What is / are the primary criticism(s) against Einstein-Cartan-Evans field theory? What is / are the primary criticism(s) against Einstein-Cartan-Evans (ECE) field theory?
On Wikipedia the references provided were: arXiv:physics/0607186, MR2372785 (2008j:83049b), MR2218579 (2007a:83003), MR1832162 (2002d:78002).
I'm searching for an undergraduate-level* explanation of the criticisms / details.
(The first link is openly accessible [i.e. arXiv], but the others are from MathSciNet; (as I don't have access to MathSciNet, I'm wondering if someone with access may provide their take on the contents.))
(*To be specific, I have background in the following courses: Physics: First-year Mechanics, Basic E&M (but not electrodynamics); Math: Calculus I, Linear Algebra, some Differential Equations.)
 A: Evan's work is not worth reading. The easy crticism is that the theory is not Lorentz invariant, which is in the paper you linked. To understand this, it is enough to see that the E and B field together form an antisymmetric tensor in relativity (this is the F tensor discussed in Volume 2 of the Feynman Lectures, if you need a good source). Adding a new component to B, without doing anything to E, violates this principle.
But the real problem here is that Evans is introducing a new component of B without giving any procedure to detect or measure this new field. It is important when defining a field to give a positivistic formulatin, to tell someone how to detect it. In this case Evans is simply postulating a field so that he can rotate B fields into each other with ordinary rotations.
The theory is either vacuous or wrong, depending on interpretaiton.
A: I asked Dr. Evans for his remarks on this topic, particularly the singular answer that was posted so far.  I wanted to share his input simply for the sake of your own reference and understanding:

This kind of nonscientific activity has been answered thoroughly many
  times for almost fourteen years, for example UFT89. 
The first remark is a famous epitaph "Evans' work is not worth
  reading". It is read by tens of millions. The second remark shows that
  this vessel of profound wisdom knows nothing about covariance. ECE is
  based directly on Cartan geometry, which is generally covariant, so it
  is automatically Lorentz covariant. So I stopped reading at that
  point.

He expounded further, but I felt that it was on a more personal note and I did not feel it appropriate to share.
He did note that:

I now expect people to apply ECE themselves to a problem of interest,
  then send me their paper. This is what was expected of me as a
  graduate student. The AIAS Fellows are able to do this because they
  all have an understanding of ECE theory. By monitoring the
  scientometrics it can be seen that these pseudocriticisms have come to
  a halt. By extrapolating the scientometrics, it is seen that ECE is
  going to be read by the best in the world for the indefinite future.

He also notes how critical it is for people who are sincerely interested to actually invest the resources to read and understand the source papers.
A: The ECE theory is a breakthrough that can be improved in certain ways. The criticisms in wikipedia and in the papers published are just B.S. Carrying out certain experiments, we reach the conclusion that the extra term is correct and experiments match the results of ECE better than the classical one. We will work on that, but convince others is not our primary goal at the moment. What mattered is that ECE helped to understand certain experimental facts that the classical theory fails.
A: ECE Theory is the only theory that can explain spiral galaxies, orbital precession, hubble red shift, photon deflection by gravity, etc, etc.  When Einstein wrote his field equation, in 1915, Riemannian torsion was not discovered yet. So, he built his theory on an incomplete model of spacetime. It is like a toy maker, who designed a children's toy, but none of the parts can rotate, they can only translate! No child would play with that toy, since it is built WRONG! Nature has translation AND rotation, but General Relativity completely omits rotation.  The last century of physics has been crippled by the omission of torsion.
