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### Definitive Proofs

Proofs of the basic mathematical principals of ECE Theory are offered here.  These proofs are proofs of Riemann and Cartan geometry, and are important to study and master. They are as follows and give much more detail than that found in any textbook that I know of.

1. Proof of the anti-symmetry of the connection
2. Proof that curvature implies torsion and vice versa
3. Proof of the tetrad postulate
4. Self checking proof of the Cartan Bianchi identity
5. Self checking proof of the Cartan Evans dual identity
6. Example of re-labeling of summation indices:
7. New General Condition for any metric:
8. Basic Hypothesis of Gravitational Physics Flow Chart

An understanding of ECE theory can only be obtained if these proofs are mastered. This study will also allow the reader to see how the harasser cell perpetrates its fraud by use of deliberate errors. Without knowledge of Cartan and Riemann geometry the reader is left in a perpetual void, or state of uncertainty, and this is what the fraudsters thrive on. This scientific fraud is similar to forgery of a painting in the art world.  Spanish translation of the proofs and flow charts can be found in the Spanish Section.

Proof of the anti-symmetry of the connection:
This is a student level exercise in Riemann geometry. By definition the commutator of co-variant derivatives is anti-symmetric in its indexes.

Proof that curvature implies torsion and vice versa:
This is definitive proof two, giving details which are left out by Carroll in his chapter three. It shows that spacetime torsion is always present in any spacetime in any dimension, irrespective of any other assumption such as metric compatibility or tetrad postulate. This proof cannot be "changed" by invoking mysterious symbolism as does Rodrigues, whose professional reputation has self-destructed as a consequence of years of misrepresentation of geometry and arrogant bombast. Similarly Bruhn and Jadczyk, or anyone who tries to misrepresent Riemann or Cartan geometry. There is no further need to read the stuff turned out by these people, but it must always be pointed out that they are misrepresenting mathematics and corrupting science. Editors who cite their stuff do not know what they are doing.

The tetrad postulate is the very fundamental requirement that the complete vector field be independent of the way in which it is defined by its components and basis elements. It has been used since 1925 and is used in any proof of Cartan geometry. Suddenly and mysteriously, the tetrad postulate started to become "debatable" as soon as I started to use it. If others such as Carroll use it, it is OK, but if my colleagues and I use it, it is not OK. So this is blatant fraud perpetrated by the same well known people for years. In definitive proof three I will give the proof given by Carroll in his chapter three, and again give more detail than Carroll, in fact complete detail of the proof. With a bit of effort and practice, these proofs are not difficult for trained mathematicians (e.g. A level students). Physicists and chemists are normally expected to have an A level in mathematics and to do university undergraduate courses in mathematics. So there is really no excuse in saying that these proofs cannot be understood. That is the kind of thing the fraudsters thrive on.

Self checking proof of the Cartan Bianchi identity:
This was first given in paper 15 of the ECE source papers about five years ago (www.aias.us) and no genuine mathematician has objected to that proof. This is hardly surprising because it is used in standard student courses in Cartan geometry. Later proofs of the identity were given in papers 99 ff. as overviewed on the ECE Sci Topics site (now also available on www.aias.us by clicking on "Myron Evans"). The homogeneous field equations of dynamics and electrodynamics are based directly on this identity, first given by Cartan in about 1925, and taught ever since. These are given in vector format in the ECE engineering model, which has been coded up. Using this model, patents have been written and applied for. In my shorthand notation the identity is

D ^ T := R ^ q := q ^ R ------------------ (1)

and brings out the fact that torsion (T) is linked ineluctably to curvature (R). So we can see that if torsion is omitted as in the now obsolete standard model then something is bound to go wrong. That something was discovered in papers 93, 95, and 120 for example using computer algebra. Again, no genuine mathematician has objected to those papers, written by four authors in total. This identity is probably Cartan's most elegant theorem, because it shows that the cyclic sum of three curvature tensors is identically equal to the same cyclic sum of the definitions of the same three curvature tensors. Eq. (1) is a most elegant expression of this result of geometry. In order to arrive at this result, the tetrad postulate is used as always in Cartan

Self checking proof of the Cartan Evans dual identity:
This will be the proof of the Cartan Evans dual identity, the basis of the inhomogeneous field equations of dynamics and electrodynamics in ECE theory. It was used in papers 93, 95 and 120 to show that the Einstein field equation and all its solutions violate geometry, a catastrophe for the standard model. Since 2007, there have been no genuine objections to this work, it is based on the use of computer algebra. Any attempted "refutation" of this standard student level work is mathematical fraud. It is of great importance to reveal this corruption of science and these five proofs are all that is needed. No attempt should be made to read the garbage being thrown at geometry by the standard fringe, as Prof. Dunning-Davies points out, this is personal baiting, a violation of ethics, those of communication and mathematics.

Example of relabeling of summation indices:
This is known as relabeling of summation indices. Sometimes these are known as dummy indices. This procedure occurs in many proofs of Riemann and Cartan geometry.

New General Condition for any metric:
Horst Eckardt and I feel that this looks like a useful new result which can be used to test metrics from the Einstein equation. It simplifies the metric compatibility condition to one where the ordinary partial derivative can be used. The whole of Cartan's geometry can be developed in this way.

Basic Hypothesis of Gravitational Physics Flow Chart
This is the second ECE hypothesis which leads to an economic description of all planar orbits, including those of galaxies, and links in to the highly developed subject of angular momentum theory (P. W. Atkins, "Molecular Quantum Mechanics", many editions, M. W. Evans and J.- P. Vigier "The Enigmatic Photon" on www.aias.us Omnia Opera, and M. W. Evans (ed.) "Modern Non-Linear Optics" (2001), two reviews of which are on the Omnia Opera). This means that the anti-symmetric connection can also be developed in as many ways as angular momentum theory can be developed, revealing after one hundred years of incorrect gravitational theory the true meaning of spacetime connection. It is the spinning of spacetime.