DFT-10: The Mathematical Structure of Projection — Formal Definitions of the S-Frame and T-Frame

MWells · Post by **MWells** » Wed Dec 03, 2025 3:44 pm

In the previous posts, we established the two central ingredients that make projection possible:

1. A single underlying scalar progression that has no geometry of its own.

2. Two independent geometric structures—space and time—each three-dimensional, each capable of hosting rotation, each constructed by abstraction from the scalar motion.

The Reciprocal System already contains these ingredients. It gives us a scalar progression; it gives us three-dimensional coordinate space and three-dimensional coordinate time; and it establishes that motion can accumulate structure in either aspect. What DFT adds now is a precise mathematical rule for how these two geometric structures can represent the same underlying scalar progression differently.

This post formalizes that rule.

1. The Underlying Object: A Scalar Progression Without Geometry

We begin by naming the primitive of the theory:

A scalar progression is a single-parameter process

$\sigma : \Lambda \to \mathbb{R}, \qquad \lambda \mapsto \sigma(\lambda)$

where:

$\lambda$ is an intrinsic parameter, not spatial or temporal,
$\sigma(\lambda)$ has no geometric content until projected,
the only requirement is that it advances uniformly in magnitude.

$\text{No inner product, metric, or affine structure is defined on }\Lambda.$

This is compatible with Larson’s natural progression: a unit motion in discrete steps, unassigned to space or time until interpreted.

DFT does not modify the RS postulate; it merely restates it in a way suitable for mathematical projection.

2. Two Target Geometries: Coordinate Space and Coordinate Time

Next, recall the unambiguous RS statement:

Space is three-dimensional.
Time is also three-dimensional.
Both are defined by the same “ordinary commutative mathematics” of Euclidean geometry.

Thus we have two 3-dimensional manifolds:

Spatial coordinate manifold: $\mathbb{R}^{3}_{S}$
Temporal coordinate manifold: $\mathbb{R}^{3}_{T}$

In RS, both originate as abstractions from the scalar motion. They are not ontological “things” but coordinate systems for describing aspects of motion.

In standard physics, time is usually treated as a single scalar parameter, but in RS and DFT the temporal aspect of motion is allowed three independent degrees of freedom, just as space is. The observable one-dimensional "clock time" of everyday physics then appears as a derived scalar measure of path length or progression in this three-dimensional temporal manifold (for example, as a norm or a preferred projection of $\mathbb{R}^{3}_{T}$ . Thus $\mathbb{R}^{3}_{T}$ is not in conflict with ordinary 1D time; it refines it, providing the internal phase/rotation structure that underlies the single time parameter used in conventional formalisms.

In DFT, these two manifolds will be the target spaces of the projection maps.

3. The Projection Maps: What the S-Frame and T-Frame Actually Are

The S-frame and T-frame are mathematical maps from scalar progression into geometric representation.

They do not change the scalar motion.
They only change how it is represented.

Definition 1 — The S-Projection

$F_{S} : \Lambda \to \mathbb{R}^{3}_{S}, \qquad \lambda \mapsto x(\lambda)$

The S-projection represents scalar progression as spatial structure, meaning:

increment in scalar magnitude appears as outward or inward spatial displacement,

rotational content is interpreted as rotation in coordinate space,

the “natural reference system” (unit progression) shows up as a spatial drift.

In RS terms, the S-frame corresponds to the familiar outward progression and the spatial interpretation of rotational displacement.

Definition 2 — The T-Projection

$F_{T} : \Lambda \to \mathbb{R}^{3}_{T}, \qquad \lambda \mapsto \theta(\lambda)$

The T-projection represents the same scalar progression as temporal structure:

increment in scalar magnitude appears as accumulation of coordinate time,
rotational content is interpreted as rotation in time,
natural progression manifests through the temporal sector rather than the spatial.

In RS, this corresponds to the “time-space” expression of motion.

The key point

The S-frame and T-frame are homomorphic representations of the same scalar object.
They preserve the structure of the changes in $\sigma(\lambda)$ but not their geometric form.

Nothing in RS is violated here:

RS already recognizes “space” and “time” as reciprocal representations of motion.
It does not provide a formal map between them.
DFT now supplies that map.

In this post we have defined $F_{S}$ and $F_{T}$ at the level of their domains, codomains, and consistency constraints. In DFT-11 and beyond, we will specify their explicit form for the concrete RS motions—linear vibration, 2-D rotation, and compound rotational patterns—so that actual trajectories $x(\lambda)$ and $\theta(\lambda)$ can be constructed and compared with known physical systems.

4. The Coupling Rule: How One Frame Constrains the Other

Projections are not independent. They are related by a rule derived from the fact that they represent the same scalar progression.

Definition 3 — Projection Consistency

For every intrinsic step $\Delta \sigma$ , the projections must satisfy:

$\bigl\| R_{S}(\lambda) \bigr\|^{2} + \bigl\| R_{T}(\lambda) \bigr\|^{2} = C^{2}\!\bigl(\sigma(\lambda)\bigr).$

In natural RS units, we can normalize the scalar progression so that

$C\bigl(\sigma(\lambda)\bigr) = 1,$

Physically, $C(\sigma(\lambda))$ encodes the scalar "intensity" or magnitude of the inward rotational content associated with the underlying motion. Different systems or energy scales correspond to different choices of this scalar. However, because $\lambda$ is an intrinsic parameter with no fixed metrical meaning, we are free to reparameterize $\lambda$ and the associated units so that, for a given class of motions under study, the effective budget $C$ is normalized to unity. This choice does not lose generality; it simply fixes the natural RS units in which the motion budget is measured. When we later compare different physical systems, the relevant scale factors will reappear in the mapping between RS units and conventional physical units (e.g., energy or frequency).

The consistency rule becomes

$\|R_{S}(\lambda)\|^{2} + \|R_{T}(\lambda)\|^{2} = 1.$

This is the core of the motion budget in mathematical form.

It says:

the S-frame and T-frame share the same scalar “budget” of change,
if one projection accumulates more geometric structure, the other must accumulate less,
total structure is limited because it originates from a single scalar source.

This is a purely mathematical requirement: two different geometric descriptions of one scalar motion cannot both take full magnitude independently.

In RS language:
“Motion in one aspect limits motion in the other.”

DFT does not add this; DFT simply formalizes it.

5. Rotational Structure Under Projection

Rotational content of scalar progression appears differently in each frame.

Let $R(\lambda)$ be the scalar rotation content (in the RS sense—an inward scalar motion).

Then:

In the S-frame, rotation becomes inward spatial displacement (magnetic/electric displacement).
In the T-frame, the same rotation becomes temporal winding.

Both are valid interpretations of a single inward scalar component.

In what follows, $F_{S}$ and $F_{T}$ are assumed to be sufficiently smooth that we can speak of their rates of change with respect to the intrinsic parameter $\lambda$. The projected motion in each frame then decomposes into a radial (progression-like) part and a tangential (rotation-like) part. We use $R_{S}(\lambda)$ and $R_{T}(\lambda)$ to denote precisely these tangential, rotation-bearing components of the projected motion in the S- and T-frames, respectively: they are the rotational parts of $\mathrm{d}F_{S}/\mathrm{d}\lambda$ and $\mathrm{d}F_{T}/\mathrm{d}\lambda$ once the purely radial progression has been factored out. In RS language, they are the projected expressions of inward scalar rotation in space and in time.

Let $R(\lambda)$ denote the scalar rotational content of the motion (in the RS sense of inward scalar rotation), and let $R_S(\lambda)$ and $R_T(\lambda)$ be its projections into the S- and T-frames, respectively.

Then we have a single scalar budget

$\|R(\lambda)\| = C\!\bigl(\sigma(\lambda)\bigr),$

This means the scalar inwardness has a single magnitude, independent of representation. The projections cannot create additional magnitude; they must divide this one magnitude between them.

$\|R_{S}(\lambda)\|^{2} + \|R_{T}(\lambda)\|^{2} = \|R(\lambda)\|^{2}.$

Geometrically, this is nothing more than the statement that $R_{S}$ and $R_{T}$ are orthogonal components of a single underlying scalar rotation when expressed in the two complementary representational frames. The Pythagorean form of the relation is not an extra assumption; it follows from treating the S- and T-frame rotational contributions as information-orthogonal ways of expressing the same inward scalar content.

This is the formal mathematical expression of Larson’s statement that rotational motion has scalar direction and can appear as inward structure in either space or time, but the total inwardness is single and cannot be duplicated.

In DFT this becomes the basis for quantization: each projection can only host an integer amount of rotational winding.

More precisely, when the projected rotational patterns are required to be single-valued up to an overall phase—so that a complete cycle of the motion returns the system to an equivalent state—the allowed configurations are classified by integer winding numbers. These integers count how many times the projected trajectory wraps around a closed loop in the appropriate S-frame or T-frame angular coordinates. The detailed derivation of the familiar RS displacement integers $(a,b,c)$ from these winding numbers will be the focus of subsequent posts.

6. Why These Definitions Matter

With these definitions in hand, three major consequences follow immediately:

1. RS displacement triplets $(a,b,c)$ become S-frame cross-sections of a single scalar winding pattern.
2. The T-frame provides the complementary winding structure needed to account for interference, phase, coherence, and quantization.
3. The motion budget becomes a general principle, not just a gravitational phenomenon.

Crucially:

RS remains intact.
RS structures are recovered exactly when the T-projection carries minimal additional structure.
Classical physics appears when the S-projection overwhelms the T-projection (or vice versa).

DFT adds nothing “on top” of RS.
It provides the formal mathematics linking the two representations that RS always treated conceptually.

7. Where We Go Next

DFT-11 will begin applying these projection rules:

How vibration appears in the S- and T-frames.
How 2-D rotation maps into spatial displacement vs. temporal winding.
How RS atomic structures become projections of a single scalar rotational pattern.
How the discrete RS rotational displacements arise naturally from projection consistency.

This will be the first point at which we can describe quantization not as an assumption, but as a geometric inevitability of projecting scalar motion into two complementary 3-D frames.