Two of the most important concepts of Larson's work, though exceedingly simple, are some of the least understood. They are the concepts of unit progression and unit speed-displacements. The progression algorithms (PAs) are even less understood, even though they are mathematical expressions of these two fundamental concepts of the RST.

Larson explains the concept of the unit progression in Chapter II of The Structure of the Physical Universe (SPU). He writes:

In beginning an examination of the consequences of the two Fundamental Postulates we note first that they involve a progression of space-time which is similar to the progression of time as it is ordinarily visualized. Let us consider a location A somewhere in space-time. During the next unit of time this location progresses to A + 1 in time and since one unit of time, on the basis of the First Fundamental Postulate, is equivalent to one unit of space the location also progresses to A + 1 in space. When n units of time have elapsed the location has progressed to A + n both in space and in time.

It should be emphasized that this statement does not refer to some object that might happen to occupy the location A; it refers to the location itself. If the hypothetical object has no independent motion of its own it will also be found at location A + n after n units of time, but this does not involve any motion of the object. It remains stationary at the same location in space-time but the location itself moves.

As previously noted, the PA of Wolfram's CA rule 254 is a 1D computer algorithm producing a graphical output of Larson's concept of space-time progression: In this PA, two, reciprocal, quantities increase with time, one increases to the left, which is identified with space, and one increases to the right, which is identified with time, as shown in figure 1 below.

Figure 1. Larson's Space-Time Progression as Produced by Wolfram's CA Rule 254

Starting with A =1, the number of units on the space side, and the number of units on the time side of the PA, will both equal A + n, when n units of time have elapsed. Larson explains further:

According to the First Postulate time and space are reciprocal quantities. Two units of space are therefore equivalent to one-half unit of time; ten units of time are equivalent to one-tenth unit of space and so on. Furthermore a single unit of space is the equivalent of a single unit of time. But when we postulate space-time as the sole component of the physical universe it follows that there can be no phenomena within the universe except those resulting from inequality between space and time; when the two are equal everything is uniformity and there are no physical phenomena of any kind. A space-time ratio of unity, indicating inequality between space and time, is therefore the condition of rest in the space-time universe, the datum level from which all action starts.

That is to say, space (time) is the reciprocal of time (space), so the appropriate understanding of the two sets of counting numbers produced by the 254 PA in figure 1 is that they are reciprocally related, as space/time = n/n.

Given the equality of the unit progression of figure 1, Larson's concept of the inequality inherent in this unit speed is also explained in Chapter II of SPU. He writes:

It is apparent from these same considerations that the significant measurement of space-time is not the total magnitude but the excess above unity. This quantity, which we will call the space or time displacement, is a measure of the divergence from the neutral unit level and hence a measure of effectiveness in physical phenomena. The occurrence of physical events depends entirely on the presence of these space and time displacements.

In other words, the reciprocal relation necessarily requires an inequality between these equal numbers on the left and on the right of the center square, on each row of the PA: That is, the ratio of 1/n is not equal to the ratio of n/1, if, and only if, the number n is greater than 1. Hence, while the n/n = 1/1 progression is the starting condition for the universe of motion, the datum level if you will, the action starts with a change from the n/n = 1/1 unit ratio to the 1/n and n/1 displacement ratios, where n > 1. 

Larson called this change from the unit datum, n/n = 1/1, to less than unity, 1/n, n > 1, or more than unity, n/1, n > 1, time speed-displacement and space speed-displacement, respectively. The numbers of this concept were discussed at the 1978 ISUS Conference, where the following chart was used to explain them:

Figure 2. 1978 Speed-Displacement Chart

Line 1 is simply the usual counting number sequence used to count units of speed-displacement, which of course starts with zero. Line 2 shows that the counting numbers can be used to count both positive and negative units of speed-displacement, where negative units are less than unity (1/n, n>1) and positive units are greater than unity (n/1, n>1), when viewed from the s/t perspective (Line 3), or where negative units are greater than unity (n/1, n>1) and positive units are less than unity (1/n, n>1) , when viewed from the t/s perspective (Line 4). This is why we can legitimately say that 1/1 = 0 units of speed-displacement.

In the next post, I will discuss some of the algebraic implications of these scalar units.