with comments by G.W. Bruhn

Evans' notes deal again with the "proof" of the ECE Lemma (formerly
denoted as Evans Lemma, now even attributed to **E**instein and **C**artan) following the same line of argumentation as
in his recent
Bruhn rebuttal.
Following a
request of Gareth Evans for pointing to "the exact point of fraud"
(in Evans' rebuttals)
I skip some well-known lines of Evans' notes and point to the essential eq. (8):

q^{a}_{λ} := R q^{a}_{λ}
(8)

Evans considers eq. (8) with a=0,1,2,3 and λ=0,1,2,3 as definition of the scalar R.
It 's hard to grasp for Evans (I can understand this well, since this objection touches
the basis of his Lemma, a question of To Be Or Not To Be) that eq. (8) contains
16 different conditions for the *one* scalar R, which will not agree in general and thus,
a value R fulfilling all 16 conditions will not exist.

The reader is therefore requested for patience when we completely list the sixteen equations now:

q^{o}_{o} := R q^{o}_{o} ,
q^{o}_{1} := R q^{o}_{1} ,
q^{o}_{2} := R q^{o}_{2} ,
q^{o}_{3} := R q^{o}_{3} ,

q^{1}_{o} := R q^{1}_{o} ,
q^{1}_{1} := R q^{1}_{1} ,
q^{1}_{2} := R q^{1}_{2} ,
q^{1}_{3} := R q^{1}_{3} ,

q^{2}_{o} := R q^{2}_{o} ,
q^{2}_{1} := R q^{2}_{1} ,
q^{2}_{2} := R q^{2}_{2} ,
q^{2}_{3} := R q^{2}_{3} ,

q^{3}_{o} := R q^{3}_{o} ,
q^{3}_{1} := R q^{3}_{1} ,
q^{3}_{2} := R q^{3}_{2} ,
q^{3}_{3} := R q^{3}_{3} ,

Evans did never show, that each of these 16 equations determines the *same* value of R,
and so the scalar R is not defined by that way, unless Evans (or someone else) proves the
*compatibility* of these 16 equations.

Subsequently Evans attempts to justify his argumentation by comparison. Though this is irrelevant for closing Evans' gap, we shall have a look at and comment on it: His (16-fold overdetermined) definition of the scalar R is:

R q^{a}_{λ} =
∂^{μ}(Γ^{ν}_{μλ}q^{a}_{ν}−ω^{a}_{μb}q^{b}_{λ})
(9)

i.e.

R := ¼q^{λ}_{a} ∂^{μ}(Γ^{ν}_{μλ}q^{a}_{ν}−ω^{a}_{μb}q^{b}_{λ})
(10)

because:

q^{a}_{λ}q^{λ}_{a} = 4 .
(11)

The definition (11) is in all ways similar to

the definition used by Einstein:

R = g^{μν} R_{μν}
(12)

In eq. (10), as is eq. (12), summation over

repeated indices is used.

__The Exact Point of Fraud__

this occurs when Bruhn states that there are

many definitions of R in eq. (10). If this were

true, Einstein's own definition of R in eq.(12) would

be many definitions. __This is of course pure nonsense.__

In eq. (12) R_{μν} is the Ricci tensor, g^{μν} the

metric and R the scalar curvature.

Eq. (12) is the analogue to eq. (10).

You are in error, since your step from eq. (9) to eq. (10) is not
allowed, unless you have proven the compatibility of the 16 eqs. (9).
However, there is no such problem with eq. (12) and hence *no similarity*
with your problem with the transition (9)→(10).
**Thus, your above objection is unjustified.**