Recall from previous chapters, Corollary 1 and Theorems 6 and 7 of the time-energy relationship:

The next theorem arises out of the quantization of potential and kinetic energy discussed in a previous chapter. This theorem relates directly to the problem of recurrence:

From Corollary 2 and Theorem 11 the next theorem in the time-energy relationship can also be written:

  This is the principle of quantization of d-particle separation in the temporal frame.

The most important effect of Corollary 1 and these four theorems is "recurrence." This is perhaps, the most informative and comprehensive of all the effects. Recurrence arises out a subject known as "probability," which shall be discussed next, beginning with a dice problem.

Suppose that we have a thousand dice. When we throw these dice, there is a certain probability that all one-thousand dice will come up sixes in the same throw. This probability is extremely remote. If we throw the thousand dice a thousand times, it is highly unlikely that all one-thousand dice will come up sixes even once in all one-thousand throws. If we throw them millions or even trillions of times, it is still unlikely that they will all come up sixes even once. But the probability does exist¹. If however, we throw the dice an infinite number of times, eventually all one-thousand dice will come up sixes. Any probability which exists on an infinite domain must occur, unless there is recurrence in the domain. Now let us play with a much larger set of dice.

Recognizing that every d-particle in the universe at any point in time, has a discrete relative q-velocity, a discrete relative q-position, a discrete relative energy state, and so on, there is a certain probability that this condition of the universe could repeat itself exactly and without imperfection. That is to say that over the course of trillions of eons, somewhere way on down the line every d-particle in the universe will be in the exact same relative q-position, with the exact same relative q-velocity that it has at this very moment. There is a certain probability that this will occur no matter how remote.

In this next statement, I am going to take some liberties with common sense and suggest that the temporal span of the universe is infinite.

Before I can go any further, there is a need to establish how it is that I can claim the temporal domain of the universe to be infinite. During our discussion of nodes in the previous chapter, it was never quite mentioned that the existence of the jump-rope in the problem is seen as the "causality" of the node. In other words, it is the shape, motion and existence of the rope that creates the "appearance" or perception of a non-moving intersection which we call a node. If the node ceases to exist, i.e. the children drop the rope to go play another game, the rope still exists but it is simply not being operated upon (spun).

In this sense, the jump-rope represents the causality required of existence before the node is even able to exist. In the metaphorical sense, the ends of the rope represent the beginning and end of time in this universe-as a completed causality constructed of many d-particles. From the viewpoint of the node, the ends of the rope (and this can be shown mathematically) are identical to each other. While one end of the rope is the beginning, the other is the end. But since these two are the same from the node's viewpoint, the effective length of the rope is infinite. (It is also incidentally, quantized-which statement coincides with previous arguments and helps to support subsequent conclusions about the nature of the universe.)

The object of probability in this case, is whether or not the universe will repeat itself. If the universe temporally continues throughout infinity, then all conditions and configurations of the universe must eventually occur.

The only thing that will prevent all conditions and configurations in the universe from occurring is a recurrence. No other occurrence, including the annihilation of the universe, will prevent all probable configurations from existing. From our understanding of nodes then, we can easily see that in a cyclic universe, annihilation implies creation. Creation and annihilation may go on an infinite number of times (as could a node in a jump-rope). (The proposed "Big Bang" may occur trillions of times between creation and annihilation-or not.)

Put simply, the life-span of the universe is fixed. While it may exist in the infinite domain, it has a cycle. The universe repeats itself on a fixed cycle.

I may conclude then, that we and our universe have been here many times in the past, and will be here again many times in the future (from the viewpoint of any individual d-particle). I can now write the next theorem relating to the Time-Energy Theory.

As a final note to this chapter, I examined the quantization of d-particle separation and found it to be a means of establishing recurrence within a closed system (i.e. the universe).

Obviously, since d-particles within the nucleus of an atom are closely packed together, there should be some basic unit of quantum separation. Naturally, electrons "orbiting" a nucleus also obey this rule of basic quantum separation. Since space does not exist, I therefore refer to this as a quantum separation in time as is recalled from Theorem 12.

When a closed system of d-particles exists in which the d-particles are in q-motion and q-moving quantumly or in discrete steps, the number of d-particles must be fewer than the number of q-positions that the d-particles can occupy. This being the case, a three-dimensional grid or matrix can be used to map the possible q-locations of d-particles.

With the introduction of the Time-Energy Theory, there are now two coordinate systems, transformations between which will be developed at a later point in time (not in this book). The first and most obvious is the quasi-spatial (q-spatial) matrix, which has its hold in the relative frame. The second and most important is the temporal matrix which holds itself to the absolute frame. This is the frame in which the q-speed of light is always the same (as opposed to the q-speed of light in a vacuum).

The temporal matrix and the q-spatial matrix are counterparts in the absolute and relative frames respectively. The q-spatial matrix appears to be a myriad jumble of individual activities, but is in fact, a result of the inner workings of the temporal matrix. The temporal frame is perfectly inelastic and d-particles conform to the potentials of their respective q-positions. Instability occurs when the temporal matrix is "loaded up." While the temporal matrix cannot be overloaded (because this would violate the existence of the universe), crowding of d-particles in a temporal zone can cause instability. In the heavier elements, part of this effect is seen as spontaneous emission (radiation), and can be predicted.

Large particles such as protons and neutrons have the appearance of being "bunched up" in the q-spatial matrix. They (or their constituent d-particles) are in fact, discretely separated in a temporal matrix.

This temporal spacing follows the same rules in the nucleus of an atom as it does outside the nucleus. Temporal electron spacing is the same as proton and neutron separation. Accounting for the difference in quasi-spatial separation is partly the subject of the next chapter.

Go to Chapter 12

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