DFT-17: Composite Systems, Interaction, and Multi-Particle Projection

MWells · Post by **MWells** » Wed Dec 03, 2025 9:41 pm

Why interaction is not mediated, but arises from shared projection constraints on multiple scalar trajectories.

One of the most striking departures of the Reciprocal System from conventional physics is its claim that “forces” do not exist as intermediating entities. Instead, interactions reflect the geometric and relational constraints imposed by motion itself. Larson understood this intuitively but lacked the formal framework needed to express it in a modern, mathematical way.

Dual-Frame Theory provides that missing framework.

In DFT, interaction is what happens when multiple scalar trajectories must share S-frame and T-frame projections in a way that maintains consistency with:

the same underlying motion budget, and
the same geometric rules of projection that apply to each trajectory individually.

This yields a clean picture:

composite systems arise naturally,
interaction is contextual rather than mediated,
multi-particle behavior is a consequence of projection geometry,
and apparent “forces” emerge from the requirement that projections of different trajectories remain compatible.

In this post we will examine:

how multiple scalar trajectories coexist in the NRS,
how S-frame projection couples multiple trajectories,
how T-frame phase structure generates correlation and coherence,
why “forces” appear even though nothing is transmitted,
how interaction energy arises from projection adjustments,
why multi-particle systems require configuration-space-like structure,
and How DFT recovers RS gravitational and electromagnetic behavior at the structural level.

The aim here is conceptual and geometric: to show how interaction can be understood as a constraint of shared projection, not to derive full quantitative force laws. Explicit Coulomb/Newton potentials, scattering cross-sections, and strong/weak interactions lie beyond the scope of this post.

1. Multiple scalar trajectories in the NRS

A single scalar trajectory is:

$\sigma(\lambda) \in \mathbb{R}^3_{\text{NRS}}$

For a system of two motions, we have:

$\sigma_1(\lambda_1)$

$\sigma_2(\lambda_2)$

with 𝜆₁,𝜆₂ playing the role of intrinsic parameters along each trajectory.

In the NRS, these are kinematically independent. There is no direct coupling in the scalar domain. Locality in the NRS is strict:

$d_{\text{NRS}}(\lambda_i,\lambda_j) \to 0 \quad \Longrightarrow \quad \text{adjacency}$

while finite NRS separation implies no direct connection at the scalar level.

Interaction does not take place in the NRS.
It arises only after projection, when multiple scalar processes must coexist within the same S-frame and T-frame structures.

2. How S-frame projection couples multiple trajectories

Each scalar trajectory projects into S-frame coordinates:

$F_S(\sigma_1) = x_1^\mu(\lambda_1), \qquad F_S(\sigma_2) = x_2^\mu(\lambda_2), \qquad \mu = 0,1,2,3.$

Although the NRS motions are independent, their S-frame projections exist in the same geometric space. That means they must jointly satisfy:

the same Minkowski-like embedding,
the same invariant interval structure,
and the same local geometry and motion budget rules as in DFT-13 and DFT-15.

The interval assigned to each worldline segment obeys

$\Delta s_a^2 = c^2(\Delta x_a^0)^2 - (\Delta x_a^1)^2 - (\Delta x_a^2)^2 - (\Delta x_a^3)^2 = \alpha^2\, d_{\text{NRS}}(\sigma_a(\lambda_a))^2, \qquad a = 1,2.$

Here 𝛼 is the same projection scale that ties NRS distances to S-frame intervals for all trajectories.

If two projected worldlines come into proximity or alignment in spacetime, the way the projection map assigns S-frame coordinates to NRS increments must adjust so that both worldlines remain consistent with:

the shared spacetime metric, and
the same mapping from NRS distance to interval.

The S-frame does not know about 𝜆₁ and 𝜆₂ as separate intrinsic parameters; it only sees the resulting 𝑥1𝜇 and 𝑥2𝜇. Ensuring that both sets of coordinates remain compatible with a single spacetime metric and a single NRS–interval map induces constraints on their allowed relative accelerations and configurations.

This is the first root of interaction:

When multiple scalar trajectories are embedded into a single S-frame metric, the requirement of metric compatibility induces constraints that appear, in the S-frame, as forces acting between worldlines.

Nothing has been mediated between the scalar trajectories. The “coupling” arises because a single S-frame geometry must accommodate them all.

3. T-frame: shared phase structure and correlation

Now consider the T-frame projections:

$F_T(\sigma_1) = \theta_1^i(\lambda_1), \qquad F_T(\sigma_2) = \theta_2^i(\lambda_2), \qquad i = 1,2,3.$

Even though the scalar motions are independent, their T-frame phase structure must exist on a single global phase manifold:

$\theta_a \in T^3 = S^1 \times S^1 \times S^1, \qquad a = 1,2.$

Because 𝑇³ is compact and shared by all trajectories:

phase relationships among multiple scalar trajectories cannot be “localized” into disjoint phase spaces, and
the global structure of 𝑇³ naturally supports correlation patterns that look “nonlocal” from the S-frame point of view (as discussed in DFT-14).

When two trajectories share phase dimensions (e.g., overlapping or commensurate components of 𝜃^𝑖, they can:

accumulate relative phase differences,
lock into harmonic relationships,
settle into stable or unstable phase relationships,
and redistribute motion budget between S and T components in correlated ways.

A schematic stability condition for a simple bound system might be written as

$\theta_1^i(\lambda_1) - \theta_2^i(\lambda_2) = \text{constant (mod } 2\pi\text{)},$

for one or more components 𝑖, over long stretches of scalar progression.

This is the second root of interaction:

Here DFT locates the geometric origin of:

coherence and decoherence,
phase locking and orbital resonance,
angular momentum exchange,
orbital coupling and rotational selection rules.

Nothing travels between particles in the T-frame. Their projections must coexist on the same finite phase manifold, and compatible phase patterns are selected dynamically.

Interaction arises because multiple scalar trajectories must share a single compact T-frame manifold, whose global phase structure constrains their relative evolution.

4. Interaction as projection consistency, not mediation

Putting the S-frame and T-frame views together:

S-frame: multiple worldlines must coexist in a single spacetime metric, with a shared interval–NRS relation.
T-frame: multiple phase trajectories must coexist in a single compact phase manifold, with shared motion-budget constraints.

Consider two S-frame worldlines that approach each other. Their segments must satisfy

$\Delta s_1^2 = \alpha^2 d_{\text{NRS}}^2(\sigma_1), \qquad \Delta s_2^2 = \alpha^2 d_{\text{NRS}}^2(\sigma_2),$

but they must also remain compatible with the same spacetime metric and the same motion-budget rules for splitting scalar progression into S-frame translation and T-frame rotation.

In the T-frame, a bound or correlated configuration can be characterized by conditions of the form

$\theta_1^i - \theta_2^i \approx \text{locked value (mod } 2\pi),$

possibly up to small fluctuations, for one or more components 𝑖.

There is no force carrier, no exchange particle, no classical field that “pushes” or “pulls.” There is only the requirement that:

multiple projected trajectories coexist within the same S-frame geometry and the same T-frame phase space,
while continuing to represent legitimate scalar trajectories with a fixed NRS norm and a shared motion budget.

The resulting compatibility conditions appear in the S-frame as forces:

Acceleration, binding, repulsion, and orbital motion are the S-frame manifestations of projection adjustments required to keep multi-trajectory embeddings consistent with the scalar progression and the shared phase structure.

This sharpened versions of the RS intuition that “the force is an effect of motion”:

the motion is the scalar progression,
the “effect” is the adjustment of projections needed to maintain consistency in S and T frames.

5. Interaction energy as projection adjustment

When two trajectories enter a composite system, the S-frame representation must typically adjust. Let the initial translational budgets be

$\mathcal{B}_{S,1}, \qquad \mathcal{B}_{S,2},$

and the corresponding rotational budgets

$\mathcal{B}_{T,1}, \qquad \mathcal{B}_{T,2},$

with

$\mathcal{B}_{S,a} + \mathcal{B}_{T,a} = \mathcal{B}_{\text{total},a}, \qquad a = 1,2.$

After forming a composite configuration (e.g., a bound orbital or a stable molecule), the “effective” translational budget seen in the S-frame becomes

$\mathcal{B}_{S,\text{new}} = \mathcal{B}_{S,1} + \mathcal{B}_{S,2} - \Delta \mathcal{B}$

with a corresponding adjustment in the T-frame budgets. The quantity Δ𝐵 does not represent a transferred substance. It represents the reallocation of motion budget between translational and rotational components forced by:

S-frame metric compatibility, and
T-frame phase-locking conditions.

In the S-frame, this reallocation shows up as interaction energy:

negative Δ𝐵 (relative to separated configurations) corresponds to a bound state with lower S-frame energy,
positive Δ𝐵 corresponds to repulsive configurations or excited states that can decay.

In the T-frame, the same adjustment is seen as a change in relative phase and winding relationships—e.g., a transition from an unstable phase relation to a stable locked pattern.

In the NRS, nothing is exchanged; only the embedding of the scalar motions into S and T frames changes.

This reinterpretation provides a conceptual explanation for:

binding energy and potential wells,
repulsive potential energy,
changes in mass-energy during bonding or decay,
orbital quantization and stability of molecular structures.

The stabilizing condition can be summarized as:

the S-frame projection minimizes the required Δ𝐵 consistent with the motion budget and interval,
while the T-frame projection minimizes relative phase instability.

Stability arises as a joint optimum of projection effort across S and T frames, not as the cancellation of arbitrarily chosen forces.

6. Configuration space emerges from multi-trajectory projection

In standard quantum mechanics, multi-particle systems require a wavefunction defined over a high-dimensional configuration space, 𝜓(𝑥₁,𝑥₂,…,𝑥_𝑁). In DFT, we can see why such a structure is needed, without postulating it.

Each scalar trajectory projects into its own T-frame phase vector:

$F_T(\sigma_k) = \theta_k^i, \qquad i = 1,2,3,\quad k = 1,\dots,N.$

When $N$ trajectories coexist, their combined T-frame projection is:

$(\theta_1^i, \theta_2^i, \ldots, \theta_N^i) \in T^3 \times T^3 \times \cdots \times T^3 = (T^3)^N.$

This product manifold is not directly visible in physical space; it is the natural consequence of:

multiple independent scalar progressions,
sharing a single compact phase manifold 𝑇³.

Constraints on this product manifold—e.g., allowed joint phase patterns, symmetry restrictions, and motion-budget limits—are exactly what we describe, in standard QM, as structure in configuration space.

Thus:

Configuration space is not a mysterious quantum invention; it is the image of multi-trajectory T-frame projections. The “wavefunction over configuration space” is a way of encoding constraints on the allowed joint phase patterns.

From this perspective:

entanglement-like correlations reflect global structure on (𝑇³)^𝑁,
superselection rules reflect symmetry and budget constraints on allowed joint regions of (𝑇³)^𝑁,
multi-particle interference reflects how S-frame measurements sample patterns on this product phase space.

DFT therefore explains why quantum mechanics must use configuration space: because multiple scalar trajectories must be jointly represented on a shared compact phase manifold.

7. Recovery (at the structural level) of RS gravitation and electromagnetism

A complete quantitative treatment of all four fundamental interactions in DFT will require:

explicit mappings from phase/winding structure to charge, mass, spin, etc.,
derivations of effective potentials and force laws,
and comparison with precise experimental data.

That work lies ahead. Here we only sketch how RS interpretations of gravity and electromagnetism arise [*]structurally in DFT.

Gravitation

In RS, gravitation is modeled as inward scalar motion (time displacement). In DFT, inward motion corresponds to increased T-frame rotational dominance, as discussed in DFT-11 and DFT-13:

$\text{T-frame dominance} \quad\Rightarrow\quad \text{reduced outward S-frame progression}.$

For multiple trajectories, regions of enhanced T-frame curvature reduce the available S-frame outward component for each worldline. The resulting S-frame behavior—worldlines bending toward such regions—appears as an attractive gravitational interaction.

At this stage, DFT provides:

a geometric mechanism for gravitational attraction via T-frame dominance,
a reinterpretation of gravitational “forces” as S-frame manifestations of budget and curvature constraints.

It does not yet derive numerical values for 𝐺, strong-field GR effects, or gravitational waves; those require further development.

Electromagnetism

In RS, electric charge and magnetism arise from rotational patterns and secondary rotations. In DFT, such patterns are encoded in:

winding numbers (𝑛1,𝑛2,𝑛3) for each trajectory, and
relative phase configurations in shared T-frame components.

Stable phase-locked relationships between these patterns impose S-frame constraints that look, at a coarse level, like Coulomb attraction/repulsion and magnetic interactions. Schematically:

$\text{compatible phase-lock pattern} \quad\Rightarrow\quad \text{attractive S-frame adjustment},$

$\text{incompatible phase-lock pattern} \quad\Rightarrow\quad \text{repulsive S-frame adjustment}.$

Again, at this stage DFT provides:

a geometric origin for RS associations of charge with rotation,
a path to re-derive electromagnetic effects as projection constraints.

It does not yet produce explicit 1/𝑟² laws, numerical values of the fine-structure constant, or full QED-level predictions. Those are targets for later, more detailed posts (especially those on fine structure and Lamb shifts).

8. Summary

DFT explains interaction without mediators:

[*Scalar trajectories are independent and strictly local in the NRS.
S-frame projection forces geometric compatibility among these trajectories in a single spacetime metric tied to NRS distance.
T-frame projection forces phase compatibility among these trajectories in a single compact phase manifold 𝑇³.
Projection adjustments needed to maintain joint compatibility appear in the S-frame as forces, energies, correlations, and potentials.
Configuration space emerges naturally as the product of T-frame phase spaces for multiple trajectories.
RS “forces”—gravitation and electromagnetism—are reinterpreted as different manifestations of these projection constraints.

Interaction is not a process that happens between things.
Interaction is a constraint that arises when many scalar processes must be embedded into one shared S-frame and one shared T-frame without violating the motion budget or the scalar progression norm.

What DFT-17 does not yet provide:

closed-form expressions for Coulomb or Newton potentials,
explicit formulas for scattering cross-sections or binding energies,
a treatment of strong and weak interactions,
a full mapping to field-theoretic language or QFT precision.

Those are essential future steps. The role of this post is to:

clarify the geometric origin of interaction in the dual-frame picture, and
set up the conceptual bridge to quantum-like behavior and configuration-space dynamics.

Next: DFT-18 — Emergence of Quantum-Like Behavior

With interaction understood as a constraint of shared projection, we are ready to show:

why projection ambiguity produces uncertainty,
why T-frame phase structure produces interference,
why S-frame trajectories appear wave-like at appropriate scales,
and why quantum-like laws emerge from scalar motion and dual-frame geometry.

DFT-18 will take the next step: interpreting interference, superposition, and uncertainty as natural consequences of representing scalar trajectories through S/T projections and the configuration-space structure described here.