = (0,0,0)$$ to $$(t_0, +x_0, +\eta_0)$$ whenever certain geometrical constraints are met, but ignores the question of what "to the right" means in system K.

First, what is the minimum value of v such that in system K the light pulse to the detector in purely in the $$+\eta$$ direction? That would mean $$x_0 = 0$$. Thus

$$v_0 = \frac{c \xi_0}{\sqrt{ \xi_0^2 + \eta_0^2}} < c$$

Then for any v such that $$v_0 < v < c$$ and assuming $$-\xi_0 < 0, \; \eta_0 > 0, \; \tau_0 = \frac{1}{c} \sqrt{\xi_0^2 + \eta_0^2} > 0$$ we have :

$$x_0 = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( -\xi_0 + v \tau_0 \right) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( -\xi_0 + \frac{v}{c} \sqrt{\xi_0^2 + \eta_0^2} \right) { \Large \quad > \quad } \frac{1}{\sqrt{1-\frac{v_0^2}{c^2}}} \left( -\xi_0 + \frac{v_0}{c} \sqrt{\xi_0^2 + \eta_0^2} \right) = \frac{1}{\sqrt{1-\frac{v_0^2}{c^2}}} \left( -\xi_0 + \frac{\xi_0}{\sqrt{ \xi_0^2 + \eta_0^2}} \sqrt{\xi_0^2 + \eta_0^2} \right) = 0$$

But that, importantly, still doesn't answer if the light comes into the left or the right of the detector, which is answered by the sign of the cross product of the light ray movement and an extension of the detector (finite or infintesimal) in the $$\eta$$ direction.

Equivalently, we may look at the sign of the cross product of light striking two ends of the detector.

So we have a detector which runs from $$(-\xi_0, \eta_0)$$ to $$(-\xi_0, \eta_0 + \Delta)$$ in system k where the detector is stationary.

Thus we have events O, A and B where the light flash occurs and is received at the two points respectively. These same events have representations in the system K (with coordinates t, x and y).

$$\begin{array}{c|ccc} & O & A & B \\ \hline \\ \tau & 0 & \frac{1}{c} \sqrt{\xi_0^2 + \eta_0^2} & \frac{1}{c} \sqrt{\xi_0^2 + \eta_0^2 + 2 \Delta \eta_0 + \Delta^2} \\ \xi & 0 & -\xi_0 & -\xi_0 \\ \eta & 0 & \eta_0 & \eta_0 + \Delta \\ \hline \\ t & 0 & \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( \frac{1}{c} \sqrt{\xi_0^2 + \eta_0^2} - \frac{v}{c^2} \xi_0 \right) & \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( \frac{1}{c} \sqrt{\xi_0^2 + \eta_0^2 + 2 \Delta \eta_0 + \Delta^2} - \frac{v}{c^2} \xi_0 \right) \\ x & 0 & \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( -\xi_0 + \frac{v}{c} \sqrt{\xi_0^2 + \eta_0^2} \right) & \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( -\xi_0 + \frac{v}{c} \sqrt{\xi_0^2 + \eta_0^2 + 2 \Delta \eta_0 + \Delta^2} \right) \\ y & 0 & \eta_0 & \eta_0 + \Delta \end{array}$$

So the cross product in system k is $$\xi_A \eta_B - \xi_B \eta_A = - \Delta \xi_0 < 0$$
And the cross product in system K is $$x_A y_B - x_B y_A = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( - \Delta \xi_0 + \frac{v}{c} \left( ( \eta_0 + \Delta) \sqrt{\xi_0^2 + \eta_0^2} - \eta_0 \sqrt{\xi_0^2 + (\eta_0 + \Delta)^2} \right) \right) < 0$$
This does not undergo a sign change at any speed $$0 < v < c$$ including $$v_0$$.

Thus because of the finite propagation speed of light, the expanding beam of light sweeps across the object in the same way, hitting the same face, a face which effectively has been rotated.

This interaction between ray-tracing beams of propagating light and the Lorentz transformation is known as Terrell rotation, not to be confused with Thomas precision which is another relationship between rotation and Lorentz transforms.

http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2049244
http://prola.aps.org/abstract/PR/v116/i4/p1041_1
http://math.ucr.edu/home/baez/physics/Relativity/SR/penrose.html

So not only is Andrew Banks completely wrong about Einstein's approach in 1905, but the effect he ignores was demonstrated as a general geometric effect in 1959. Any legitimate scientific venue for publication of matters related to the math of special relativity would have realized this along with many other flaws of this paper.

The article proves SR results in a contradiction.

No, it doesn't , wacko. The "article" is written by just another wacko and it is wrong.

Any legitimate scientific venue for publication of matters related to the math of special relativity would have realized this along with many other flaws of this paper.

Bur "Asian...." is not a legitimate journal, it is just another Chinese scam that takes 100$and publishes any crap in return. Equivalently, we may look at the sign of the cross product of light striking two ends of the detector. So we have a detector which runs from $$(-\xi_0, \eta_0)$$ to $$(-\xi_0, \eta_0 + \Delta)$$ in system k where the detector is stationary. Thus we have events O, A and B where the light flash occurs and is received at the two points respectively. These same events have representations in the system K (with coordinates t, x and y). $$\begin{array}{c|ccc} & O & A & B \\ \hline \\ \tau & 0 & \frac{1}{c} \sqrt{\xi_0^2 + \eta_0^2} & \frac{1}{c} \sqrt{\xi_0^2 + \eta_0^2 + 2 \Delta \eta_0 + \Delta^2} \\ \xi & 0 & -\xi_0 & -\xi_0 \\ \eta & 0 & \eta_0 & \eta_0 + \Delta \\ \hline \\ t & 0 & \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( \frac{1}{c} \sqrt{\xi_0^2 + \eta_0^2} - \frac{v}{c^2} \xi_0 \right) & \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( \frac{1}{c} \sqrt{\xi_0^2 + \eta_0^2 + 2 \Delta \eta_0 + \Delta^2} - \frac{v}{c^2} \xi_0 \right) \\ x & 0 & \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( -\xi_0 + \frac{v}{c} \sqrt{\xi_0^2 + \eta_0^2} \right) & \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( -\xi_0 + \frac{v}{c} \sqrt{\xi_0^2 + \eta_0^2 + 2 \Delta \eta_0 + \Delta^2} \right) \\ y & 0 & \eta_0 & \eta_0 + \Delta \end{array}$$ So the cross product in system k is $$\xi_A \eta_B - \xi_B \eta_A = - \Delta \xi_0 < 0$$ And the cross product in system K is $$x_A y_B - x_B y_A = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( - \Delta \xi_0 + \frac{v}{c} \left( ( \eta_0 + \Delta) \sqrt{\xi_0^2 + \eta_0^2} - \eta_0 \sqrt{\xi_0^2 + (\eta_0 + \Delta)^2} \right) \right) < 0$$ This does not undergo a sign change at any speed $$0 < v < c$$ including $$v_0$$. Thus because of the finite propagation speed of light, the expanding beam of light sweeps across the object in the same way, hitting the same face, a face which effectively has been rotated. This interaction between ray-tracing beams of propagating light and the Lorentz transformation is known as Terrell rotation, not to be confused with Thomas precision which is another relationship between rotation and Lorentz transforms. http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2049244 http://prola.aps.org/abstract/PR/v116/i4/p1041_1 http://math.ucr.edu/home/baez/physics/Relativity/SR/penrose.html So not only is Andrew Banks completely wrong about Einstein's approach in 1905, but the effect he ignores was demonstrated as a general geometric effect in 1959. Any legitimate scientific venue for publication of matters related to the math of special relativity would have realized this along with many other flaws of this paper. Here is your simple error. "in system k where the detector is stationary.' We already agree in k, the light strikes the mirror and reflects. What we are talking about is the system K. It does not reflect there. So, yet again you have been refuted. Bur "Asian...." is not a legitimate journal, it is just another Chinese scam that takes 100$ and publishes any crap in return.

So, can you prove this claim crank?

And the cross product in system K is $$\left( - \Delta \xi_0 + \frac{v}{c} \left( ( \eta_0 + \Delta) \sqrt{\xi_0^2 + \eta_0^2} - \eta_0 \sqrt{\xi_0^2 + (\eta_0 + \Delta)^2} \right) \right) < 0$$
This does not undergo a sign change at any speed $$0 < v < c$$ including $$v_0$$.

Ignoring chinglu for a while, how did you get that the expression is smaller than 0? It isn't obvious .

So, can you prove this claim crank?

rpenner already did. As predicted, you are wacko enough never to admit. Go away.

Equivalently, we may look at the sign of the cross product of light striking two ends of the detector.

So we have a detector which runs from $$(-\xi_0, \eta_0)$$ to $$(-\xi_0, \eta_0 + \Delta)$$ in system k where the detector is stationary.

Thus we have events O, A and B where the light flash occurs and is received at the two points respectively. These same events have representations in the system K (with coordinates t, x and y).

$$\begin{array}{c|ccc} & O & A & B \\ \hline \\ \tau & 0 & \frac{1}{c} \sqrt{\xi_0^2 + \eta_0^2} & \frac{1}{c} \sqrt{\xi_0^2 + \eta_0^2 + 2 \Delta \eta_0 + \Delta^2} \\ \xi & 0 & -\xi_0 & -\xi_0 \\ \eta & 0 & \eta_0 & \eta_0 + \Delta \\ \hline \\ t & 0 & \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( \frac{1}{c} \sqrt{\xi_0^2 + \eta_0^2} - \frac{v}{c^2} \xi_0 \right) & \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( \frac{1}{c} \sqrt{\xi_0^2 + \eta_0^2 + 2 \Delta \eta_0 + \Delta^2} - \frac{v}{c^2} \xi_0 \right) \\ x & 0 & \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( -\xi_0 + \frac{v}{c} \sqrt{\xi_0^2 + \eta_0^2} \right) & \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( -\xi_0 + \frac{v}{c} \sqrt{\xi_0^2 + \eta_0^2 + 2 \Delta \eta_0 + \Delta^2} \right) \\ y & 0 & \eta_0 & \eta_0 + \Delta \end{array}$$

So the cross product in system k is $$\xi_A \eta_B - \xi_B \eta_A = - \Delta \xi_0 < 0$$
And the cross product in system K is $$x_A y_B - x_B y_A = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \left( - \Delta \xi_0 + \frac{v}{c} \left( ( \eta_0 + \Delta) \sqrt{\xi_0^2 + \eta_0^2} - \eta_0 \sqrt{\xi_0^2 + (\eta_0 + \Delta)^2} \right) \right) < 0$$
This does not undergo a sign change at any speed $$0 < v < c$$ including $$v_0$$.

Thus because of the finite propagation speed of light, the expanding beam of light sweeps across the object in the same way, hitting the same face, a face which effectively has been rotated.

This interaction between ray-tracing beams of propagating light and the Lorentz transformation is known as Terrell rotation, not to be confused with Thomas precision which is another relationship between rotation and Lorentz transforms.

http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2049244
http://prola.aps.org/abstract/PR/v116/i4/p1041_1
http://math.ucr.edu/home/baez/physics/Relativity/SR/penrose.html

So not only is Andrew Banks completely wrong about Einstein's approach in 1905, but the effect he ignores was demonstrated as a general geometric effect in 1959. Any legitimate scientific venue for publication of matters related to the math of special relativity would have realized this along with many other flaws of this paper.

"Thus because of the finite propagation speed of light, the expanding beam of light sweeps across the object in the same way, hitting the same face, a face which effectively has been rotated."

How did you get the mirror to rotate when it is fixed?

Did God do it or does some unknown flat earth technology do it?

You need to prove this assertion.

How did you get the mirror to rotate when it is fixed?.

It is a relativistic effect called Terrell-Penrose rotation. Go away.

Ignoring chinglu for a while, how did you get that the expression is smaller than 0? It isn't obvious .

Crackpot, rpenner is now claiming the mirror is rotated.

you are behind he times.

It is a relativistic effect called Terrell-Penrose rotation. Go away.

That is fine.

Let me see the math that proves the mirror rotates in the unprimed frame while it remains fixed in the primed frame.

I think all would like to see your proof.

That is fine.

Let me see the math that proves the mirror rotates in the unprimed frame while it remains fixed in the primed frame.

I think all would like to see your proof.

Crank. Incurable.

Do you have a proof from your flat earth world yes or no?

otherwise, you must submit to the conclusions of the article of this thread.

if you do not, then you are a crank and crackpot.

if you do not, then you are a crank and crackpot.

I agree, you are a crank and a crackpot. Your only consolation is that you are incurable, as well.

If you use LT and take the partial derivative assuming z=0 and a fixed y as per the article, you will find the horizontal intersection speed with the y line is greater than the speed of light when the light pulse is in between the 2 origins.

This means, along any y-line, a sphere is above that line before the mirror strikes it so that it runs into it. That would mean there is a light beam along a y line that is in front of any other light beam on that line.

Two people have separately pointed out that the mirror in the unprimed frame hits the light in front of it, not the light behind it. These have been your responses. They have no meaning. I can parse them as English, but they have no semantics - just piles of physics words strung together. From the first response, at least, it sounds like you think you can express this concept with math, so please do so.

"Thus because of the finite propagation speed of light, the expanding beam of light sweeps across the object in the same way, hitting the same face, a face which effectively has been rotated."

How did you get the mirror to rotate when it is fixed?

Did God do it or does some unknown flat earth technology do it?

You need to prove this assertion.

This thread, and the trolling, reminds me of why physforum is dead and populated by intellectual zombies. Clearly the science threads are no longer being moderated in a meaningful way. These science boards are so boogered up with nonsense it's a trolls paradise.

Hi Aqueous Id.

Here is my polite exchange with billvon regarding his rash 'maths' example to chinglu...

Undefined to billvon said:
No, it doesn't, because it is based on a false assumption.

Let's use another example. Let's say you take two atoms of hydrogen and two atoms of antihydrogen. What would you have once they combined? You might correctly think they would annihilate each other and produce energy.

"Wrong!" a dolt could say. "2+2=4. The math PROVES that if you combine them you just get more hydrogen!"

Is his math wrong? No. Are his assumptions? Yes.

Specifically, if it's matter and anti-matter entities, then it should be "(+2H) + (-2H) = (zeroH) + (energy equivalent to 4H).

Take care not to sound like you put personal ridicule before obnjective answers to the article's mathematics and conclusions as posted by chinglu, else he will win the debate on this OP by default. Good luck, and enjoy friendly objective on-topic discussion, everyone. Bye.

Can you please tell me what I said to billvon therein that prompted you to post the following remark?

Undefined said:

Specifically, if it's matter and anti-matter entities, then it should be "(+2H) + (-2H) = (zeroH) + (energy equivalent to 4H).

Take care not to sound like you put personal ridicule before obnjective answers to the article's mathematics and conclusions as posted by chinglu, else he will win the debate on this OP by default. Good luck, and enjoy friendly objective on-topic discussion, everyone. Bye.

Birds of a feather.

No-one has indicated any objections to what I said to billvon therein, except you. Your comment seems a non-sequitur and unjustified by the substance of my friendly urging to caution there to billvon. Can you please clarify what reason you found in it for posting your comment? Thanks.

I know. That's why you don't understand.

The OP contains an article that cannot be refuted.

And my example contained math that cannot be refuted! 2+2=4. Do you deny that? The math stands.

Chinglu, do you understand the difference between an effective rotation and a real rotation?

First off, let's correct rpenner on the experiment proposed by Einstein.
It's ridiculous when you start making mistakes in the sentence before you attempt to "correct" me. Einstein was proposing no experiment. Einstein was addressing educated people who fully well could work out the implications of basic geometric and electromagnetic assumptions.
Andrew Banks said:
To develop the Lorentz transformations (LT), Einstein placed a mirror in the primed frame at some location (x’,0).
....
So already in sentence one, Andrew Banks has botched it by misunderstanding the 108-year-old paper that every physics baccalaureate understands the conclusions of. Einstein was not using primes to distinguish different coordinate systems as is common in relativity textbooks today. He used Latin letters for one system (K) and Greek letters for the other system (k).
[Andrew Banks' vanity-press-published] article claimed Einstein used (x',0) in the context of the moving frame for his LT equations.
That is absolutely what it looks like Andrew Banks is claiming. While Einstein makes it clear that the system called K uses Latin labels x, y, z, and t for coordinates, while the system called k uses Greek labels $$\xi, \; \eta, \; \zeta, \; \textrm{and} \; \tau.$$. Also in sections 1 and 2 Einstein establishes that he is using primed coordinates for different values in the same coordinate system. Thus when he eventually writes x' = x - v t, this only makes sense with x, x', t and v being defined in the same coordinate system -- the only system he has fleshed out at that point, the stationary system K.

Albert Einstein (translated) said:
§ 3. Theory of the Transformation of Co-ordinates and Times from a Stationary System to another System in Uniform Motion of Translation Relatively to the Former

Let us in “stationary” space take two systems of co-ordinates, i.e. two systems, each of three rigid material lines, perpendicular to one another, and issuing from a point. Let the axes of X of the two systems coincide, and their axes of Y and Z respectively be parallel. Let each system be provided with a rigid measuring-rod and a number of clocks, and let the two measuring-rods, and likewise all the clocks of the two systems, be in all respects alike.

Now to the origin of one of the two systems (k) let a constant velocity v be imparted in the direction of the increasing x of the other stationary system (K), and let this velocity be communicated to the axes of the co-ordinates, the relevant measuring-rod, and the clocks. To any time of the stationary system K there then will correspond a definite position of the axes of the moving system, and from reasons of symmetry we are entitled to assume that the motion of k may be such that the axes of the moving system are at the time t (this “t” always denotes a time of the stationary system) parallel to the axes of the stationary system.

We now imagine space to be measured from the stationary system K by means of the stationary measuring-rod, and also from the moving system k by means of the measuring-rod moving with it; and that we thus obtain the co-ordinates x, y, z, and $$\xi$$, $$\eta$$, $$\zeta$$ respectively. Further, let the time t of the stationary system be determined for all points thereof at which there are clocks by means of light signals in the manner indicated in § 1; similarly let the time $$\tau$$ of the moving system be determined for all points of the moving system at which there are clocks at rest relatively to that system by applying the method, given in § 1, of light signals between the points at which the latter clocks are located.

To any system of values x, y, z, t, which completely defines the place and time of an event in the stationary system, there belongs a system of values $$\xi$$, $$\eta$$, $$\zeta$$, $$\tau$$, determining that event relatively to the system k, and our task is now to find the system of equations connecting these quantities.

It would have been better if Andrew Banks had quoted Einstein or at least properly cited the 1905 paper. Also, I would like to point out that I linked to this document first at the bottom of my initial post and I did quote Einstein from the same source you advocate now.

"To any time of the stationary system K"

So, capital K is the stationary system.
That is largely undisputed, with only the minor quibble that scare quotes should be placed around stationary as the whole point of the 1905 paper is to establish that the coordinate system k is physically indistinguishable from "stationary" in light of the assumptions made in the paper. But the four paragraphs at the top of section 3 of Einstein's 1905 paper establish that system K uses lowercase Latin labels (x,y,z,t) while system k uses lowercase Greek labels ($$\xi, \eta, \zeta, \tau$$). At no point does Einstein establish a "primed coordinate system" as that is a convention used by later authors who set up their coordinates in different ways.

Section three then continues like this:
Albert Einstein (translated) said:
In the first place it is clear that the equations must be linear on account of the properties of homogeneity which we attribute to space and time.

If we place $$x' = x - v t$$, it is clear that a point at rest in the system k must have a system of values x', y, z, independent of time. We first define $$\tau$$ as a function of x', y, z, and t. To do this we have to express in equations that $$\tau$$ is nothing else than the summary of the data of clocks at rest in system k, which have been synchronized according to the rule given in § 1.

From the origin of system k let a ray be emitted at the time $$\tau_0$$ along the X-axis to x', and at the time $$\tau_1$$ be reflected thence to the origin of the co-ordinates, arriving there at the time $$\tau_2$$; we then must have $$\frac{1}{2}(\tau_0+\tau_2)=\tau_1$$, or, by inserting the arguments of the function $$\tau$$ and applying the principle of the constancy of the velocity of light in the stationary system:—
$$\frac{1}{2}\left[\tau(0,0,0,t)+\tau\left(0,0,0,t+\frac{x'}{c-v} + \frac{x'}{c+v}\right)\right]= \tau\left(x',0,0,t+\frac{x'}{c-v}\right)$$.
Hence, if x' be chosen infinitesimally small,
$$\frac{1}{2}\left(\frac{1}{c-v}+\frac{1}{c+v}\right)\frac{\partial \tau}{\partial t} = \frac{\partial \tau}{\partial x'}+\frac{1}{c-v}\frac{\partial\tau}{\partial t}$$,
or
$$\frac{\partial\tau}{\partial x'}+\frac{v}{c^2-v^2}\frac{\partial\tau}{\partial t}=0$$.

It is to be noted that instead of the origin of the co-ordinates we might have chosen any other point for the point of origin of the ray, and the equation just obtained is therefore valid for all values of x', y, z.

Then we have,

"From the origin of system k let a ray be emitted at the time t'0 along the X-axis to x', and at the time be reflected thence to the origin of the co-ordinates, arriving there at t'1"
You misquote Einstein, in the important way that he did not write "t" but "$$\tau$$". This is as basic an error as mistaking "丁" for "十". Further the use of x' is an example of metonymy where what is meant is "the point stationary in system k, but uniquely identified uniquely such that at time t=0 the point has coordinates (x', y, z) in system K and at any time t it has coordinates in system K given parametrically as (x' + v t, y, z)." The problem is one of English comprehension because you have to read more than one paragraph to understand what Einstein is saying, so your tiny quote (actually a misquote) does not convey that which was meant.

So, we have lower case k as the moving system as indicated in the article. Then, in the moving system k, a light pulse is emitted from the origin of k along the X axis to x' which means (x',0) just as the article says, and there is a mirror there which reflects back to the origin of the k moving system.

This is exactly what the article says and this is exactly what Einstein says.
Einstein didn't say mirror, he said "a point at rest in the system k" which he endows with the property of reflecting light. Lots of things reflect light other than mirrors. The context around your tiny quote demonstrates your interpretation of Einstein's words to be in error, as he develops a functional relationship between coordinates in system K (t and x') with $$\tau$$ a coordinate in system k.

This proves rpenner is in absolute error.
Reasonable and competent people have disagreed with your interpretation for 108 years.

The only next relevant claim by rpenner is that the unprimed frame claims that the moving mirror strikes the light sphere on the front side so that reflection occurs for both frames even though the mirror is on the positive side of the unprimed frame x-axis when light strikes it and the back side is the non-reflective side is facing the unprimed origin.
That is my conclusion based on the Lorentz transform and the finite propagation speed of light. For the same reason when you are driving in a rainstorm all the rain seems to come at the front windshield, as you are literally driving into the rain. The same phenomenon is seen in astronomy where the motion of the Earth about the sun changes the apparent angular position of the stars in the sky. This is known as stellar aberration and is a predicted consequence of the finite propagation speed of light even in Newtonian mechanics. That Andrew Banks ignores it demonstrates his lack of competence and the lack of competence of the purportedly scientific journal.

Therefore, this beam would exceed c and contradict SR.
That is not the case.

....
Thus because of the finite propagation speed of light, the expanding beam of light sweeps across the object in the same way, hitting the same face, a face which effectively has been rotated.

This interaction between ray-tracing beams of propagating light and the Lorentz transformation is known as Terrell rotation, not to be confused with Thomas precision which is another relationship between rotation and Lorentz transforms.

http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2049244
http://prola.aps.org/abstract/PR/v116/i4/p1041_1
http://math.ucr.edu/home/baez/physics/Relativity/SR/penrose.html

So not only is Andrew Banks completely wrong about Einstein's approach in 1905, but the effect he ignores was demonstrated as a general geometric effect in 1959. Any legitimate scientific venue for publication of matters related to the math of special relativity would have realized this along with many other flaws of this paper.

We already agree in k, the light strikes the mirror and reflects.

What we are talking about is the system K. It does not reflect there.

So, yet again you have been refuted.
That's not a refutation -- that's a repetition of an assertion, but it ignored the argument, geometry and calculation of [post=3096606]my second post[/post].

"Thus because of the finite propagation speed of light, the expanding beam of light sweeps across the object in the same way, hitting the same face, a face which effectively has been rotated."

How did you get the mirror to rotate when it is fixed?
I didn't say rotated, I said "effectively .. rotated" because of the finite speed of light the motion of the mirror affects how light arrives at the mirror. Stellar aberration is the same effect. http://en.wikipedia.org/wiki/Aberration_of_light

You need to prove this assertion.
Thus the argument, geometry and math which you ignore.
Let me see the math that proves the mirror rotates in the unprimed frame while it remains fixed in the primed frame.
It's still there in [post=3096606]my second post[/post].

Chinglu, do you understand the difference between an effective rotation and a real rotation?
Good question!