The Larson Research Center

The biggest mystery of nature, as far as physicists are concerned, is how she can be both discrete and non-discrete, or continuous, at the same time. In the development of the LRC’s RST-based physical theory, we are using the Reciprocal System of Mathematics (RSM) to develop the consequences of the fundamental postulates of the RST, which introduces the science of discrete units of scalar motion to the existing science of continuous, or vector, motion. The motivation for developing a new math here is to improve the language of the LRC’s new science, which now includes both discrete and continuous magnitudes of motion, so that it can be more effectively used in the LRC’s theoretical development efforts.

To undertake this task requires an intimate understanding of how the language is to be used and how it refers to the physical world, but now that our theoretical world consists of magnitudes of motion that are simultaneously discrete and continuous, it necessarily implies solving the biggest mystery of nature. “How can we be so pretentious?” one might ask. Well, for one thing, when we started, we didn’t understand that what we were doing could be characterized like this. It’s only in hindsight that we can articulate the effort in these terms. For most of the time, the left hand didn’t know what the right hand was doing.

The first big breakthrough was the development of the Progression Algorithm (PA), which was adopted from rule 254 in Stephen Wolfram’s list of cellular automata rules, the most uninteresting of all 256 rules, as far as Wolfram is concerned. However, by reinterpreting the progression of space and time in this rule, in order to incorporate them as two, reciprocal, aspects of motion, we were able, for the first time, to graph the scalar progression and the “direction” reversals, which enabled us to visualize the meaning of these concepts and think about their implications in terms of numbers.

From this, it soon became apparent that only two space|time ratios were possible, 1|2 and 2|1, and that these had to be combined to get anything interesting, but combining them only led us back to the unit progression, 1|1! However, we eventually recognized that the sum of the numbers 1|2 and 2|1 was actually the sum of 2|2 and 2|2 in terms of total units of progression; that is, there is one unit of inward motion, for every unit of outward motion, in a cycle of two “direction” reversals. Hence, the complete equation for summing two, reciprocal, units of scalar space|time oscillation had to be

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

where num is an acronym for “natural units of motion.” However, at that point in time, we were still using the division symbol “/” in the equation, so the normal mathematics didn’t work! Using normal rules for adding rationals, the sum equation would be

ds/dt = 1/2 + 1/1 + 2/1
= (1/2 + 1/1) + 2/1
= (1+2)/2) + 2/1
= 3/2 + 2/1
= (3+4)/2
= 7/2 = 3.5 num.

Nevertheless, it was apparent that the normal rules for adding rationals was incorrect, and that the new rules, adding numerators to numerators, and denominators to denominators, gave the correct answer in every case. We knew what the correct answer had to be, given the PA. Therefore, we proceeded with the development with the new rules for adding rational numbers, but having no idea why this strange way of adding fractions worked.

With this new math, we were able to add (and subtract) SUDRs and TUDRs in various numbers, which led to SUDR|TUDR combinations like

ds/dt = (1/2 + 1/1 + 2/1) + 1/2 = 5/6 num, and

ds/dt = (1/2 + 1/1 + 2/1) + 2/1 = 6/5 num,

where there is an unequal number of SUDRs and TUDRs. We immediately recognized that this is just like adding integers, except that 0 is replaced by 1, in the form of 1/1. However, in terms of Larson’s concept of speed-displacement, 1/1 is 0 speed-displacement! Duh, clearly we were working with a reciprocal, or rational, form of integers here! The conventional rational rules didn’t work, precisely because this is a rational form of positive and negative integers, not a rational form of a decimal number, or a fraction, of a whole.

Nevertheless, while this enabled us to explore sums of SUDRs and TUDRs, as discrete combinations of motion, there was no way we could find to multiply these reciprocal numbers (RNs), which was strange, because integers can certainly be multiplied. Well, we could multiply RNs in a sense, but it was equivalent to repeatedly adding a unit of 1/2 or 2/1 motion. For example,

3/1 * 1/2 = (1/2 + 1/2 + 1/2) = (-1 + -1 + -1) = -3,

but this required interpreting 3/1 as 3 not -2. In other words, we could conceptualize what it meant to multiply three units of -1, and that the result was equivalent to the sum of three negative units, but we couldn’t show how to consistently employ the RN form to do it. Then we realized that we could multiply RNs by positive integers, if we multiplied both the numerator and the denominator by the number. For example,

ds/dt = 3(1/2+1/1+2/1) = 3 * 4/4 = 12/12,

which is the same result as summing three S|T units: 4/4 + 4/4 + 4/4 = 12/12,

but this seems like cheating and imbues the new RN equations with an invalid connotation. Thus, we didn’t like it and seldom used it to express sums of RNs. Not only that, but we didn’t really need a multiplication operation, as long as we were satisfied that it was a shorthand method for repeated addition. Along the way, we had discovered that the RN form of integers were what David Hestenes would call an operational interpretation (OI) of number, as opposed to the quantitative interpretation (QI) of number, which gave us a legitimacy that we didn’t expect in the beginning, so we sort of hid the multiplication issue under the rug, so-to-speak, while we enjoyed our legitimate status as having found a new form of integers that can be summed.

It was not long after this phase that we began studying the properties of groups and this seemed to offer us insight into why we couldn’t multiply our OI RNs: integers form a group under addition; They don’t form a group under multiplication, and they don’t form a field under addition and multiplication. The importance of group properties is that they enable combinations. Combinations of a group elements are mathematically sound, in that the elements of the group can be combined into elements that can be combined into other elements of the group mathematically, in all the ways that combining is important. So, we were very happy campers to discover that our set of SUDRs and TUDRs were a group under addition.

Then we came across the preon model of Sundance Bilson-Thompson, wherein he employs the concept of twisted ribbons, as a topological approach to explaining the properties of the standard model entities. It was almost immediately clear that we could implement this same model in the form of S|T units, and we quickly did so, but, while this constituted a major breakthrough in identifying the triple combinations of S|T units that would serve the purpose, it also introduced a need for a multiplication operation to represent the relation between the constituent S|T units, and not only this, but the operation had to be in the other sense of multiplication, the one that raises and lowers (through its inverse) the degree of freedom, not the one that merely expresses repeated addition.

It’s not like we went out to discover such an operation directly, but we were studying how to form the triplet preons. We took our clue from Sundance that the bosons were simply stacks of S|T units, representing superimposed S|T units, but the fermions were quite different. In his twisted ribbon model, the two possible directions of the twists of the ribbons correspond to the two possible polarities of our RNs, positive and negative. An untwisted ribbon corresponds to our neutral RN, so this was really good news, but then Sundance represents the different configurations of these twisted ribbons, which correspond to the different entities of the standard model, by braiding three of them together in unique ways. Of course, we couldn’t braid three S|T units!

What we soon discovered, however, is that S|T units could combine in the form of triangles, which was doubly fortunate, because in this form, the associated space|time locations of the S|T units cannot be superimposed, but they must remain as partially merged, adjacent, locations! Wow. This means that the theoretical bosons and the theoretical fermions, constructed in this way, have the same properties that distinguish physical bosons from physical fermions! That was really the headline of the good news, but the caveat in the fine print was that coupling adjacent locations raises the dimensions, or the degree of freedom, of the triplet, from the one-dimensional set of a stack of S|T units (really a line of points), to the two-dimensional set of a triangle of S|T units (really an area of adjacent points).

This means that the binary relation between the constituent S|T units of the fermion triplet is fundamentally different than the binary relation of the constituent S|T units of the boson triplet, which is just the sum relation of the OI RNs that we already understand fairly well. In other words, we were looking for another group to use with the fermions, but this time it had to be a group under multiplication.

It seems remarkable in retrospect, but the conventional rational numbers didn’t come to mind immediately. I guess this is because we sort of felt that we had to have a set of OI RNs, because QI RNs, the set of rationals, weren’t discrete, and we more or less relegated them to the domain of the LST community and vector motion. They can take any value, and thus are non-discrete, and we were thinking in terms of combining the discrete units of motion, the SUDRs and TUDRs at the apexes, or nodes of the triplets, so non-discrete values didn’t’ seem appropriate.

However, no matter how hard we tried, we couldn’t see how to multiply, or divide, OI RNs in a way that raises or lowers a dimension of the number. Then, once again, something remarkable happened that opened the way. We came across a CAT scan of a photon! This is the work of Lvovsky et al, wherein they do something that is charmingly described by Rebecca Slayton in her article on it entitled “Golfing with a Single Photon.”

The connection here was not immediately apparent, but it involved a very important aspect of SUDRs, TUDRs, and SUDR|TUDR combos, which we had never really discussed in detail before, because there was never a good opportunity to do so. It has to do with the weirdness of quantum mechanics. Slayton writes:

Where quantum mysteries are concerned, Schrödinger’s cat has nothing on a single photon—at least you’d have some chance of finding the feline, whether dead or alive. In contrast, if you looked for a photon in a small space, within a limited range of momentum, you’d seem to have a negative chance of finding it.

Lvovsky shows that the photon’s wave property, expressed as a probability of locating its position, or determining its momentum, is actually a combination of positive and negative probability, but what is a negative probability? The only answer Lvovsky has for this question is that it is “a strongly non-classical character” of a photon.

However, the scalar motion of the S|T combo is just this sort of property; that is, it’s an oscillation of locality to non-locality, in that it’s a transformation of a point (localized point) into a sphere (non-localized point) and back again. There are also two, reciprocal, aspects of this oscillation in the S|T unit, the SUDR and the TUDR. In other words, the S|T unit is precisely a combination of a positive and negative oscillation of probability amplitudes, where one is the inverse of the other.

This means that these two oscillations are instances of two, reciprocal, harmonic motions, which can be represented by two, reciprocal, rotations, the same rotation that is present in two meshed gears. The bottom line of all this is that the ratios of gear rotations, in a set called a gear train, are multiplied straight across, from numerator to denominator, just like we add OI RNs! In fact, if we attribute the size of the gear to the number of SUDRs and TUDRs in a given S|T unit, coupled with another S|T unit, to the number of SUDRs or TUDRs contributed by the respective S|T units, we get a new RN, that is quantitatively interpreted; that is, we get a QI RN that we can multiply in the same way that we add OI RNs, straight across. Thus,

ds/dt = 2:1 * 1:2 = 2:2 = 1:1 = 1

is the coupling of two QI RNs, where each QI RN is composed of two OI RNs. In other words, a QI RN is a ratio of ratios, but since the OI RNs, in the numerator and denominator of the QI RN, are equivalent to positive and negative integers, the set of QI RNs is isomorphic to the conventional rationals, the fractions of ordinary mathematics!

This is a remarkable and welcome insight, because it not only will permit us to handle the mathematics of the fermion’s nodes, but it also sheds a great light on the fundamental issue of how nature can be both discrete and non-discrete at the same time. To understand this, we need only see that the set of OI RNs is the inverse of the set of the QI RNs. This may seem strange and impossible at first, because it’s tantamount to saying that the integers are the inverse of the fractions, but we can see that, as a set, or group, this is true. To show it, we simply compare the respective elements of the sets.

Clearly, the elements in the set of integers extend from 0 to infinity in the positive “direction,” and from 0 to infinity in the negative “direction.” At the same time, the non-zero rationals extend from 1 to 0 in the positive “direction,” and from 1 to 0 in the negative “direction.” Illustrating this graphically, we can see the inverse relationship of the two sets:

1. OI RNs (discrete): -infinity <——————-> 0 <——————> + infinity
2. QI RNs (non-discrete): -0 <————————-> 1 <————————> + 0

Since the only difference in the OI RNs and the QI RNs is the quantitative and the operational interpretations of the reciprocally related quantities, we are free to choose either interpretation, depending on the situation. In other words, the RNs are both discrete and continuous, depending on our perspective. Thus, it seems that the perfect symmetry of reciprocity explains the dual nature of reality through chirality.

Selah