DFT-5: Scalar Progression, Coordinate Time, and the Motion Budget

MWells · Post by **MWells** » Tue Dec 02, 2025 11:14 pm

One of the most important clarifications that Dual-Frame Theory (DFT) attempts to make concerns how the same unit of scalar motion can express two different kinds of structure—translational and rotational, outward and inward, S-frame and T-frame—without ever ceasing to be a single, indivisible quantity of motion in the sense Larson insisted on.

Larson built the Reciprocal System on a strict foundation:
motion is fundamental, discrete, three-dimensional, and has two reciprocal aspects, space and time. In his view, every physical phenomenon—including matter, radiation, gravitation, and atomic rotation—is nothing more than a compound of motion in various combinations of these aspects. What Larson did not fully formalize, however, is how the same scalar magnitude distributes itself between different manifestations. His explanations are qualitatively correct and often profoundly insightful, but the mechanism linking different “expressions” of the same scalar motion—simple vibration, scalar rotation, inward gravitation, outward progression—remained qualitative.

The conceptual gap is this:
if space and time are reciprocal aspects of a single scalar quantity, then any structure appearing in one aspect must reduce what can appear in the other.
This relationship is never spelled out in RS, but it is implied everywhere—from the photon’s fixed inward/outward oscillation to the gravitational rotation of the atom.

This is where a crucial insight from Prof. K.V.K. Nehru becomes relevant.

Nehru’s Clarification: Coordinate Time and Accumulated Structure

In his paper Precession of the Planetary Perihelia Due to the Coordinate Time, Nehru explained something implicit in RS but not explicitly stated:
certain motions accumulate “coordinate time”—an added temporal component distinct from clock time—that alters how the total scalar motion must be distributed.

His argument begins with a refinement of Larson’s point that time has both:

clock time — the uniform progression (1 unit per unit)
coordinate time — a 3-dimensional temporal analog to coordinate space

Nehru then shows that independent motion at speed 𝑣 adds a definite amount of coordinate time, proportional to:

$\Delta t_c = 3 \frac{v^2}{c^2}$

This additional time component is not vectorial and does not have a direction in space. It is a scalar increase in the temporal aspect of the motion, and because it is scalar, it must be accounted for in the total balance of motion.

Nehru’s key contribution was to show, by explicit calculation, that an independent motion such as gravitation necessarily adds coordinate time to the total temporal structure experienced by an orbiting body. Because this additional time is real in the RS sense—even though it is not clock time—the orbital motion must be evaluated relative to this enlarged temporal component, not relative to uniform clock progression alone.

In DFT, I take the structural relationship revealed by Nehru’s calculation and broaden it: whenever motion contributes scalar structure in one aspect—whether interpreted as spatial or temporal—the complementary aspect must adjust, because both arise from the same underlying scalar progression. This reciprocal rebalancing of effective S-frame and T-frame structure is what DFT refers to as the motion budget.

The reason Mercury precesses more than Newtonian theory predicts is not because space “curves,” but because the motion contains an additional scalar temporal component that must be accommodated.

This is the first real appearance—within the RS literature—of the idea that different components of motion draw on the same underlying scalar resource.

And that is exactly the conceptual seed from which the motion budget emerges.

From Coordinate-Time Accumulation to the Motion Budget

In DFT, the motion budget is simply a fully formal expression of the principle implicit in both Larson’s postulates and Nehru’s refinement:

There is only one scalar progression. Any structure appearing in one aspect of the progression reduces what can appear in the other.

This applies to:

simple linear vibration
scalar rotation
inward gravitational rotation
outward progression
atomic rotation
orbital motion
coordinate-time accumulation
phase winding
harmonic modes
S-frame translation
T-frame coherence

All of these must be consistent with the same unit of scalar motion and its three scalar dimensions.

Thus:

If more structure appears as coordinate time (Nehru’s case), the corresponding spatial representation must adjust.
If more structure appears as rotation (Larson’s atoms), less is available for translational displacement.
If more structure appears in T-frame coherence, less can appear in S-frame geometry.
If the S-frame becomes highly constrained (e.g., gravitational binding), more structure can and must appear in the T-frame.

The motion budget becomes a precise rule:

All S-frame structure and all T-frame structure must sum to the same scalar magnitude of the underlying progression. Neither frame is primary; neither can exceed the scalar limit.

Nehru’s clarification about coordinate time accumulation provided the missing conceptual bridge. It demonstrated, in a rigorous RS-internal way, that motion can “use up” part of its allocation in one aspect (coordinate time) in a way that forces a compensating expression elsewhere (additional orbital advance).

This is exactly the kind of cross-frame constraint DFT formalizes.

The NRS Reinterpreted with the Motion Budget

Larson defined the Natural Reference System (NRS) as the three-dimensional progression of scalar motion—“one unit of space per unit of time”—from which all phenomena arise.

DFT maintains that definition unaltered but adds:

The NRS is the unique point at which S-frame and T-frame projections coincide.
Each projection is selective, capturing only part of the scalar structure.
Each projection incurs a cost: whatever structure it expresses must be debited from the common scalar magnitude.

Thus the outward unit progression is not “lost” when gravitation, rotation, or phase structure appears; it is simply redistributed through the two projection rules.

This resolves longstanding RS puzzles:

why rotational motion produces a translational inward motion
why gravitating atoms remain at rest relative to the progression
why photons have no independent motion
why orbital motion modifies temporal structure
why atomic rotation fixes quantized energy levels
why RS scalar rotation looks “3D” in T-frame projection but “inward” in S-frame projection

Every one of these features becomes a consequence of the motion budget applied to the NRS.

Total Coordinate-Time Increase in Nehru’s Derivation (for reference)

These are the primary equations from Nehru’s argument, all provided as plain LaTeX:

The coordinate-time rate introduced by gravitational speed 𝑣:

$\frac{dt_c}{dt} = 3 \frac{v^2}{c^2}$

The gravitational speed (scalar inward motion magnitude) at distance 𝑟 from mass 𝑀:

$v^2 = \frac{GM}{r}$

The orbital ellipse (polar form):

$l = r(1 + e \cos \theta)$

Total coordinate-time accumulation per revolution:

$\Delta t_c = \frac{3GM}{lc^2} , 2\pi$

Larson/RS perihelion advance:

$\delta_{\text{RS}} = \frac{3GM}{a c^2 (1 - e^2)}$

GR equivalent (shown only for comparison):

$\delta_{\text{GR}} = \frac{12 \pi^2 a^2}{P^2 c^2 (1 - e^2)}$

These equations are not “DFT additions”; they are historically part of RS2 analysis. They appear here only because they helped clarify the idea that certain motions accumulate structure in one aspect (coordinate time), requiring compensating adjustments elsewhere.

That is the essence of the motion budget.

Closing Thought

The goal of DFT is not to reinterpret RS through a foreign lens, but to clarify what was already latent in it:
a single scalar progression whose different expressions are not independent, but coordinated.

Larson showed the richness of the components.
Nehru showed that their interdependence can be precisely calculated.
DFT shows that this interdependence is universal.

The motion budget is simply the missing rule that makes the whole structure explicit and mathematically tractable.