Einstein uses his distance to the mirror x' with which
to derive the differential equation and tau function from
which he derives the t' and x' transforms of Special Rela-
tivity. The greater that distance, the more time it takes
for the light to travel either direction, and roundtrip.
But Einstein concludes that the slope of tau wrt the dist-
ance to the mirror is the inverse of the slope wrt the time
it takes.

Einstein's x' is the distance to the mirror, which also
defines the distance back to the source at the moving origin.
This distance shows up in the time expressions in his un-
known tau functions, and when differentiated wrt x' gives
a value of 1.00, proving that the x' of the @tau/@x' term
is indeed the distance to the mirror and not the other x'
in his model (yes, there are two; the other is the location
of the light and/or the clock in use at the time).

The greater the distance, the greater time it takes
for light to cover the total and part-wise distances.

But Einstein's differential equation and his resultant
tau equation say that although tau increases when the
distance increases, tau decreases when time increases,
and vice versa.

His differential equation is:

    (@tau/@x') + (v/(cc-vv))(@tau/@t) = 0.

We put the two terms on opposite sides:

    (@tau/@x') = - (v/(cc-vv))(@tau/@t).

Thus, either v must always be negative or the slope
of tau with respect to x' is the negative of the slope
of tau with respect to t. Yet, his model - for that
very x' - is that x' and v together fully define t,
and that the time - with a constant v, which is how
Einstein treated v - increases as x' increases.

This aburdity is repeated in his immediately
consequent tau function:

    tau = a(t-vx'/(cc-vv)).

There can be no doubt that the x' in the differential
equation and the resultant tau function are the x'
that is the distance to the mirror. When he different-
iates the time expressions in his unknown taus wrt x',
the slope of that distance x' is 1 wrt to the differen-
tiating x'.

QED, by reduction to the absurd, his derivation of the
SR transformations is nonsense. It is based on a model
in which tau increases with a greater x' and/or a greater
t - t being an increasing function of an increasing x'
- but Einstein's conclusion is that tau increases with
one when it decreases with the other.

Objection:

But, you say, you said there were two x' usages.
Surely the tau at the time the light returns to the
moving origin, at location L=0, is later than the tau when
light reaches the mirror at L=x'. That's a negative
relationship.

OK. That is saying tau is an obvious inverse function of
the location coordinate.

But the tau at emission is surely less than at either
of the other occasions, and its L is zero also, making
it a direct function of the location coordinate, by
the same argument.