DFT-2: Scalar Progression and the Natural Reference System
Posted: Tue Dec 02, 2025 10:26 pm
DFT: Scalar Progression and the Natural Reference System
In the first post I talked about how this work was developed—using AI tools iteratively, testing formulations against RS constraints, and stabilizing something I’m now calling Dual-Frame Theory (DFT). Before saying anything further about “frames” or “projections”, I want to step back and lay out the foundations exactly where Larson put them: in a universe of motion, not a universe of matter.
Everything that follows in this post is meant to be faithful to that starting point. DFT is not a replacement for these ideas; it is built on them.
Larson’s thesis is stark and simple: the universe in which we live is not fundamentally made of “stuff” in a container, but of motion. All of the familiar entities—atoms, radiation, gravitation, electric charge—are different manifestations or combinations of motion. This is not just saying “motion causes things”; it’s stronger. It is saying that what we call “things” are themselves motions.
Once you take that seriously, space and time can no longer play the role they usually do in physics—some sort of pre-existing stage, “a vast world-room” on or in which matter sits. If motion is the basic reality, and motion is defined in the usual scientific way as a relation between space and time, then space and time are not independent substances or a background. They are the two aspects of motion. They only exist as aspects of motion.
In ordinary physics you can say: space and time are what things are in. In a universe of motion you cannot. You have to say instead:
That’s the first crucial shift.
From that point, Larson adopts a very specific and very constrained set of postulates. In words: the physical universe is made entirely of motion, that motion comes in discrete units, it exists in three dimensions, and it has two reciprocal aspects—space and time. The mathematics is assumed to be ordinary commutative math with absolute magnitudes, and the geometry is Euclidean. None of that is exotic. The only really new piece is the insistence that motion, not matter, is the sole component—and that space and time are just its two faces.
Discrete units of motion do not mean that motion happens in jerks or jumps. Larson emphasizes the word progression. The basic motion is a continuous progression where each completed unit is immediately followed by the next, like links in a chain: discrete as units, continuous as a whole. Each such unit is one unit of space associated with one unit of time. This gives the natural unit speed: one unit of space per unit of time.
That unit speed—this “one-to-one” progression—is the most primitive, featureless state. If every unit of motion was just that, with no deviations, the universe would be an undifferentiated uniformity. To get anything physically observable—any phenomenon at all—there has to be deviation from unit speed, either upward or downward. Physical activity always extends from unity, not from zero.
It is here that Larson introduces a key idea: the natural reference system is not a stationary grid. Because every unit of motion involves one unit of space and one of time, continuation of motion necessarily implies a continued increase in both. If you pick out what we might naïvely call a “fixed point” and then watch what happens as unit progression continues, you find that, relative to that fixed coordinate system, every absolute location is continually moving outward at unit speed. What we normally take as “at rest” in a coordinate system is, in the natural system, participating in this outward progression.
So the natural system of reference is a moving system. Every absolute location is continually being carried outward, away from other locations, by the basic scalar progression.
This is where the distinction between scalar motion and vectorial motion becomes essential.
Larson spends a lot of effort clarifying this. Scalar motion has magnitude but no inherent geometric direction. It is simply “outward” or “inward” in a scalar sense when represented in a stationary frame—not a direction along x, y, or z. Vectorial motion, by contrast, has a definite direction in a coordinate system. The familiar motions of daily life—moving a car east or west, orbiting a planet, etc.—are vectorial. The basic progression of the natural reference system is scalar.
A standard analogy Larson uses is a balloon. If a bug crawls on the surface, its motion is vectorial: it moves from A to B along a definite path on the surface. If instead the balloon itself expands, every point on the surface recedes from every other point, not because those points are “choosing a direction”, but because the whole surface is stretching. That expansion is scalar. In that second case, if you dress it up in a coordinate system, any local “direction” you assign to the expansion is coming from the reference frame, not from the motion itself.
The outward motion of the natural reference system is like that expansion. It has a scalar magnitude and a scalar polarity (outward). It does not pick out a special geometric axis.
This is also where absolute location enters. Each absolute location stays where it is in the natural system, but relative to a fixed coordinate grid it appears to move outward at unit speed. Objects that have no independent motion—photons in their simplest description—remain at the same absolute locations and are carried outward by the progression. Relative to a stationary reference system, that looks like a motion away from an origin. As Larson shows later, an emitting source stationary in a fixed frame becomes the center of an expanding sphere: not because “space” as a container is expanding, but because the natural reference system is progressing while we insist on describing things in a fixed one.
At this point, it’s also important to be clear about “space”. Larson separates what he later calls extension space—the three-dimensional coordinate system we use to represent positions—from the spatial aspect of motion itself. Extension space is a reference system, a bookkeeping grid. It has no physical action; it is not a medium. The real “space” of physical processes is the space that appears in the motion ratio 𝑠/𝑡 s/t. The balloon-like spheres that the basic progression defines around emission points are not a material or energetic medium; they are the geometry of a reference system when you insist on describing scalar progression against a fixed grid.
The same goes for time: it is not a flowing substance, nor a quasi-spatial dimension joined onto three spatial ones. It is the reciprocal aspect of motion, just as space is the direct aspect. Both exist only in association as motion.
From there, Larson moves to what he calls independent units of motion. These are real units of motion—just like the units of progression—but they are not merely part of the background, they exist as discrete additions superimposed on that background. Their existence is permitted by the postulates, and since anything not excluded exists, they must be present. However, the postulates provide no mechanism for creating or destroying such units, so the total number of effective units of independent motion is fixed. This is the root of the conservation laws: if all physical entities are motions, and the independent motion units that make them up cannot be created or annihilated by any internal process, then conserved quantities are ultimately counting constraints on such units.
Now we can talk about specific kinds of basic motion. Under the postulates, a few are allowed at this most primitive level.
First, there is the ever-present unit progression itself, which outwardly is always there as part of the natural reference system. Because that is always present, any inward continuous motion must exist as a deviation superimposed on that outward component. You cannot have a net effective outward independent motion at this stage, because the discrete-unit requirement prevents you from adding a fractional bit of space or time to the unit speed. The only effective independent continuous motion that can appear is inward, and it must be uniform and continuous because no mechanism for discontinuity has been introduced yet.
Second, Larson identifies simple harmonic motion—linear vibration—as an admissible basic motion. This is a motion in which the scalar direction reverses at each end of a unit. It is scalar in the same sense as the progression: it is a relation between space and time with reversals of scalar sign. Viewed in a fixed reference system, this appears as a vibration back and forth over a unit of space (or time), with a definite path that can be described because the unit itself, like a link of a chain, has internal distinguishable portions.
Larson spends a lot of time being careful about the word “direction” here. Scalar direction is “inward” or “outward” in the representational sense; vectorial direction is the usual directional information in a coordinate system. Scalar motion has only inward/outward polarity and no inherent vectorial orientation; any apparent vectorial direction we assign to it comes from how we embed it in a reference frame. Indiscriminately using “direction” to cover all these meanings causes confusion, and he tries to avoid that by qualification.
The important point for our purposes is that simple harmonic motion is allowed by the postulates, it is continuous and uniform (in the sense of the projection from circular motion), and at this basic level it is confined to a single unit. The reversals of scalar direction all happen within that unit; they do not jump across unit boundaries.
Now, combine that scalar vibration with the unit progression of the natural reference system, oriented perpendicularly in the representational sense, and what you get—when described in a stationary coordinate system—is a sine-curve-like path moving outward. This is precisely the behavior we associate with radiation.
Larson identifies this independent unit of simple harmonic motion as the photon, and the outward unit progression as the “speed of radiation” or speed of light. The famous wave-particle duality becomes almost trivial: as a discrete unit of motion, the photon is particle-like in emission and absorption; as a vibratory motion combined with progression, its path in a fixed coordinate system is wave-like. What looked like a paradox is resolved by recognizing that what we call “radiation” is not a single thing but a combination of scalar vibration and scalar progression represented in a spatial frame.
This is all still within Larson’s universe of motion. We haven’t added anything beyond what he already spelled out.
So where does DFT come in?
At this stage, what I am calling DFT does not change any of this. The ontology is exactly as Larson described it:
Specifically, I introduce a neutral symbol for the underlying scalar progression—something like σ(λ)—to stand for “the bare scalar motion before any geometric interpretation.” This symbol is not a trajectory, not a vector, not a worldline. It is just a way of saying: here is the scalar change that exists in the Natural Reference System, prior to us mapping it into extension space or any other frame.
Later posts will then take that underlying scalar progression and show how two different kinds of projection recover the structures we already know from RS: one that produces the spatial, translational picture (extension, positions, distances), and another that produces cyclic, rotational, and vibrational structures (frequency, phase, spin-like behavior). The key point for now is that those projections are not replacing any of Larson’s work—they are a way of making explicit the step he left implicit: how we go from “nothing but motion” to the familiar geometrical and physical structures.
In other words, this post is still wholly inside Larson’s territory. DFT enters only as a slight sharpening of language and notation, so that when we talk about S- and T-frame projections later, we are absolutely clear what we are projecting from.
In the next installment, I plan to take this foundation and begin to talk about how scalar motion appears when we interpret it as spatial structure—that is, how the S-frame (in DFT language) lines up with Larson’s use of extension space and translational motion, and why the moving natural reference system plays such a central role in phenomena like cosmic expansion and gravitational balance.
As always, questions, corrections, and challenges are welcome—especially if you spot anywhere I’ve misrepresented Larson’s own words. The goal here is to preserve his foundations exactly, then build DFT on top of that, not sideways.
In the first post I talked about how this work was developed—using AI tools iteratively, testing formulations against RS constraints, and stabilizing something I’m now calling Dual-Frame Theory (DFT). Before saying anything further about “frames” or “projections”, I want to step back and lay out the foundations exactly where Larson put them: in a universe of motion, not a universe of matter.
Everything that follows in this post is meant to be faithful to that starting point. DFT is not a replacement for these ideas; it is built on them.
Larson’s thesis is stark and simple: the universe in which we live is not fundamentally made of “stuff” in a container, but of motion. All of the familiar entities—atoms, radiation, gravitation, electric charge—are different manifestations or combinations of motion. This is not just saying “motion causes things”; it’s stronger. It is saying that what we call “things” are themselves motions.
Once you take that seriously, space and time can no longer play the role they usually do in physics—some sort of pre-existing stage, “a vast world-room” on or in which matter sits. If motion is the basic reality, and motion is defined in the usual scientific way as a relation between space and time, then space and time are not independent substances or a background. They are the two aspects of motion. They only exist as aspects of motion.
In ordinary physics you can say: space and time are what things are in. In a universe of motion you cannot. You have to say instead:
- motion is primary,
- and space and time are the numerator and denominator of the motion ratio.
That’s the first crucial shift.
From that point, Larson adopts a very specific and very constrained set of postulates. In words: the physical universe is made entirely of motion, that motion comes in discrete units, it exists in three dimensions, and it has two reciprocal aspects—space and time. The mathematics is assumed to be ordinary commutative math with absolute magnitudes, and the geometry is Euclidean. None of that is exotic. The only really new piece is the insistence that motion, not matter, is the sole component—and that space and time are just its two faces.
Discrete units of motion do not mean that motion happens in jerks or jumps. Larson emphasizes the word progression. The basic motion is a continuous progression where each completed unit is immediately followed by the next, like links in a chain: discrete as units, continuous as a whole. Each such unit is one unit of space associated with one unit of time. This gives the natural unit speed: one unit of space per unit of time.
That unit speed—this “one-to-one” progression—is the most primitive, featureless state. If every unit of motion was just that, with no deviations, the universe would be an undifferentiated uniformity. To get anything physically observable—any phenomenon at all—there has to be deviation from unit speed, either upward or downward. Physical activity always extends from unity, not from zero.
It is here that Larson introduces a key idea: the natural reference system is not a stationary grid. Because every unit of motion involves one unit of space and one of time, continuation of motion necessarily implies a continued increase in both. If you pick out what we might naïvely call a “fixed point” and then watch what happens as unit progression continues, you find that, relative to that fixed coordinate system, every absolute location is continually moving outward at unit speed. What we normally take as “at rest” in a coordinate system is, in the natural system, participating in this outward progression.
So the natural system of reference is a moving system. Every absolute location is continually being carried outward, away from other locations, by the basic scalar progression.
This is where the distinction between scalar motion and vectorial motion becomes essential.
Larson spends a lot of effort clarifying this. Scalar motion has magnitude but no inherent geometric direction. It is simply “outward” or “inward” in a scalar sense when represented in a stationary frame—not a direction along x, y, or z. Vectorial motion, by contrast, has a definite direction in a coordinate system. The familiar motions of daily life—moving a car east or west, orbiting a planet, etc.—are vectorial. The basic progression of the natural reference system is scalar.
A standard analogy Larson uses is a balloon. If a bug crawls on the surface, its motion is vectorial: it moves from A to B along a definite path on the surface. If instead the balloon itself expands, every point on the surface recedes from every other point, not because those points are “choosing a direction”, but because the whole surface is stretching. That expansion is scalar. In that second case, if you dress it up in a coordinate system, any local “direction” you assign to the expansion is coming from the reference frame, not from the motion itself.
The outward motion of the natural reference system is like that expansion. It has a scalar magnitude and a scalar polarity (outward). It does not pick out a special geometric axis.
This is also where absolute location enters. Each absolute location stays where it is in the natural system, but relative to a fixed coordinate grid it appears to move outward at unit speed. Objects that have no independent motion—photons in their simplest description—remain at the same absolute locations and are carried outward by the progression. Relative to a stationary reference system, that looks like a motion away from an origin. As Larson shows later, an emitting source stationary in a fixed frame becomes the center of an expanding sphere: not because “space” as a container is expanding, but because the natural reference system is progressing while we insist on describing things in a fixed one.
At this point, it’s also important to be clear about “space”. Larson separates what he later calls extension space—the three-dimensional coordinate system we use to represent positions—from the spatial aspect of motion itself. Extension space is a reference system, a bookkeeping grid. It has no physical action; it is not a medium. The real “space” of physical processes is the space that appears in the motion ratio 𝑠/𝑡 s/t. The balloon-like spheres that the basic progression defines around emission points are not a material or energetic medium; they are the geometry of a reference system when you insist on describing scalar progression against a fixed grid.
The same goes for time: it is not a flowing substance, nor a quasi-spatial dimension joined onto three spatial ones. It is the reciprocal aspect of motion, just as space is the direct aspect. Both exist only in association as motion.
From there, Larson moves to what he calls independent units of motion. These are real units of motion—just like the units of progression—but they are not merely part of the background, they exist as discrete additions superimposed on that background. Their existence is permitted by the postulates, and since anything not excluded exists, they must be present. However, the postulates provide no mechanism for creating or destroying such units, so the total number of effective units of independent motion is fixed. This is the root of the conservation laws: if all physical entities are motions, and the independent motion units that make them up cannot be created or annihilated by any internal process, then conserved quantities are ultimately counting constraints on such units.
Now we can talk about specific kinds of basic motion. Under the postulates, a few are allowed at this most primitive level.
First, there is the ever-present unit progression itself, which outwardly is always there as part of the natural reference system. Because that is always present, any inward continuous motion must exist as a deviation superimposed on that outward component. You cannot have a net effective outward independent motion at this stage, because the discrete-unit requirement prevents you from adding a fractional bit of space or time to the unit speed. The only effective independent continuous motion that can appear is inward, and it must be uniform and continuous because no mechanism for discontinuity has been introduced yet.
Second, Larson identifies simple harmonic motion—linear vibration—as an admissible basic motion. This is a motion in which the scalar direction reverses at each end of a unit. It is scalar in the same sense as the progression: it is a relation between space and time with reversals of scalar sign. Viewed in a fixed reference system, this appears as a vibration back and forth over a unit of space (or time), with a definite path that can be described because the unit itself, like a link of a chain, has internal distinguishable portions.
Larson spends a lot of time being careful about the word “direction” here. Scalar direction is “inward” or “outward” in the representational sense; vectorial direction is the usual directional information in a coordinate system. Scalar motion has only inward/outward polarity and no inherent vectorial orientation; any apparent vectorial direction we assign to it comes from how we embed it in a reference frame. Indiscriminately using “direction” to cover all these meanings causes confusion, and he tries to avoid that by qualification.
The important point for our purposes is that simple harmonic motion is allowed by the postulates, it is continuous and uniform (in the sense of the projection from circular motion), and at this basic level it is confined to a single unit. The reversals of scalar direction all happen within that unit; they do not jump across unit boundaries.
Now, combine that scalar vibration with the unit progression of the natural reference system, oriented perpendicularly in the representational sense, and what you get—when described in a stationary coordinate system—is a sine-curve-like path moving outward. This is precisely the behavior we associate with radiation.
Larson identifies this independent unit of simple harmonic motion as the photon, and the outward unit progression as the “speed of radiation” or speed of light. The famous wave-particle duality becomes almost trivial: as a discrete unit of motion, the photon is particle-like in emission and absorption; as a vibratory motion combined with progression, its path in a fixed coordinate system is wave-like. What looked like a paradox is resolved by recognizing that what we call “radiation” is not a single thing but a combination of scalar vibration and scalar progression represented in a spatial frame.
This is all still within Larson’s universe of motion. We haven’t added anything beyond what he already spelled out.
So where does DFT come in?
At this stage, what I am calling DFT does not change any of this. The ontology is exactly as Larson described it:
- motion as the sole constituent
- space and time as reciprocal aspects of motion
- discrete units
- unit progression
- a moving natural reference system
- scalar vs vectorial motion
- independent units of motion
- linear vibration as the first independent unit (the photon)
Specifically, I introduce a neutral symbol for the underlying scalar progression—something like σ(λ)—to stand for “the bare scalar motion before any geometric interpretation.” This symbol is not a trajectory, not a vector, not a worldline. It is just a way of saying: here is the scalar change that exists in the Natural Reference System, prior to us mapping it into extension space or any other frame.
Later posts will then take that underlying scalar progression and show how two different kinds of projection recover the structures we already know from RS: one that produces the spatial, translational picture (extension, positions, distances), and another that produces cyclic, rotational, and vibrational structures (frequency, phase, spin-like behavior). The key point for now is that those projections are not replacing any of Larson’s work—they are a way of making explicit the step he left implicit: how we go from “nothing but motion” to the familiar geometrical and physical structures.
In other words, this post is still wholly inside Larson’s territory. DFT enters only as a slight sharpening of language and notation, so that when we talk about S- and T-frame projections later, we are absolutely clear what we are projecting from.
In the next installment, I plan to take this foundation and begin to talk about how scalar motion appears when we interpret it as spatial structure—that is, how the S-frame (in DFT language) lines up with Larson’s use of extension space and translational motion, and why the moving natural reference system plays such a central role in phenomena like cosmic expansion and gravitational balance.
As always, questions, corrections, and challenges are welcome—especially if you spot anywhere I’ve misrepresented Larson’s own words. The goal here is to preserve his foundations exactly, then build DFT on top of that, not sideways.