## Dimensions <<<———>>>>>>>>> UPR   Material Sector Formation                                  Displacing the Progression Ratio

Description:
The UPR can be displaced in two “directions,” the space “direction,” where the progression of time, t, is greater than that of space, s, and the time “direction,” where the progression of space, s, is greater than time, t.  In the material sector, as Larson labeled the velocity sector, the physical dimensions of the UPR are those of velocity, s/t.  When a constant change occurs in the “direction” of the progression of one of its two aspects, with respect to the reciprocal aspect, a space/time displacement is created in the unit progression, at the location of the constantly changing aspect.

The change in the progression is a change from continuous increase, to alternating increase/decrease.  This change in the progression pattern of one aspect of the motion, relative to its reciprocal aspect, creates a space/time displacement in the progression ratio of n units and changes the unit progression ratio from ds/dt = n/n to ds/dt = n/2n, in the case of a change in the space aspect’s progression pattern.

In the case of a change in the time aspect’s progression pattern, from the continuously increasing to the alternating increase/decrease pattern, what Larson termed “reversals,” a displacement of n units in the opposite “direction” is created, and this alters the unit progression ratio from ds/dt = n/n to ds/dt = 2n/n. (Note that the use of ‘d’ in the notation refers to a fixed delta of changing space or time, not the infinitesimal of the differential calculus.)

Since both aspects of the UPR are increasing scalar quantities, the only possible change in this increasing “direction” is a change to a decreasing “direction;” that is, the scalar progression in either case can be a continuously alternating increase/decrease of quantity, as well as a continuously increasing quantity.  When the progression of the space aspect is continuously increasing and decreasing, between two adjacent spatial locations in the progression, while the time aspect is continuously increasing, a unit displacement in the space/time progression ratio is created.

In the LRC terminology, the new progression ratio, ds/dt = n/2n is a unit (n = 1) displacement caused by reversals in the space aspect of the progression, and it is therefore referred to as the space unit displacement ratio, or SUDR, because it is created by continuous “direction” reversals in the space aspect of the UPR, at a given location. In the case where the change to a continuous “direction” reversal pattern occurs in the time aspect of the progression, the new progression ratio, ds/dt = 2n/n, created by a unit (n =1) displacement, in the opposite “direction,” is referred to as the time unit displacement ratio, or TUDR, because it is created by continuous “direction” reversals in the time aspect of the UPR.

Once SUDRs and TUDRs are in existence, the possibility exists that they can interact and combine with each other, due to the fact that a stationary SUDR progresses in time, while a stationary TUDR progresses in space. The combination of one SUDR and one TUDR creates the basic theoretical entity of the material sector, referred to as the SUDR|TUDR combo, or S|T for short.  The mathematical equation of the S|T combo constitutes a reciprocal number (RN) in the Reciprocal System of Mathematics (RSM).  The form of the RN is:

((1/2) + (1/1) + (2/1)) = (4/4) num,         [1]

where “num” is short for natural units of motion, with physical dimensions s/t.  The terms of the RN are operationally interpreted ratios, or proportions of space and time progression rates.  Therefore, numerators are added to numerators, and denominators are added to denominators, in summing the terms.  The most general form of the RN is

((xn/2xn) +  (xn/yn) + (2yn/yn)) = (2xn + 2yn)/(2xn+ 2yn),         [2]

where x is the number of SUDRs and the y is the number of TUDRs in the S|T combo. When n = x = y = 1, equation [1] is obtained.

The SPU document (SPUD) hyperlinks, at the top of the page, allow the user to navigate to the documentation of the different possibilities of these combinations in the SPUD development.

Discuss the Material Formation Description
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The RSt Concept:
Larson describes the SUDR in Chapter II of the preliminary edition of his The Physical Structure of the Universe, even though he doesn’t use the LRC terminology:

…The reciprocal postulate includes the further requirement that under certain conditions associations of n units of one component must exist and that under those conditions the n units of this kind are equivalent to 1/n units of the other component.

We are then led to inquire how it can be possible for n units of space or time to act as an association when each of the individual units in this association is required to Progress uniformly with a unit of the opposite kind as an integral part of the general space-time progression…

… It is apparent that where n units of one component replace a single unit in association with one unit of the other kind in a linear progression, the direction of the multiple component must reverse at each end of the single unit of the opposite variety. Since space-time is scalar the reversal of direction is meaningless from the space-time standpoint and the uniform progression, one unit of space per unit of time, continues just as if there were no reversals. From the standpoint of space and time individually the progression has involved n units of one kind but only one of the other, the latter being traversed repeatedly in opposite directions.

Discuss the RSt Concept
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The LRC Concept:
The major difference here is that, where Larson refers to the word direction, as if these scalar values had direction, as in “the latter being traversed repeatedly in opposite directions,” at the LRC, we make the crucial distinction between “direction” and direction.  Direction is a property of vectors, while “direction” is a quite different property of scalars.  A scalar quantity increase is in the opposite “direction” of a scalar quantity decrease, but the meaning of the word “direction” is very different from the meaning of the word direction, as used in connection with vectorial concepts of motion.

A good example of this difference is found in the financial world, where prices are commonly referred to as “going up,” or “going down” like an elevator, but these two opposite “directions” of up and down are understood as simply increases or decreases in the price of a stock or commodity, a scalar value.  There is no concept of a literal movement in a linear direction, with respect to scalars.

Likewise, in the case of a spatial concept of scalar, an increase, or decrease, in space is not a literal movement of “space” in a spatial direction, but simply an increase, or decrease, of a quantity of space relative to a previous, or subsequent, quantity of space.  For example, the increase in the volume of a balloon, is a “movement” in the “direction” of increasing volume, which is opposite to the “movement” in a decreasing “direction,” when its volume is decreased.  Obviously, this use of the word “direction,” in connection with the scalar increase/decrease of the volume of a balloon, has a different meaning than the meaning of the word direction in connection with the vectorial motion of objects.

Discuss the LRC Concept
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The LST Concept:
Again, the LST community has no concept of scalar motion; that is, it has no physical concept of space as the reciprocal of time, in the equation of motion, but the mathematics community has an important mathematical concept that is similar.  This mathematical concept is called the operational interpretation of number, and it refers to the value of a number that is found in its relationship to another number.  Thus, the value of an operationally interpreted rational number, such as  1/2 can be interpreted as -1, as opposed to the value of the operationally interpreted rational number, 2/1, which can be interpreted as +1.

This is nothing more than the ancient concept of “balance” used in monetary trade and finance. A negative balance is in the opposite monetary “direction” of a positive balance, and a zero balance is achieved when negative magnitudes are balanced by positive magnitudes, as when the weight on one side of a pan balance is equal to the weight on the other side.  Hence, the balance of operationally interpreted rational numbers, as in

1/2 = 2/1 = 1,

is the same, in the absolute sense, as

|-1| = |+1| = 1

in the quantitative interpretation of integers.

However, in the LST community, the operational interpretation of rational numbers, and their scalar “direction” property, has been disregarded in favor of the direction property of vectors, achieved through the use of imaginary numbers.  This is understandable, given that the LST is a vector based system, but the importance of operationally interpreted rational numbers, to the scalar-based RST, is obvious.

Discuss the LST Concept