DFT-8: Why Projection Becomes Necessary — From Scalar Motion to Observable Structure

[MWells]

By the time one reaches the rotational combinations of RS—vibration, two-dimensional rotation, the rotational base, and the entire hierarchy of atomic and sub-atomic displacements—one has already accepted a deeply non-Newtonian premise: the physical universe consists of nothing but motion, and everything we observe is some expression of that underlying scalar change. But there is a subtle point, running quietly beneath all these developments, that tends to remain implicit in Larson’s presentation. It is this point that DFT must eventually make explicit. The distinction is not a matter of convention; it is a structural fact that arises the moment one attempts to express an intrinsically non-geometric motion in a geometric coordinate frame.

The moment we say that a unit of scalar motion can appear as a linear vibration, or as a two-dimensional rotation, or as a deviation of coordinate time, or as a stable atomic configuration, we are implicitly admitting that the same underlying motion can be interpreted differently depending on how we view it. RS already demonstrates this repeatedly. A rotation that is purely inward in the scalar sense manifests outward or inward depending on whether it is interpreted as a motion in space or a motion in time. A photon’s progression depends entirely on which component—space or time—is taken as “unidirectional.” Gravitation, which is scalar inward motion, becomes an inward spatial acceleration only because the reference system is taken to be stationary in space rather than stationary in the natural system of motion. Even the universal recession is not a vectorial motion; it is simply the spatial aspect of progression viewed from a static frame.

In other words, every RS phenomenon presupposes an act of interpretation. Larson himself makes this clear in many places, but he never gives a formal rule describing how those interpretations emerge. The theory proceeds by switching frames when needed—sometimes looking from the fixed spatial reference frame, sometimes from the moving natural reference frame, sometimes treating space as the unidirectional component, sometimes treating time that way—but without a general mathematical principle governing these switches. In RS this switching is perfectly consistent, but the rule governing it is implicit rather than formal. His insights are correct; the missing piece is the unifying rule.

At this juncture in the series, now that linear and rotational motions are fully in place, it becomes impossible to proceed further without addressing the following question directly:

How does a scalar motion become an observable physical structure?
Or more precisely:
By what rule does the underlying scalar progression give rise to the spatial and temporal quantities we measure?
RS implicitly answers that question by relational reasoning, whereas DFT will explicitly answer it through projection maps.

This question arises naturally in RS, though it is never treated as a separate problem. The very act of assigning displacements to an atom presupposes that rotational magnitudes of scalar motion can be seen as spatial displacements, even though the motion itself exists in neither space nor time. Calling σ(λ) “pre-geometric” does not imply a lack of structure; it means that σ(λ) possesses neither coordinates, metric relations, nor dimensional decomposition until represented. Geometry is not intrinsic to σ(λ); it arises when σ is projected into a reference frame. The transition from scalar relations to spatial magnitudes is conceptually clear—Larson repeatedly explains what each quantity corresponds to—but mathematically undefined. RS has the ingredients of a projection theory but not a formal projection map, not because such a map contradicts RS, but because RS functions entirely in the interpreted domain and never needed to formalize the act of interpretation.

When rotational combinations become more intricate, this implicit interpretive act becomes increasingly important. The difference between a sub-atomic particle and an atom, the difference between material and cosmic structures, even the distinction between inward and outward motion—each hinges on how scalar motion is viewed from a spatial or temporal standpoint. Nothing in the scalar progression itself specifies a privileged interpretation. The distinction enters only when a frame is chosen.

This is the exact point where the necessity of projection enters—not as a modification of RS, but as its formal completion. DFT proposes that what RS has been calling “space” and “time” are in fact two complementary interpretive frames applied to a deeper scalar process. Neither frame is ontologically prior; together they form a complete representation of the scalar progression. A projection, in the DFT sense, is not merely a change of coordinates. It is a map from σ(λ) into a representational manifold that preserves some structural features of σ while suppressing others. Formally, a projection selects which relational aspects of σ(λ) are made observable. Thus the S-frame and T-frame do not add structure to σ; they partition and represent different aspects of the same scalar evolution. The S-frame corresponds to the interpretation in which spatial geometry is emphasized; the T-frame corresponds to the interpretation in which phase, rotation, and harmonic structure are emphasized. The two frames are not additive views of separate motions; they are orthogonal projections of the same scalar content. "Orthogonal" here does not mean orthogonal within a spatial metric. Rather, the orthogonality is informational: the representational degrees emphasized by the S-frame are not those emphasized by the T-frame, and vice versa. Each frame exposes complementary structure that cannot be recovered from the other alone, but neither contradicts the other. Neither frame contradicts RS; rather, each isolates structure RS inherently requires. Both represent aspects RS has been invoking since the beginning—Larson alternates between them whenever the situation requires—but RS never treats the distinction itself as mathematically consequential.

In DFT, the distinction becomes explicit. A scalar progression , which in the NRS is simply a unit change per unit change, admits two complementary representations Here σ(λ) is not embedded in spatial or temporal geometry. It is a parameterized path in a pre-geometric scalar domain: σ: Λ → ℝ, where λ ∈ Λ tracks intrinsic progression but does not itself carry spatial or temporal meaning. Observable geometric content is not present in σ(λ), but rather appears only after projection into a representational frame. When the representation emphasizes extension, orientation, and outward vs inward displacement, one obtains the familiar spatial picture of RS—the S-frame. When the representation emphasizes rotational structure, harmonic winding, and the quantization inherent in cyclic phases, one obtains the complementary temporal-phase picture—the T-frame. The two frames are not separate motions. They are distinct projections of one and the same scalar progression.

This duality is implicit in every major RS result. Gravitation becomes inward spatial acceleration only when the scalar inward rotation is read through the spatial frame; coordinate time accumulates only when the same rotation is read through the temporal frame. Sub-atomic particles differ from atoms not because they possess fundamentally different types of motion, but because the scalar content available for projection into three spatial dimensions is insufficient to generate a complete double rotation. This is a projection constraint, not a dynamical one. Even the distribution of rotational displacements implicitly assumes a way of “looking at” the scalar motion: we assign certain components to spatial dimensions and others to temporal ones.

Thus the need for projection is not a DFT overlay on RS; it is a structural fact already latent within RS mathematics, made explicit here for the first time. What DFT adds is the recognition that projection is a mathematically definable operation. The S-frame and T-frame are not metaphors; they are explicit maps from scalar motion into observable structure. And once one states this openly, consequences follow immediately: each projection can only reveal certain aspects of the scalar motion, neither projection can show all of it, and the two projections together impose constraints on each other. Those constraints, in turn, will become the motion budget, complementarity relations, and the eventual explanation of quantization.

But we are not yet there. DFT-8 has a more modest goal: to show why projection theory becomes unavoidable once rotation exists. In the following posts, we will formalize how the two frames arise, how they relate, and how they jointly recover the full RS displacement structures without contradiction.

In DFT-9 we will make this precise by defining the projection maps explicitly, specifying their domains and codomains, and showing how the scalar progression constrains what can appear simultaneously in the S-frame and T-frame. This will reveal why the motion budget and quantization follow as geometric consequences of projection, not as added assumptions.