The Larson Research Center

Now that we have seen that combining our SUDRs and TUDRs into SUDR|TUDR combos (S|T units) leads to theoretical entities with interesting QM-like properties such as probability densities, and measurement-dependent reality, and that these entities subsequently may be combined together to form preons, which appear to have the “directed” magnitudes of observed physical properties, such as photons, neutrinos, electrons, positrons, protons, and neutrons, as understood in the QM standard model of fermions and bosons, we want to delve into the quantitative aspects of these concepts.

The beginning point of our quantitative development is deciding what the natural units of the system are. Following Larson, we choose the Rydberg frequency of hydrogen and the speed of light, as the two physical constants we need to define the natural unit of the system. The reason that we don’t need three natural constants to do this is that we interpret the Rydberg frequency as a velocity, due to the central role of the definition of motion in the system, and this gives us both a space and a time constant, not just a time constant. Larson explains this in Chapter 13 of NBM:

From the manner in which the Rydberg frequency appears in the mathematics of radiation, particularly in such simple relations as the Balmer series of spectral lines, it is evident that this frequency is another physical manifestation of a natural unit, similar in this respect to the speed of light. It is customarily expressed in cycles per second on the assumption that it is a function of time only. From the explanation previously given, it is apparent that the frequency of radiation is actually a velocity. The cycle is an oscillating motion over a spatial or temporal path, and it is possible to use the cycle as a unit only because that path is constant. The true unit is one unit of space per unit of time (or the inverse of this quantity).

So, what we are assuming here is that the Rydberg frequency of hydrogen atom gives us the unit of time (space) it takes to complete an expansion/contraction cycle in a SUDR (TUDR); that is, it takes one unit of time (space) to expand the point in all directions to the unit sphere, and it takes another unit of time (space) to contract the unit sphere in all directions to the point, completing one cycle. Thus, “this is the equivalent of one half-cycle per unit of time (space) rather than one full cycle, as a full cycle involves one unit of space (time) in each [‘direction’],” as Larson explained it. In other words, since we are assuming that a cycle of the Rydberg frequency is the natural unit of space per natural unit of time once in each “direction,” the unit of time (space) of the frequency is twice the natural unit of time (space) in the velocity (inverse velocity). Thus, we need to double the value of the Rydberg constant, as Larson explains:

For present purposes the measured value of the Rydberg frequency should therefore be expressed as 6.576115 x 10¹⁵ half-cycles per second [2 x Ry = 3.28… x 10¹⁵].

The final step, then, is to take the reciprocal of this number, which gives us the natural unit of time, and to multiply the newly obtained natural unit of time times the speed of light, assuming the speed of light is the natural unit of motion, to give us the natural unit of space, as Larson explains:

The natural unit of time is the reciprocal of this figure, or 1.520655 x 10^-16 seconds. Multiplying the unit of time by the natural unit of speed, we obtain the value of the natural unit of space, 4.558816 x 10^-6 centimeters.

Of course, the accuracy of the measurements of these physical constants has improved since Larson’s day, and the new values will be used. They are

1. Rydberg constant for hydrogen, Ry = 1.09678 x 10⁵ cm^-1
2. Speed of light, c = 2.99792458 x 10¹⁰ cm/sec
3. Rydberg frequency of hydrogen, R_H = cRy = (2.99792458 x 10¹⁰ cm/sec) x (1.09678 x 105 1/cm) = 3.2880637208524 x 10¹⁵ Hz
4. Double R_H = 2 x 3.2880637208524 x 10¹⁵ Hz = 6.5761274417048 x 10¹⁵ Hz
   
5. Natural unit of time, t_n = 1/2R_H = 1.520652 x 10^-16 sec
   
6. Natural unit of space, s_n = c x t_n = 1.520652 x 10^-16 sec x 2.99792458 x 10¹⁰ cm/sec = 4.558799 x 10^-6 cm
7. Summary: t_n = 1.520652 x 10^-16 sec and s_n = 4.558799 x 10^-6 cm

What we discovered in the combination of SUDRs and TUDRs that leads us to the preons of the SM, consisting of combined S|T units, is that bosons consist of one compound space|time location, while fermions consist of more than one, adjacent, space|time location.  In fact, we are initially assuming that subatomic fermions consist of three adjacent space|time locations, but more or less than this may also be interesting.  In general, however, this concept has the following consequence:

If adjacent locations have the same space|time progression, they will remain adjacent, if not, they will separate. 

In combining adjacent space|time locations into one, partially merged, location, as we do in the preon triplets, we introduce the notion of distance between locations. Hence, if the space|time magnitude of the constituent S|T units of a given S|T triplet are equal, the space|time progression of adjacent space|time locations, defined by the constituent SUDRs and TUDRs of the S|T units making up the triplet, is equal.  Therefore, like three aircraft flying in formation, through a climbing turn, the distance between the relative locations of the three S|T units remains the same.

Of course, the relative distance between the aircraft of the flight formation is the space aspect of M₂ motion, whereas the relative distance between the S|T units of the S|T triplet is the space aspect of M₄ motion.  Nevertheless, the concept of preserving the relative space difference of adjacent locations is the same in both instances, disregarding the dimensions of motion involved.  In the case of the aircraft flight formation in a climbing turn, the 1D altitude is changing at the same time the 2D heading is changing, which means that all three dimensions, x, y, and z, are affected.  As every pilot knows, if the changes in altitude and heading of the three aircraft are not carefully coordinated by the pilots, the distance between the adjacent aircraft, in the dimension where a difference exists, will change accordingly, and if it is great enough, the integrity of the flight formation, as three adjacent flight paths, will be lost.

Similarly, if a difference in the 3D scalar motion of the S|T triplets exists, the integrity of the S|T triplet, as three adjacent locations in the space|time progression, will be lost. Thus, we immediately see that the only triplets that would normally remain intact are those where all three S|T units in the triplet possess displacement in the same “direction,” with the same magnitude.  Assuming the same magnitude in each S|T unit, the only stable fermion triplets, then, are the neutrino, electron and positron (see here), those where all three center colors are the same.  The quark triplets would not be stable.

If all the S|T triplets, corresponding to theoretical quarks and anti-quarks, are unstable, this would theoretically explain why the physical quarks are never observed in isolation, something the LST physicists explain in terms of “asymptotic freedom.”  This is a concept wherein the strong force gets stronger, as the distance between quarks increases, and, therefore, the energy required to separate them is equal to the energy required to create a new set.  Thus, quarks are never observed in isolation, only as constituent triplets of baryons.

However, in the development of our theory, the challenge is just the opposite of that of the LST physicists who need to explain why quarks can only be observed together.  Our challenge is to explain why quarks can exist together.  We know that they can’t exist apart, because the “directions” of the speeds of their three progressing locations don’t match.  For example, the “direction” of the speed magnitude of two S|T units in the down quark is neutral (green), while the “direction” of the third S|T unit in the triplet, is an “odd man out;” that is, the “direction” of the third S|T unit of the down quark is negative (red), which means, over time, the space|time location of that unit will move away from the adjacent locations of the other two S|T units, destroying the integrity of the triplet.

If we look at the constituent speed displacements of the triplet in terms of speed, we can see this more clearly. For example, assuming unit displacement in each unit of the neutrino, we get (ignoring the inward component of the motion):

1. A = (1|2) + (2|1) = 3|3 = 1|1 = 0
   
2. B = (1|2) + (2|1) = 3|3 = 1|1 = 0
   
3. C = (1|2) + (2|1) = 3|3 = 1|1 = 0
   

where A, B, and C, are the three combined space|time progression ratios of the three merged space|time locations of the neutrino triplet, represented by the apexes of the triangle.  In this case, the change in time and the change in space is unit change in each instance, thus maintaining the triplet “formation.” From the perspective of the unit progression, the magnitude of the outward space progression is equal to the magnitude of the outward time progression; that is, it is a one for one progression, and therefore the three space and time locations are synchronized, as the progression proceeds.

However, if one of the S|T units of the triplet picks up an additional SUDR, which would change the color of that unit from green to red, changing the color combo of the neutrino triplet to that of the down quark triplet (2 green, 1 red), the speeds of the nodes change to

1. A = (2|4) + (2|1) = 4|5 = -1
   
2. B = (1|2) + (2|1) = 3|3 = 0
   
3. C = (1|2) + (2|1) = 3|3 = 0
   

assuming that the A node is the SUDR connection of the affected S|T unit. Now, we see that the progression of the A node, from the unit progression perspective, changes.  It changes, from unit space|time progression, with no speed-displacement (3|3 = 0), to non-unit space|time progression, with a negative unit of speed-displacement (4|5 = -1).

What this speed-displacement difference between the nodes translates to is a difference in the length of the cycle of the A node relative to the B and C nodes.  The speed of the change in the SUDR side of the affected S|T unit is still the same, because two units of space quantity in each “direction” in four units of time quantity in one “direction”, is the same speed, or space|time ratio (2|4 = .5c), as one unit of space in two units of time (1|2 = .5c), but it takes twice as long to complete one cycle of oscillation.  It’s the same as saying that the speed of two cars traveling two different distances is the same, but if one has to travel twice as far as the other, at that speed, the duration of the longer trip doubles, relative to the shorter trip, but not the speed of travel.

Equivalently, we can say that the frequency of node A has decreased relative to the frequency of nodes B and C, because one of the two S|T units, connected to node A, picked up an additional SUDR.  Therefore, the relative frequency “distance” between the nodes, which was zero, is now non-zero. However, we are not accustomed to thinking of frequency in terms of space|time progression. After all, we don’t think of two separate piano keys as moving away from one another over time, because their frequencies differ.  Hence, we need to quantify the difference in a more useful manner.

Fortunately, there is a great way to do this, I think, but it requires a subtle notion of representation.  In fact, I think it is the same mathematical idea of a “representation” of a group, which LST physics uses in connection with Lie group theory, but I may be mistaken. Someone more proficient in abstract algebra and group theory might be able to enlighten us on this.

Regardless, taking the notion of the 3D, or scalar, oscillation of SUDRs and TUDRs, from a space (time) point to a space (time) sphere and back to space (time) point again, as a reciprocal relation of space|time locality|non-locality in the S|T unit (see previous post below), we can construct a mathematical representation of this scalar motion in a vector space; that is, the set of mathematical values that the reciprocal oscillations of the S|T unit take, in our RN equations, form a symmetrical group that acts on the complex vector space of rotations in the same way that the solutions to Maxwell’s equations form a group that acts on the rotations of the SO(3) group (or the other way around!).

To tell you the truth, I don’t know a thing about these obtuse mathematical concepts, and I’m sure I have the concepts all mangled up here, but intuitively, I think the following concept is “isomorphic” (LOL) to group representations: The S|T unit is a representation of rotation similar to the SO(3) group, because the “oscillation” from spatial local to non-local is the reciprocal of the “oscillation” from temporal local to non-local. I don’t know if the rotation group has a name, or if it’s simply isomorphic to a well known existing group, but I can identify the rotations: they are gear ratios!

I know this must sound like the height of crackpottery to professional mathematicians and physicists, but if it is, we will soon find out, and I’ll stand corrected.  However, as things now stand, this conclusion is inescapable.  As shown in figure 1 below, the identity gear ratio is the identity element of the group. 

 

Figure 1. The Identity Element of the Group of Reciprocal Rotations, or the “Gear Group”

Clearly, just as the “direction” of the space “oscillation” of the SUDR, is always the reciprocal of the “direction” of the time “oscillation” of the TUDR (i.e. outward in time is equivalent to inward in space and vice-versa), the direction of the rotation of the SUDR gear is always the reciprocal of the direction of the rotation of the TUDR gear; that is, if the SUDR gear rotation is clockwise, the TUDR gear rotation must be counterclockwise, and vice-versa.

In figure 1, the unit ratio of the identity element is expressed by the equal radius of the two gears.  The reciprocal, gear, ratio is 1:1, just as it is in the reciprocal, unit, SUDR|TUDR ratio; that is, the 1|2:2|1 S|T unit ratio is the equivalent of a -1:+1 ratio of reciprocal speed-displacement, where the signs of the magnitudes represent the inverse magnitudes of operationally interpreted rational numbers.

The phase of the “oscillation” is represented by the position of the orange dots on the gears.  In the illustration of figure 1, the phase indicates that both components are local (points), or both 0, at the same time, which is convenient for comparing the relative number of revolutions of the reciprocal gears, but it is not what we want, since this would not represent the reciprocal phase of symmetry that constitutes the law of conservation, according to the Noether theorem. 

Thus, we must modify the relative phase of the representation, as shown in figure 2, below.

 

Figure 2. S|T Representation of the Identity Element of the “Gear Group”

Of course, to show that the “gear group” meets the criteria of a mathematical group, is going to take some explaining, which I will undertake in the next post, but it’s clear that the operationally interpreted rational numbers of the new reciprocal system of mathematics (RSM), should enable us to do just that, since we already know that they form a group under binary addition.

However, combining the S|T units, as conserved space|time speed-displacement ratios, will necessarily lead us to consider a group under binary multiplication.  I hope we can work the required RSM math operation of multiplication out, but I really don’t know if we can, at this point. I haven’t given it any thought, because, before the discovery of the “gear group,” I didn’t know why we needed it, but now that has changed.

 

 

The basics of quantum mechanics, the main feature that distinguishes it from classical mechanics, is Heisenberg’s uncertainty principle; that is, in quantum mechanics, unlike in classical mechanics, the position and the momentum of a moving object cannot both be determined precisely, at any given moment in time. This is understood in terms of wave (uncertain position) and particle (certain position) duality, and quantum superposition: Superimposing the quantum states of many waves realizes the particle.

For example, in the case of classical mechanics, the x,y coordinates of a point moving uniformly around the circumference of a circle can be determined precisely, at any given moment in time.  Moreover, if the x,y momentum of an orbiting classical object is plotted against its x,y position on the orbit, at a given time, the plot obtained (phase-space) is also precise (see illustration here).  However, in quantum mechanics, this is not possible, because the more precisely the momentum is known, the less precisely the position is known and vice-versa. So, the idea of probability amplitude was born.  The probability amplitude is associated with the wave function in the phase-space of quantum phenomena, the famous quantum state of the microcosmic world.

In this case, the probability amplitude of the wave function differs from the combined probability distribution of a classical phase-space, where a series of measurements on identical oscillators defines the probability that the definite point will be found at a given location, with a given momentum.  In contrast, in the case of the probability amplitude of the wave function, a given location, for a given momentum, does not even exist. It is not defined, not just unobservable.

At one time, this idea was easily dismissed by skeptics, but not today.  The reason for this is that, today, quantum optics permits these things to be visualized experimentally (see explanation here). The visualizations rely on a few mathematical principles like Wigner functions, Fock states, etc.  Wigner functions (WFs) are the analogs of classical phase-space probability distributions.  In other words, they are like the probability distributions of classic phase-space in some ways, but they are different in other, important, ways. 

The most important distinction is that while WFs are similar, they are not phase-space probability distributions.  This is because they are measurements of only one, or the other, of the two, reciprocal, aspects of the phase-space. They are a measurement of either the momentum or the position aspect of the quantum state, never both.  The “probability” distribution of only one of the reciprocal pair of quantum parameters of phase-space is called a phase-space quasiprobability distribution, or density.  It is the WF of a given parameter in a quantum state.

Another, important, distinction between classical probability distributions and WFs is that, since the WF is not a probability density per se, its magnitude doesn’t have to be a positive definite magnitude; that is, it can be a negative quasiprobability!  This leads to the concept where, no matter what the energy of the Fock state is, the phase-space has regions where the WF, or quasiprobability, of the quantum state is negative. 

Using these two properties of WFs, the states of quantum systems can be studied in real experiments with lasers and optics.  A most interesting experimental result is reported by Alex Lvovsky et al in a paper entitled “Quantum State Reconstruction of the Single-Photon Fock State” (see here).  In the introduction to this paper, they write:

States of quantum systems can be completely described by their Wigner functions (WF), the analogs of the classical phase-space probability distributions. Generation of various quantum states and measurements of their WFs is a central goal of many experiments in quantum optics [1–3]. Of particular interest are quantum states whose Wigner function takes on negative values in parts of the phase space. This classically impossible phenomenon is a signature of highly nonclassical character of a quantum state.

That is to say, in contrast to the classical state, such as the coherent state of laser emissions, the quantum state can be both positive and negative. But how can a probability be both positive and negative?  What exactly is a negative probability?

Rebecca Slayton, writing in the Physical Review Focus, called Lvovsky’s experiment a “strange result” that goes way beyond the strangeness of Schrödinger’s cat.  In her article, entitled “Golfing with a Single Photon”, she explains

…in the quantum world, where a photon’s position and momentum cannot be determined simultaneously, this “elevation” [the WF] can only be understood as an approximation to probability [i.e. quasiprobability]. The new experiments by Alex Lvovsky and his colleagues at the University of Konstanz in Germany show that the photon’s phase space contains a circular ridge, where the photon is likely to be found, and a deep crater in the center, where your chances of finding the photon seem to be negative.

A graphic, depicting the result, is included in the article, as shown below:

 

As can be seen by the illustration above, the probability density of a single photon goes way below zero, like a volcanic pipe, extending below ground level. 

Slayton explains how Lvovsky’s team exploited the WF of the single photon, by using pairs of photons generated by down-conversion.  The down-converted photons share the same quantum state, so the wave property (the WF of non-local position) of one set of photons can be measured, while the particle property (WF of local momentum) of their corresponding twins can be measured separately, and then the WFs are combined together mathematically, like the slices of a CAT scan, to give the resulting phase-space probability distribution, depicted in the graphic above.

This has to be startling to LST physicists, whose only available interpretation of this is simply as a visualization of the uncertainty principle.  In reality, however, it’s much more than this; It’s tantamount to a CAT scan of a single photon, where we can visualize the photon as a combination of positive and negative oscillations, plotted in the reciprocal relation of phase-space, as probabilities. Clearly, this is a perfect picture of an S|T unit composed of positive and negative 3D oscillations.

But this is not the first time we’ve seen this.  Recall the discussion of Harry Swinney’s “oscillons” in his experiments at the University of Texas, at Austin.  Oscillons are oscillating groups of small brass beads, formed in a bed of such beads, vibrated at a given frequency. The peaks and valleys of oscillons are called crater (or dish) states and peak states.  Two crater states, or two peak states, repel each other, but unlike states attract.  Moreover, when a oscillon in a crater state collides with an oscillon in a peak state, they don’t cancel each other, but rather form a bound sytem of combined crater and peak states, which is stable over time.

I don’t know if anyone in the LST community has connected these two experimental results together yet, as I am doing here, but I think it would be unlikely without the understanding of the RST-based concepts of SUDRs, TUDRs, and the combination of these two theoretical entities into the theoretical SUDR|TUDR (S|T) units, which then are grouped as preon elements of the standard model entities. We’ve discussed, in several of the previous posts below, how combining S|T units into triplets yields the bosonic and fermionic configurations of standard model entities.

Nevertheless, we have yet to discuss the uncertainty principle in connection with these S|T units, and, as we are seeing from the work of Lvovsky and others, the uncertainty principle plays a crucial role in observing and calculating the values of quantum properties.  Clearly, this seems to present a problem, at first blush, because the LST uncertainty principle is expressed in terms of momentum and position, the phase-space of quantum mechanics, and the associated wave function of quantum states, which are quantities within the purview of M₂, or vector, motion, but not in that of M₄, or scalar, motion.  How, then, can we understand these seemingly weird ideas in RST-based theories?

The answer is surprising, because not only does it tell us how to work with these things, in the new system, but it also explains why they exist, which is, of course, something the LST community can’t offer, much to its chagrin. To understand the uncertainty principle, we first need to understand how it arises fundamentally from the particle/wave duality of the discrete world, and also that this duality is, in reality, a principle of reciprocity.  In the LST community, the reciprocal relation of two quantities is a central pillar of the science of physics.  Called conjugate quantities, the relationship between them in quantum mechanics is described in the Wikipedia article on the “Uncertainty principle.”  In part, it reads:

In quantum mechanics, the Heisenberg uncertainty principle is a mathematical property of a pair of canonical conjugate quantities - usually stated in a form of reciprocity…. It therefore mathematically limits the accuracy with which it is possible to measure (actually even define) such pairs.

In classical mechanics, the most familiar conjugates (literally joining two entities together) are the concepts of potential and kinetic energy, regarded as two, reciprocal, aspects of the total energy of a dynamic system, such as the swinging pendulum, or bouncing spring.  These two, reciprocal, aspects of the pendulum’s, or spring’s, energy, express the law of energy conservation, as they transform into one another, because no matter what the quantity of either is, at a given point in time, they always add up to the total energy, in a lossless system (e.g. see here).

The same principle is expressed by the sine and cosine numbers of the changing angle that the arm of the pendulum makes with the vertical norm at the bottom of the swing: Their sum is always equal to 1, at any given angle of rotation. The identical relation can also be seen in the two, reciprocal, aspects of radiation, frequency and wavelength, where the total energy can be expressed in terms of either. Consequently, to increase, or decrease, the radiative energy of a system, the frequency must increase, or decrease, but the wavelength must decrease, or increase, and vice versa, because frequency and wavelength are reciprocals in the radiation energy equation.

However, in de Broglie’s hypothesis, this familiar relation of reciprocal quantities starts to seem very strange, because, as he reasoned, if the wave energy of light is discrete, then the discrete entities of matter must be waves.  Of course, this is exactly how it turns out to be, but the big question immediately becomes, how can matter (or anything) be both discrete and continuous at the same time, knowing that waves are continuous, and matter is discrete? 

Modern physics’ initial answer to this perplexing question was found in the Born statistical interpretation of the wave function, where the squared amplitude of the wave at a certain time yields the probability of finding an electron at a certain location in the vicinity of the nucleus of an atom. However, this initial interpretation shortly evolved into the form of Schrodinger’s wave equation, based on a so-called dispersion relation, where the concepts of energy, momentum and velocity are manipulated to yield a reciprocal relation between momentum and position; that is, the greater the momentum of a particle (locality), the more its wave is spread out (non-locality).  Thus, de Broglie’s relation of wavelength, Planck’s constant and momentum (velocity of mass), as discrete units of energy per unit of momentum, leads to a reciprocal relationship between momentum and position that doesn’t exist in classical mechanics (see online article here).  Fundamentally, however, it is clearly a reciprocal relation of the value of “locality” versus the value of “non-locality” that we are dealing with here.

Calling this unclassical relationship (all classical objects are always local objects) the “uncertainty principle” is therefore misleading, in a way, because it’s really only a reciprocal relation between locality and non-locality.  And, as we know, these two properties just happen to be the two, reciprocal, aspects of volume: As we’ve seen time and time again in these posts, the point and the sphere are reciprocals, both numerically and geometrically, where the numerical magnitude, corresponding to the three-dimensional, geometric, unit point is 1³ = 1, and where the numerical magnitude, corresponding to the three-dimensional, geometric, unit sphere is 2³ = 8. 

These are the scalar and pseudoscalar of the octonions, respectively, that exhibit the well known reciprocal symmetry, 1331, in the number of its linear spaces, as generated in the binomial expansion. In the reciprocal system of mathematics (RSM), the magnitude of the M₁ sphere is the (1|1)³ point, and the magnitude of the M₂ sphere is the (2|2)³ cube, and, in the theoretical development of the LRC, these numerical magnitudes correspond to the physical magnitudes of the SUDR and TUDR, based on the reciprocal system of physical theory (RST).

More relevant, however, when one visualizes the point expanding to a sphere, it’s easy to visualize the local vs. non-local nature of these two, reciprocal, aspects: The outward expansion of the point turns its locality (1³) into non-locality (2³), as the expansion is measured along one of the eight diagonals of the (2|2)³ = 8 discrete cubes (or continuous sphere). To measure the expansion of the point into the sphere requires a selection of a definite (local) point on the surface of the sphere, or a specific corner of the cube (a wave function collapse), which can be expressed in terms of these eight diagonals, or octutures.

Of course, motion in the opposite “direction,” a contraction of the sphere to the point, is a transition from non-locality to locality.  In the case of the S|T unit, two of these volume oscillations are bound together, one positive and one negative.  If we plot these changing values of volume, as changing degrees of locality, constituting a change of quantum state, then the result implies the scalar motion of a SUDR|TUDR combo, just as surely as plotting the changing values of kinetic and potential energy, constituting a change of mechanical state, implies the vectorial motion of a pendulum, or bouncing spring.

Obviously, we can observe the motion of a pendulum, or spring, that constitutes its changing state of kinetic|potential energy, but we can also describe it mathematically, in terms of the sine and cosine of the changing angle of the arm, or changing length of the spring.  Unfortunately, we can’t observe the motion of the S|T unit that constitutes its changing state of locality|non-locality, but we can describe it mathematically, in terms of the space and time magnitudes of the changing volume of the SUDR and TUDR.

However, the first order of business is to describe what is being conserved in the oscillations themselves. In the case of the moving pendulum, or the bouncing spring, the motion, or momentum, of the weight at the end of the arm, or spring, changes, but the total energy is conserved;  that is, at the top of the swing, or bounce, the motion, and thus the kinetic energy, is zero, but the total energy of the system is conserved, in the potential energy of the weight’s position, which is relative to the pivot point and the length of the arm, in the case of the pendulum, and relative to the tension and length of the spring, in the case of the bouncing spring.  On the other hand, the motion, or momentum, at the bottom of the swing, or the middle of the bounce, represents the total energy of the system, at that point, and the potential energy of the weight’s relative position is zero, at this point. In between these two points, a portion of the total energy is potential energy, and a portion is kinetic energy, which, when added together, invariably equals the total energy of the system.

In the case of the S|T unit, an analogous situation exists, only there is no mass, and, thus, there is no momentum, and there is no position involved, in the dynamics of the system. The conserved quantity of this system, therefore, is not a given quantity of mechanical energy, expressed in terms of the changing energy of momentum and a changing reciprocal of this, the energy of position, but the conserved quantity is a given value of scalar quantity, expressed in terms of a changing volume of space and a changing reciprocal of this, the “volume” of time.

In this way, the oscillating, or pulsating, volume is analogous to the swinging pendulum, or the bouncing spring, but the two, reciprocal, aspects of the change of the volume are the space|time point and the space|time sphere; that is, the change is from purely local space|time (point) to purely non-local space|time (sphere).  Because space is the reciprocal of time, in the RST, more space is equivalent to less time, and vice-versa. Hence, as the unit space point is transformed into the unit space sphere, its reciprocal aspect, the unit time sphere, is transformed into a unit time point, corresponding perfectly to the reciprocal relation of conjugate quantities.

The constant, unchanging, quantity that is being conserved in the transformation of these two, reciprocal, aspects of volume, is what we can call the volume’s “locality density;” that is, as one space|time aspect of the unit point increases to a unit sphere, its “locality density” decreases, but at the same time, the reciprocal aspect’s “locality density” increases, as it transforms from a unit sphere to a unit point. Hence, as the point’s locality value (probability density) becomes less “dense” as a sphere, the reciprocal sphere’s locality value (probability density) becomes more dense as a point, and the sum of the two “locality densities” is always constant, throughout the cycle.

Consequently then, the analog of the SUDR is the swinging pendulum, or bouncing spring, while the analog of the TUDR is a negative swinging pendulum, or negative bouncing spring, which, though conceptually straightforward, in an abstract sense, is impossible to actually implement with moving masses. 

If we plot the oscillation of the SUDR, and the oscillation of the TUDR, in terms of transitions from point to sphere (local to non-local), and from sphere to point (non-local to local), we get the expected result of two, reciprocal, aspects of the total motion, constantly changing, but, unlike in the case of the pendulum, a coordinate position has no role to play at all in the system, even though the property of non-locality|locality could be expressed in terms of an increasingly/decreasingly “uncertain” position, if that’s what we wanted to do.

Of course, we have no need to do this in our scalar system.  What we want to do instead is to express the local|non-local transitions of the SUDR|TUDR combo in terms of the radius of its spheres, the pulsating spatial and temporal spheres, which constitute the unit space and unit time expansions/contractions of the SUDR and TUDR. These are the two parameters of the scalar quantum system, or scalar phase-space, of the S|T unit. Unfortunately, like the momentum|position phase-space of the LST’s quantum mechanics, these two parameters of the RST-based scalar phase-space are mutually exclusive, as far as measurement is concerned, because, even though, now, non-locality will not be expressed in terms of the probability of defining a location, it will still be expressed in terms of two reciprocal quantities; that is, the magnitude of a radius of a space volume, is the inverse of the radius of a time volume, and as one grows, the other shrinks, and it’s impossible to measure something that is both growing and shrinking at the same time.  The more accurately you measure the change in the space volume, the less accurately you can know the change in the time volume, and vice versa.

Fortunately, however, each of the inverse quantities can be measured separately, as Lvovsky et al and others in the laser physics field have demonstrated. What we end up with, in the final analysis, is simple harmonic motion; that is, the S|T unit is a unit of propagating, simple, harmonic, motion.

After posting the previous post below, I could almost hear the guffaws, and the last words I wrote, asking if there is any other way to make the space|time dimensions of the force equations work out, were ringing in my ears. Then, with a sudden flash of insight, I realized that, in fact, there is another way.

In the meantime, walking the dog, I kept thinking about the idea in the post below.  Certainly, the dimensions are correct, but how could a time interval substitute for an area.  Intuitively, we know that the affect of the electrical charge (inverse speed) is omni-directional, or scalar.  The force, due to the charge, is a manifestation of this scalar motion in a given direction, or multiple directions, in the case of multiple charges, just as I pointed out with the expanding galaxies analogy. Since spatial distance, is a measure of the space aspect of a given vectorial motion, disregarding its time aspect, then it follows that temporal distance is a measure of the time aspect of a given vectorial motion, disregarding its space aspect. 

Nevertheless,  since time does not have direction in space, vectorial time motion is not possible and thus, the new force equation, with its scalar time interval, even if it is valid, would be hard to conceptualize, let alone use.  So, I’ll probably just have to take my lumps on that one, even though there’s much more that I can say about it that I think is useful.

As I mentioned, however, I realized soon after posting the previous article that there is another way to make the space|time dimensions of the force equations consistent, and, to boot, it sheds a great deal of light on the whole idea of what force and acceleration are.  In this approach, we have to challenge Larson’s dimensional assignments to units of inverse speed, which is something that I’ve seen coming for a while now, because they are inconsistent with the new understanding of numbers in the chart of motion (CM).

Recall that the CM contains the four numbers 1-4 and the dimensions of these from 0 to 3, forming a four by four matrix of magnitudes (it probably should be called the chart of magnitudes, instead of the chart of motion.)  We’ve mostly referred to M₂ and M₄ motion, and sometimes M₃ motion in the CM, but not M₁ motion, probably because it isn’t motion; that is (1|1)ⁿ cannot be displaced, regardless of its dimensional exponent, n. 

However, be this as it may, there is a difference between a unit point, (1|1)⁰, a unit line, (1|1)¹, a unit area, (1|1)², and a unit cube, (1|1)³.  The difference is the degree of duality in each dimension, or maybe what we could call potential duality.  The point is (no pun intended), in the mathematics of dimensions, there is the scalar magnitude, and then there is the pseudoscalar magnitude; that is, (1|1)⁰, is a point, or magnitude with no direction, and (1|1)³ is a sphere, or magnitude with all directions, one being the inverse of the other, in a sense.  Therefore, the dimensions of a point charge must be one or the other, but as we saw below, LST physicists can’t make the zero-dimensional point charge work, because of infinities (currently driving string theorists), and they can’t make the three-dimensional spherical charge work either, because of the need for “Poincare stresses.”

However, the situation with Larson’s RST-based theory is not much better, even though it doesn’t face the same challenges as the LST theories.  Nevertheless, the reason is, again, due to dimensions, but this time the problem is not reflected in unwanted infinities or the need for compensating forces, but rather because the dimensional analysis of the force equations shows a problem with the assignment of the space|time dimensions in both the charge and gravitational equations.  In the charge equation,

F = (Q1Q2)/d², 

the space|time dimensions of the charges are the same as energy (t/s).  However, both energy and charge are scalar quantities, not vector quantities.  Hence, they should have the dimensions of scalars, which is dimension zero, the dimensions of a point, not dimension one, the dimensions of a line.  The problem is that the dimensions of scalars, n⁰, hasn’t been carefully thought out by physicists, probably because they have been taught that n⁰ is always equal to 1, while n¹ is equal to n, which is true enough, but that’s not the whole story.

In the quantitative interpretation (QI) of number, setting the dimension of a number to zero is tantamount to setting it to one, but in the operational interpretation (OI) of number, the interpretation used in the reciprocal system of mathematics (RSM), one has a different meaning, because there are an infinite number of ones, from 1|1 to ∞|∞, and one is always greater, or lesser, than another one. This has huge ramifications, but one effect (LOL) on our theory development is that the dimensions of charge and energy must change from t/s, to t⁰/s⁰.  Because these two concepts are scalar concepts, they must have the dimensions of scalars, but without a knowledge of the OI numbers, the LST physicists had to find a way around this dimensional inconsistency to avoid problems, and they did it through the definition of work, an ingenious workaround, to say the least (but, again, no pun intended, really).

However, in the new system, we have no need for the expediency used by the LST community to define energy in terms of work.  Therefore, when we correct the dimensions of the scalar charges, the force equation between charges becomes dimensionally consistent:

F = (Q1Q2)/d² =  ((t/s)⁰(t/s)⁰)/s² = (t/s)⁰(1/s²) = t⁰/s²,

and energy has the dimensions of a scalar, while space has the dimensions of area, which properly describes force, as energy per unit area, (t⁰/s⁰)/s².

When we turn to the gravitational force equation, things get even more interesting, because we not only change the number of the exponents of the space|time dimensions, but we actually invert the dimensions themselves.  The reason is subtle, but so are the concepts involved, which is why they have been so troublesome for so long. Recall that, according to all observation in the LST community, inertial mass and gravitational mass are exactly equivalent.  No difference has ever been detected, mystifying the physicists.  In the RST community, however, there is no mystery to this at all, because they are the same; that is, mass is simply a measure of the inherent inward scalar motion that constitutes matter.  Matter consists of discrete units of three-dimensional, inward, speed, while mass is the three-dimensional opposition, or 3D outward speed that it takes to cancel, or overcome, the inward speed.

Thus, if we substitute the dimensions of the inward motion of matter, for the dimensions of the outward resistance of matter, we also switch from a concept of force (a quantity of resistance to motion), to a concept of acceleration (a quantity of motion), without affecting, in the least, the quantitative variables of the equation.  Thus, the gravitational equation of acceleration, A, becomes:

A = G(M1M2)/d² = K((s/t)³(s/t)³)/s² = K(s/t)⁶/s² =  K(s⁴/t⁶),

and, applying our Bott periodicity theorem correction, this becomes

A = K(s^4-4/t^6-4) = K(s⁰/t²),

where the dimensions of the universal gravitational constant, Big G, drop out, and only a dimensionless constant, Big K, remains, representing the magnitude difference between the first and second tetraktys of the RSM.  This also correctly shows the scalar view of acceleration, where it is a speed “density,” so-to-speak, or scalar speed per square unit of time, (s⁰/t⁰)/t².

Of course, all this is so new, it’s totally subject to revision, but for now, at least, it seems to make an awful lot of sense.

In the previous post below, we began our discussion of force and acceleration concepts, in the RST, noting that, although Larson redefined motion in terms of scalar magnitudes, he didn’t redefine force and acceleration in scalar terms. Yet, charge, gravitational as well as electrical and magnetic, is a scalar magnitude, not a vectorial magnitude. Force, on the other hand is not a scalar magnitude, but a vector magnitude; that is, a Coulomb force (charge), or a gravitational force (mass), affects another charge, or another mass, only in the direction defined by the distance between them.

Thus, the scalar magnitude of charge is manifest as a vectorial magnitude of force, whenever it interacts with another charge. We see the same thing with scalar motion in general. Galaxies A, B, and C, on a line, are moving away from each other, because of the scalar motion affecting them, but the scalar motion, expanding in all directions, is manifest as a vectorial motion of the galaxies, in a direction along the line between them.

However, in the LST concept, force is measured in units called Newtons, and given autonomous status, as if it were something independent of motion. Larson pointed out emphatically that this is a mistake (see especially The Neglected Facts of Science), but he didn’t go so far as to actually undertake to reexamine the force equation in light of this conceptual error.

Nevertheless, when we consider the so-called self-force concept, which is regarded as the force generated by a charge on itself, we get to the the crux of the major difficulty in LST theory that plagues it to this day. The electromagnetic theory of force is inconsistent. Because of this, classical electromagnetic theory was abandoned in favor of the quantum mechanical approach, but, in the end, this major change didn’t help solve the problem. As Richard Feynman explains it in his Lectures on Physics, page 28-4, the problem ultimately comes down to the idea of autonomous force, which Larson criticized, even though Feynman doesn’t use those words:

In deriving our equations of energy and momentum, we assumed the conservation laws. We assumed that all forces were taken into account and that any work done and any momentum carried by other “nonelectrical” machinery was included. Now, if we have a sphere of charge, the electrical forces are all repulsive and an electron would tend to fly apart.  Because the system has unbalanced forces [read autonomous forces here], we can get all kinds of errors in the laws relating energy and momentum.  To get a consistent picture, we must imagine that something holds the electron together.  The charges [discrete fractions of charges distributed over the surface] must be held to the sphere by some kind of rubber bands - something that keeps the charges from flying off.  It was first pointed out by Poincare that the rubber bands - or whatever it is that holds the electron together - must be included in the momentum and energy calculations. For this reason the extra nonelectrical forces are also known by the more elegant name “Poincare stresses.” If the extra forces are included in the calculations, [they] are consistent with relativity; i.e., the mass that comes out from the momentum calculation is the same as the one that comes from the energy calculation.  Both of them contain two contributions; an electromagnetic mass and contribution from the Poincare stresses.  Only when the two are added together do we get a consistent theory.

Thus, the LST’s autonomous forces, generated by the distributed electrical charges, on the surface of the sphere, have to be held together by some mechanism of restraint, a mechanical force of some kind, and if this is included in the calculations, then the mass of the electron, calculated from the energy of the electrical field from the charge, or from the mass component of the momentum associated with the velocity of the field, is the same. The bottom line is that the mass cannot arise solely from the electron’s charge; another, non-electrical, non-explainable, autonomous, force is required to hold the electron together against the strong electrical forces pulling it apart, and this leads to trouble.  As Feynman observes:
 

Clearly as soon as we have to put forces on the inside of the electron, the beauty of the whole idea [i.e. deriving mass from Maxwell’s equations], begins to disappear.  Things get very complicated.  You would want to ask: how strong are the stresses? How does the electron shake? Does it oscillate? What are all its internal properties? And so on. It might be possible that the electron does have some complicated internal properties.  If we made a theory of an electron along those lines, it would predict odd properties, the modes of oscillation, which haven’t apparently been observed.  We say “apparently” because we observe a lot of things in nature that still do not make sense.  We may someday find out that one of the things that we don’t understand today (for example the muon), can, in fact, be explained as an oscillation of the Poincare stresses…there are so many things about fundamental particles that we still don’t understand.  Anyway, the complex structure implied by this theory is undesirable, and the attempt to explain all mass in terms of electromagnetism…has led to a blind alley.

Feynman goes on to explain in more detail why this is so, but it comes down to the same thing: if you allow the electron to be a point and, at the same time, act on itself, something Feynman characterizes as “perhaps a silly thing,” you have to modify Maxwell’s theory of electrodynamics.  “Many attempts have been made,” writes Feynman, “but all of these theories have died.”  Moreover, quantum mechanics doesn’t help, because there is no model of the electron in quantum mechanics. Quantum mechanics only describes a quantum state of the electron, not its structure.  Feynman explains:

 

It turns out, however, that nobody has ever succeeded in making a self-consistent quantum theory out of any of the modified [classical] theories…We do not know how to make a consistent theory - including the quantum mechanics - which does not produce an infinity for the self-energy of the electron, or any point charge.  And, at the same time, there is no satisfactory theory that describes a non-point charge.  It is an unsolved problem.

Of course, in developing our RST-based theory, we escape the horns of this dilemma, by recognizing that electrical, magnetic, and gravitational forces, are properties of scalar, not vectorial, or M₂, motions, and that they are certainly not autonomous entities that can exist independently on the surface of a sphere, as a consequence of interacting charges.  In the new system, we can define force and acceleration (mass) in space|time terms in a way that eliminates the infinities of point particles and the need for Poincare stresses, in non-point particles.

That is to say, we can redefine force, just as we redefine motion: In the universe of motion, like everything else, force and acceleration must be motions, a combination of motions, or a relation between motions.   Clearly, then, they are relations between motions, and/or relations between combinations of motions. As pointed out in the previous post below, we can see from their dimensions that force, dt/ds²,  is one-dimensional energy (inverse speed), dt/ds, per unit of space, 1/ds, and acceleration, ds/dt², is one-dimensional speed, ds/dt, per unit of time, 1/dt.

However, things are complicated quite a bit by the fact that numerical dimensions, are not necessarily geometric dimensions, that, while geometric dimensions are limited to three, numerical dimensions are not limited to three, even though they are related to the three geometric dimensions, in a special manner. Consequently, we have to be very careful in our expressions of multi-dimensional magnitudes, such as t/s², or s/t², in order not to confuse geometric dimensions with numerical dimensions.

For instance, while force, t/s², can be interpreted as

F = ma,

because the dimensions are consistent, in this case, giving us

F = ma = (t³/s³)(s/t²)  = t/s²,

it doesn’t necessarily follow that force should be defined this way in general. In the case of the electrical force equation,

F = Q1Q2/d², 

for instance, the force equation does not involve the dimensions of mass or acceleration:

F = (t/s)(t/s)/s² = (t²/s²)(1/t)  =  t/s² (substituting 1/t for s², explained below), while,

in the gravitational equation, it does involve the dimensions of mass, but not of acceleration,

F = GM1M2/d² = G(t³/s³)(t³/s³)/s² = G(t⁶/s⁶)(1/t) = G(t⁵/s⁶).

Clearly, t⁵/s⁶, makes no sense geometrically, because there are only three geometric dimensions, not six (we don’t subscribe to a concept of hidden dimensions). Yet, when we recognize that the numerical dimensions, higher than three (four, counting zero) are geometrically mapped, so-to-speak, by the Bott periodicity theorem, into repeating groups of four dimensions, called tetrakti, in the reciprocal system of mathematics (RSM), we see that x⁵ corresponds to x^5-4, and x⁶ corresponds to x^6-4, geometrically.  Therefore, t⁵/s⁶ is equivalent to t/s², adjusted by a constant, presumably the observed gravitational constant G.

However, if this is true, then what is the 1/t term in the force equation, and where do the higher dimensions come from? Physically, 1/dt is a frequency, and it’s hard to see how time, t, can be equivalent to distance squared. Of course, this will take some more thought, but it’s possible that t is related to s², because the temporal period of a cycle is related to the spatial area of a square, as the time it takes to complete one cycle defines a definite circumference and diameter, at unit speed.  So, increasing/decreasing the distance between two locations, as the index, d, of a square area variable, might also be equvalently expressed as an increasing/decreasing period, as an index, t, of a circular area variable (π r²), even though this seems like a long shot at this point.

As far as where the higher dimensions come from, the separation of geometric from numerical dimensions makes this clear.  The dimensions are actually the number of points involved; that is, two points (two S|T units), form the basis for a line, three an area and four a volume, but 5 is a point in the volume, 6 is a line in the volume, 7 is enough for an area, and 8 is enough for a second volume in the initial volume.  Then 9 is a point in the volume of the volume, 10 is a line in the volume of the volume, etc, ad infinitum.  Again, this is what Raul Bott proved as the periodicity theorem that bears his name.

I know this sounds crazy, but it’s where the path leads. The obvious conclusion is that what we have here, in order to make the dimensions come out right, is a new force equation, the self-force equation, if you will, except that the self-force is actually a scalar force.  In other words, we can redefine the vector force equation, the equation that holds for the vectorial force between two charges, or two masses, in terms of the scalar motion equation that the vector force is part of. Just as the galaxies are moving apart scalarly, but a vector motion is part of that motion, as explained above in the beginning of this post, the charges/masses are moving scalarly, but the force property of that motion is directed along the line between them.

Thus, the equations,

F = Q1Q2/t, and F = GM1M2/t,

express the scalar force, because t, the period of oscillation, determines the radius of the sphere that in turn determines the 4 pi r² area of the sphere, a part of which, is directed along a line between the charges, and is subject to the inverse square law along the line between them.

I might be all wet, but is there any other way to make the space|time dimensions come out right?

 

 

 

 

In changing from a quantitative interpretation of number (QI), to an operational interpretation (OI), we found, in the previous post below, that we can speak in terms of multi-dimensional magnitudes that are not products, but sums of two, or more, one-dimensional, magnitudes. In this way, the three space|time magnitudes of a fermion S|T triplet, represented by the vertices of the triangle, constitute the three dimensions of the triplet, even though three points (S|Ts) can only lie in a plane, geometrically. By the same token, two points (S|Ts), constitute a two-dimensional entity, even though two points can only lie in a line, geometrically speaking.

If we add another point (S|T) to the triplet, we form a four-dimensional entity, which is three-dimensional geometrically, and adding a fifth point (S|T), we form a five-dimensional entity, which is also three-dimensional geometrically, but extended in one of the three dimensions. Therefore, while the evolution of the geometric dimensions of these S|T combinations is limited to three degrees, from a point to a line, from a line to a plane, and from a plane to a tetrahedron, which is the first 3D geometry, the evolution of their numerical dimensions is unlimited.

Moreover, with the reciprocal number (RN) of the reciprocal system of mathematics (RSM), we can express the total number of speed-displacements of an S|T unit, or a multi-dimensional combination of S|T units, in terms of the total number of such displacements, beginning with an absolute minimum value of an initial S|T unit:

ds|dt = 1|2 + 1|1 + 2|1 = 4|4 num

By adding SUDRs and/or TUDRs to this initial, or first, number, we can increase the value, or compound the magnitude, we might say, of the point. For instance, adding a SUDR to the initial unit increases the total magnitude of the above point (S|T) from four units of progression to six units:

ds|dt = (1|2) + [(1|2) + (1|1) + (2|1)] = (2|4) + (2|1) + (2|1) = 6|6 num

(recalling that the minimum number of displaceable units of progression is two units, an outward and an inward unit), where “num” denotes natural units of motion. Subsequently adding a TUDR to this number, increases the total magnitude of the point to eight natural units of motion:

(2|1) + [(2|4) + (1|1) + (2|1)] = (2|4) + (2|2) + (4|2) = 8|8 num

We can also add two or more points directly, forming a combined point of higher magnitude:

ds|dt = [1|2 + 1|1 + 2|1] + [1|2 + 1|1 + 2|1] = (2|4) + (2|2) + (4|2) = 8|8 num

So far so good. We can easily sum the dimensions of the S|T combos this way, but there is much more to describing the structure of the physical universe than summing the possible combinations of S|T units. The universe of motion consists of discrete units of motion, combinations of discrete units of motion, and relations between discrete units of motion. It is the relation between the units, and combination of units, that is most challenging.

Indeed, in the legacy system of physical theory (LST), the relation between physical entities is the sole means for classifying them. The goal of that program has been to develop a classification of physical entities, according to the kinds of relations between them. These have been identified by LST scientists as the gravitational, electromagnetic, and nuclear relations (including both weak and strong relations) that bind the constituent entities together and determine their behavior under varying conditions and in different environments. These interactions are described in terms of force, as defined by Sir Isaac Newton, who defined it as an “influence” that has the potential to cause a physical entity, or set of physical entities, to change positions, relative to another physical entity, or set of physical entities.

However, in Larson’s new system, motion is defined without necessarily referring to the relative positions of physical entities. In this system, space changes just as time changes, unrelentingly, as part of a pre-existing, universal, motion, and no object, with a definite location, or momentum, is included in its definition. It follows, therefore, that a new definition of force is in order, which applies to the inherent motion of physical entities. This is what is referred to as “self force,” and it is has been an enigmatic concept from the beginning in LST science, because of their definition of motion.

However, even in the development of RST-based theory, it’s a challenge, because Larson didn’t redefine Newton’s concept of force, at least not explicitly. Nevertheless, he does clarify the concept in general, which he finds distorted by the LST community’s insistence in regarding it as a fundamental entity, rather than a property of motion; that is, he asserts that force, by definition, cannot be the cause of motion in the ultimate sense. It is true that it may cause the change of position of an object, or M₂ motion, via an exchange of momentum, but this is quite different than attributing the properties of physical entities, such as electrical charge, to the existence of an autonomous force that is somehow distributed about a point, such as an electron, which has no observable physical extent.

It is clear, when we examine the dimensions of force, that they are the dimensions of energy, dt/ds, per unit of space, or dt/ds * 1/ds = dt/ds². In other words, the dimensions are an expression of the spatial density of energy, a scalar quantity, which, in the new system, is inverse speed. Thus, force is the inverse of acceleration, not in the mathematical sense, but in the conceptual sense, in that acceleration is the temporal density of speed, ds/dt, per unit of time, or ds/dt * 1/t = ds/dt². However, in the context of the limited definition of motion in the LST, these dimensions of force and acceleration can also be consistently interpreted as a change in energy per unit of length, and a change of velocity per unit of time, consequently defining them as non-scalar quantities, with the property of direction in one dimension, a vector, rather than the property of a varying scalar density.

Nonetheless, the vectorial concept of force, as the change of units of energy per unit of length is just another way at looking at the inverse square law.  As the distance, or radius, from a point (S|T), with a given quantity of energy, or units of inverse speed-displacement, increases, so does the area of the spherical surface, representing the distribution of these scalar units, at that distance, as illustrated in figure 1 below.

 

Figure 1.  Inverse Square Law as Scalar Density

In our system, the units of inverse speed may be equal to, greater than, or less than, units of speed, the two together forming the total units of progression that constitutes a given S|T unit. Thus, the inherent energy (inverse speed) density of an S|T unit (point), is the ratio of inverse speed-displacement units to the S|T’s total number of progression units, and the speed density, the acceleration, of an S|T unit (point), is the ratio of speed-displacement units, to the total number of the S|T’s progression units.  However, the total effect of the inherent motion of a given S|T unit (point), on another S|T unit (point), is determined by the distance between the two units (points), which decreases the density of the motion per unit area, as shown in the illustration above.  Thus, it obeys the inverse square law from an energy (inverse speed) spatial density perspective (force), and a speed temporal density perspective (acceleration). 

Of course, the temporal density of speed is an entirely new concept that will require much more development.  Larson found that its effect is reduced by the inter-regional ratio, a measured physical constant with a dimensionless value of 156.4444…, representing the directional properties of time relative to those of space.  More on this later.