There are several significant results that have emerged from the analytical processes pursued in this paper. The most important is that, accepting both Planck's and de Broglie's hypotheses, it has been shown that a matter particle possessing a spatial velocity can exhibit a dual existence not only in the spatial, but also in the temporal dimension of D₀, in both corpuscular and wave function form.

The spatial component of the dual existence, as determined herein, is particularly significant for two reasons. Firstly, it obviously supports this effect as the explanation for the fact that electrons, protons, neutrons and even molecular particles can produce interference patterns when fired through an appropriately sized lattice grid, in exactly the same manner as electromagnetic radiation. Secondly, and perhaps more importantly, the spatial matter wave component, when considered as that of an electron in a Bohr atom, will provide theoretical justification of the quantisation process of the orbit energy levels in that theory. This will permit the resurrection of the Bohr/Sommerfeld theory of atomic structure, as an alternative to that of modern quantum mechanics. This will form the subject of a future series of papers.

The derivation of the matter wave characteristics of a stationary particle showed that this wave propagates entirely along the temporal axis. This was also effectively demonstrated in the discussion on energy where in (3.2) if the spatial velocity is put to zero, the total energy of the stationary particle, the rest mass energy, is seen to be all temporal. This explains why the vast amount of energy contained within matter cannot presently be tapped in the spatial dimension. It also explains why the possible future extraction of this energy, as demonstrated in [3], would require the application of a temporal force. However, when a mass is in spatial motion and its Existence Velocity Vector has rotated towards the spatial dimension, it is seen that the energy projected into the spatial dimension is greater than just the kinetic energy acquired due to the motion. Perhaps this may, eventually, provide easier access to the rest mass energy of matter.
The analysis and discussion on the spatial - temporal distribution of energy in this paper has been somewhat superficial because the main subject here was the characteristics of de Broglie matter waves in Pseudo-Euclidean Space-Time. The two subjects are related via Planck's quantum energy hypothesis. A more in depth dissertation on the spatial - temporal distribution of energy will be the subject of a future paper.

The Compton effect confirms that light certainly conforms to Planck's hypothesis of energy quanta, but also confirms that it can exhibit the characteristics of a particle, i.e. a photon. The analysis here agrees with other approaches, in appearing to necessitate that the photon possesses zero rest mass energy, in order for its propagation velocity to avoid a relativistic increase in mass. However, it is considered that this creates an anomaly, because, if the photon is to be regarded as a particle as well as an electromagnetic radiation wave function, then it must possess a mass in accordance with Einstein's classic energy-mass relationship. This anomaly is however, removed if the mechanism causing photon emission is via a natural application of a combined spatial/temporal accelerative force, constant in magnitude, but with a spatial/temporal direction vector that was a function of its spatial velocity. In this case its mass rate variability would be zero. Such a process was the detailed subject of [3] Section 2.3. As a consequence, the photon could possess a rest mass energy given by the second term on the RHS of the first expression in (2.10). This energy would be completely converted to kinetic energy via the process of [3] Section 2.3 when photons attain light velocity. Whether they can exist as free particles in a spatially stationary state, or at velocities lower than that of light is unknown, but if so, they may well exhibit the characteristics of particles similar to the neutrino.

The manner in which kinetic energy was stored by an accelerated mass was, as in the literature and in [1], shown to be the relativistic increase in mass. The actual method of storage was not addressed. This paper has however, shown that this method is by an increase in the matter wave frequency of the body in motion. The frequency in question is that of the matter wave propagating along the path of the body's Existence Velocity Vector, and the projection of this into the spatial dimension. The storage of kinetic energy by this means is, on the basis of Planck's and de Broglie's hypotheses, believed to be the final answer to this issue.

The clarification provided for the so called phase velocities of matter waves, has it is believed, removed an awkward anomaly. Einstein's criterion of the maximum spatial velocity of propagation would be contravened, as would the primary criterion of existence in D₀, if these parameters truly existed. This is irrespective of statements such that these terms have no physical meaning. These parameters do not exist for the reason stated herein, and have been replaced by the projected terms as shown and discussed.

When taking the root of a squared term, of course both positive and negative results are mathematically permissible. But this is a mathematical concept only and may not apply to the physics of the problem depicted. The derivation of Einstein's relativistic energy - momentum relationship presented herein, has shown that it contains both spatial and temporal components. The mathematical magnitude of this relationship therefore can only have a positive value, because both the spatial and temporal components are positive. The consequence of this is that within the Relativistic Space-Time Domain that is D₀, Dirac's 'sea of negative energy' cannot exist. In [1] D₀ was shown to be in fact, Pseudo-Euclidean Space-Time and therefore, gravitation apart, the Domain in which all things, down to atomic level and below, exist. Consequently, if accepted, this would have serious implications in the field of quantum electrodynamics.

Finally, there is the question of the nature of matter waves themselves. It has been stated, [4], that matter waves are distinctly different from electromagnetic waves and propagate at different, (lower), velocities. However, this reference unfortunately contains a number of grammatical and mathematical errors and may not be a reliable reference. The presentation here indicates that matter waves propagate at the velocity of light. Their precise nature however, remains unclear.