It is shown below that the quantity

Quotations from Evans' articles or book appear in
**black**

In his GCUFT book Vol.1 [5;Chap.J.3] M.W.Evans considers the differential equation

o q_{λ}^{a} =
∂^{μ}(Γ_{μ}^{ν}_{λ}
q_{ν}^{a})
−
∂^{μ}(ω_{μ}^{a}_{b}
q_{λ}^{b}) .
(J.30)

As abbreviation for the right hand side of (J.30) we introduce the quantities

(1)
R_{λ}^{a} :=
∂^{μ}(Γ_{μ}^{ν}_{λ}
q_{ν}^{a})
−
∂^{μ}(ω_{μ}^{a}_{b}
q_{λ}^{b}) ,

which allows us to write the differential equation (J.30) in the much shorter form

(2)
o q_{λ}^{a} = R_{λ}^{a} .

The **statement of the Evans Lemma** is the assertion that a scalar factor *R* exists
such that

*R* q_{λ}^{a} =
∂^{μ}(Γ_{μ}^{ν}_{λ}
q_{ν}^{a})
−
∂^{μ}(ω_{μ}^{a}_{b}
q_{λ}^{b}) .
(J.31)

or in our abbreviated notation

(3)
*R* q_{λ}^{a} = R_{λ}^{a} .

However, Evans believes the existence of *R* to be obvious, which,
as we shall see below, is a fatal error.

*R* shall be the proportionality factor between the two 4×4 matrices,
Q = (q_{λ}^{a}) and
P = (R_{λ}^{a}).

Q is the invertible tetrad matrix Q = (q_{λ}^{a}),
the inverse of which is called here
Q^{−1} = (q_{a}^{λ}), hence
q_{a}^{λ} q_{μ}^{a} =
δ_{μ}^{λ}
and for later use

(4)
q_{a}^{λ} q_{λ}^{a}
= δ_{λ}^{λ}
= 1 + 1 + 1 + 1 = 4

under Einstein summation convention.

Evans' Lemma means that a factor *R* exists such that *R* Q = P,
i.e. both matrices are *elementwise* proportional
with the *same* propertionality factor *R* for *all* matrix elements.

Of course, such a factor *R* cannot exist in general, since that
*one* real value *R*
would be *overdetermined* by the matrix equation *R* Q = P,
which is equivalent to 4×4 = 16 scalar equations. Thus the
*degree of overdetermination* is 16:1.

Nevertheless, M.W. Evans knows such a factor. From [6; Equ.(2)] we learn

*R* = q_{a}^{λ} R_{λ}^{a} .
[6;(2)]

**The solution of that enigma:**

We have to distinguish between the *special* case __ a__ where the matrix P is

Case **a**

If both matrices are proportional, P = *R* Q, then the factor
*R* can be determined: We "multiply" the indirect definition of *R*,
the equation (J.31), by
q_{a}^{λ} to obtain

*R* q_{λ}^{a} q_{a}^{λ}
= q_{a}^{λ} R_{λ}^{a}

and due to (4)
we obtain *R*·4
= q_{a}^{λ} R_{λ}^{a}
or

(5)
*R*
= ¼ q_{a}^{λ} R_{λ}^{a} .

Comparison with [6;(2)] shows that the factor ¼ is missing there.
A *minor* error only, that occasionally happens when handling tensor equations.
Of course, to be corrected.

Case **b**

What, if the matrix P is NOT proportional to Q, which is the general case:
Then there doesn't exist any proportionality factor *R*. The equations
*R* q_{λ}^{a} = R_{λ}^{a}
determine different values *R* depending on the indices λ and a.

But the corrected equ. (5) determines some value *R* nevertheless.
What, if we use that value?

"No proportionality" means that however the value *R* is chosen
at least one index position (a,λ) can be found such that
*R* q_{λ}^{a} ≠ R_{λ}^{a}.
Then for that special index position we have

o q_{λ}^{a} =
R_{λ}^{a} ≠
*R* q_{λ}^{a}

This means that:

If P is NOT proportional to Q, then (5) yields an irrelevant value.

Evans' "nice" ECE Lemma does not hold with the consequence that the EVANS Field Equation is wrong too. One essential column of the EVANS-Cartan-Einstein-Theory (ECE) has broken down.

Due to a simple flaw of thinking! Think first, write later.

References

[1] M.W. Evans, SUMMARY OF BRUHN REBUTTALS,

http://www.mathematik.tu-darmstadt.de/~bruhn/asummaryofbruhnrebuttals.pdf

[2] G.W. Bruhn, Refutation of Myron W. Evans’ B^{(3)} field hypothesis,

http://www.mathematik.tu-darmstadt.de/~bruhn/B3-refutation.htm

[3] G.W. Bruhn, Myron W. Evans' Most Spectacular Errors,

http://www.mathematik.tu-darmstadt.de/~bruhn/MWEsErrors.html

[4] M.W. Evans, Generally Covariant Unified Field Theory, Vol.1,

Arima Publishing 2005

[5] M.W. Evans, Some Subsidary Relations from the ECE Lemma (pdf),

http://www.mathematik.tu-darmstadt.de/~bruhn/anecelemmacrosscheck.pdf