1) Let's start with your misunderstanding of Einstein's thought experiment which motivated LT construction. You continue to claim somehow that Einstein meant x' applies to the "stationary" K system.

It is my assertion, based on ALL of the text in sections 1, 2 and 3 of the translated 1905 paper, that lowercase Latin letters (x,y,z,t) refer to coordinates in the "stationary" K system and that the lowercase Greek letters ($$\xi, \eta, \zeta, \tau$$) refer to coordinates in the system k which has the property that things described as "at rest" in k have constant velocity of v in the X direction in system K. Further, Einstein will use both primes and subscripts to talk about different values of coordinates in the same system. Further, I assert that because you can only add or subtract values in the same coordinate system and get sensible results, Einstein never adds or subtracts x' with any coordinate in system k.

Einstein introduces in section 3 "point at rest in the system k". What is that point's description in system k?

It is : $$\xi = \xi(t) = \xi(0), \quad \eta = \eta(t) = \eta(0), \quad \zeta = \zeta(t) = \zeta(0)$$

And what is that point's description in system K?

It is : $$x = x(t) = x(0) + v t, \quad y = y(t) = y(0), \quad z = z(t) = z(0)$$.

So when Einstein introduces x', we have "x' = x - v t" with the obvious interpretation:

$$x' = x - v t = x(t) - v t = x(0) + v t - v t = x(0)$$

because v only has meaning to the point in system K, the system where the point is not at rest.

So when Einstein writes: "If we place x'=x-vt, it is clear that a point at rest in the system k must have a system of values x', y, z, independent of time." he is saying ANY point at rest in system k will have a unique description in system K coordinates if take $$(x', y, z) = \left( x(0), y(0), z(0) \right)$$.

Only after introducing this $$x' = x(0)$$ concept does he begin to explore how we might learn the values of $$\xi, \eta, \zeta, \tau$$ beginning with $$\tau$$.

Einstein wrote,

Albert Einstein (translated) said:

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:—

$$\frac{1}{2}\left[\tau(0,0,0,t)+\tau\left(0,0,0,t+\frac{x'}{c-v} + \frac{x'}{c+v}\right)\right]= \tau\left(x',0,0,t+\frac{x'}{c-v}\right)$$.

This equation only holds if x' is the location of the mirror and it is in the "stationary" lower case k system, which is moving relative to K.

Because you cut off the quote in mid-sentence, I have completed the quote for you.

Here Einstein is talking about leaving one object (the origin of k) hitting another object (our point on the X-axis which is coincident in both systems) and returning to the first object. Both objects are at rest in k, but we only have descriptions of them in system K. Thus we have for the origin:

$$x_O(t) = 0 + v t, \quad y_O(t) = 0, \quad \z_O(t) = 0$$

and for the point:

$$x_P(t) = x_P(0) + v t, \quad y_P(t) = 0, \quad \z_P(t) = 0$$.

Einstein then posits a linear functional relationship between between descriptions in K of points at rest in k and time in K with time in k

$$\tau\left( x(t) - vt, y(t), z(t), t \right) = \tau\left( x(0), y(0), z(0), t\right) = \tau(x', y, z, t)$$

because Einstein established that $$x' = x(0) = x(t) - v t$$ was a time-independent description of the point in system K.

$$\begin{array}{c|cccc|cccc|c}

Event & x & y & z & t & x' & y & z & t & \tau

\hline

0 & x_O(t) = v t & y_O(t) = 0 & z_O(t) = 0 & t & vt - vt = 0 & 0 & 0 & t & \tau(0,0,0,t)

1 & x_P(t + \Delta_1) = x_P(0) + v t + v \Delta_1 & y_P(t + \Delta_1) = 0 & z_P(t + \Delta_1) = 0 & t + \Delta_1 & x_P(0) + v t + v \Delta_1 - v ( t + v \Delta_1 ) = x_P(0) & 0 & 0 & t + \Delta_1 & \tau(x_P(0), 0,0,t + \Delta_1 )

2 & x_O(t + \Delta_1 + \Delta_2 ) = v t + v \Delta_1 + v \Delta_2 & y_O(t + \Delta_1 + \Delta_2 ) = 0 & z_O(t + \Delta_1 + \Delta_2 ) = 0 & t + \Delta_1 + \Delta_2 & v t + v \Delta_1 + v \Delta_2 - v ( t + \Delta_1 + \Delta_2) = 0 & 0 & 0 & t + \Delta_1 + \Delta_2 & \tau(0,0,0,t + \Delta_1 + \Delta_2 ) \end{array}$$

Because $$c \Delta_1 = x_P(t + \Delta_1) - x_O(t) = x_P(0) + v \Delta_1$$ it follows that $$\Delta_1 = \frac{x_P(0)}{c - v}$$ (which follows only from coordinates in system K and the speed of light in system K. Likewise $$\Delta_2 = \frac{x_P(0)}{c + v}$$, and so we have:

$$\tau_0 = \tau(0,0,0,t), \quad \tau_1 = \tau(x_P(0), 0, 0, t + \frac{x_P(0)}{c-v}), \quad \tau_2 = \tau(0, 0, 0, t + \frac{x_P(0)}{c-v} + \frac{x_P(0)}{c + v})$$ which allows Einstein to finish his sentence as a coherent inference from assumption. So not only does Einstein's textual use of subscripts and primes imply that he is using the same coordinate system (K) as the unprimed Latin letters, but his math requires it.

Therefore, as the article noted, the mirror is located at (x',0) in the primed system.

This was not the conclusion of Einstein's section 3.

So, you are wrong here as I have explained over and over.

You assert over and over, but your arguments are based on misconceptions and fail to hit the target.

2) Let's consider you silly rotating mirror hypothesis. That means when we are at rest on the earth, a moving car will rotate. That is complete stupidity. Further, if you are correct, then from system K for Einstein, the mirror at (x',0) in the primed frame must rotate so that the K system believes no light was reflected hence SR is false. Very good.

It appears you still haven't grasped what I mean by "effective".

Another way to show this is by using a parametrized circle instead of a point-like detector, and the math shows the light comes in at the same parametrized point on the circle in both frames, thus the light must hit on the same "side" as defined in a way that makes physical sense.

I see that this Andrew Banks/chinglu thread has properly been moved to pseudoscience -- the category of those that only ape the form and not the substance of science, so I believe that constitutes a formal evaluation of those worthless ideas.