DFT-18: Emergence of Quantum-Like Behavior

MWells · Post by **MWells** » Wed Dec 03, 2025 10:21 pm

Why projection ambiguity looks like uncertainty, why coherence and interference follow from T-frame phase relations, and how RS vibration/rotation become wavefunctions.

Dual-Frame Theory starts from a simple claim:

The physical world we describe in space and time is a projection of a deeper scalar progression, seen through two complementary frames:

an S-frame (geometric), and
a T-frame (phase/rotational).

Once you accept that:

the S-frame cannot represent all of the T-frame structure at once, and
the T-frame must remain consistent with the S-frame’s geometric constraints and motion budget,

quantum-like behavior becomes unavoidable:

uncertainty arises from projection ambiguity and axis selection,
coherence and interference arise from T-frame phase relations,
wavefunctions arise as S-frame encodings of RS vibration and rotation.

In this post, we focus on three questions:

Why does projection ambiguity look like uncertainty?
Why do coherence and interference follow from T-frame phase relations?
How do RS vibration and rotation become wavefunctions in the S-frame?

This post is still at the framework level: it explains structure, not yet the exact numerical value of ℏ or the full Schrödinger equation. Those require additional work built on top of what follows.

1. Scalar progression and dual projections

The underlying reality is a scalar trajectory in the Natural Reference System:

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

with an invariant progression norm

$\left\|\frac{d\sigma}{d\lambda}\right\| = \kappa$

This single definite process is seen through two projections:

S-frame (spacetime-like)

$F_S : \sigma(\lambda) \mapsto x^\mu(\lambda), \qquad \mu = 0,1,2,3,$

T-frame (phase/rotational)

$F_T : \sigma(\lambda) \mapsto \theta^i(\lambda), \qquad i = 1,2,3.$

The T-frame phases live on a compact manifold

$T^3 = S^1 \times S^1 \times S^1$

The motion budget links both projections:

$\mathcal{B}_S + \mathcal{B}_T = \mathcal{B}_{\text{total}}$

This is the fundamental constraint: S-frame and T-frame are not independent. They are coupled views of the same scalar motion.

2. Why projection ambiguity looks like uncertainty
2.1 T-frame motion as a 3-component phase velocity

In the T-frame, rotational evolution is encoded as a phase-velocity vector

$\dot{\Theta}(\lambda) = \left(\dot{\theta}^1(\lambda),\ \dot{\theta}^2(\lambda),\ \dot{\theta}^3(\lambda)\right$

All three components are present in the scalar progression. Nothing in the NRS singles out one rotational axis as intrinsically preferred.

2.2 S-frame measurement selects one effective T-frame direction

A measurement in the S-frame that resolves some spatial quantity (for example, position along a chosen axis, or a component of momentum) cannot access the full three-component T-frame structure. It can only extract one effective component of $\dot{\theta}$ that is compatible with:

the chosen S-frame observable, and
the motion-budget constraints for that configuration.

Formally, the measurement defines a projection direction in T-frame:

$\dot{\theta}^{(\text{obs})} = a_1 \dot{\theta}^1 + a_2 \dot{\theta}^2 + a_3 \dot{\theta}^3$

where the coefficients $a_1, a_2, a_3$ encode how the chosen S-frame observable aligns with the T-frame axes.

Key point:

The S-frame measurement does not read out the full T-frame motion.
It reads out one axis-selected component of it.

2.3 Projection is non-invertible and non-commuting

The map from the full T-frame structure to an S-frame observable is many-to-one:

Many different triples ( $\theta^1, \theta^2, \theta^3)$ can produce the same S-frame outcome.
different observables correspond to different effective T-frame directions, i.e. different linear functionals on $\dot{\theta}$

Let $R_A$ and $R_B$ be two different measurement projections (e.g., two different spatial directions, or position vs momentum).Each is a map that:

chooses a direction in T-frame associated with the observable,
redistributes the remaining motion budget between S and T components.

In general,
$R_A R_B \ne R_B R_A$

Intuitively:

$R_A$ aligns the S-frame with one effective T-frame direction and “forgets” the components orthogonal to it.
$R_B$ aligns with a different combination and forgets a different set of components.

Once one projection has been applied, the information needed to reconstruct what the other projection would have seen is simply not available. Doing $R_A$ then $R_B$ is not equivalent to $R_B$ then $R_A$ .

This is the geometric origin of incompatible observables in DFT: different S-frame measurements correspond to different, generally non-commuting ways of slicing the same T-frame structure under a finite motion budget.

2.4 Uncertainty as projection non-invertibility

Because projection is many-to-one and non-invertible:

Knowing an S-frame outcome (for example, a sharp position) does not uniquely determine the T-frame configuration that produced it.
Many T-frame states are compatible with the same S-frame result.

Conversely:

Insisting on sharp information about one S-frame observable requires that the T-frame state be chosen from a large equivalence class of phase configurations.
When we later measure a different S-frame observable that depends on a different T-frame direction, this underdetermination shows up as a spread in the possible outcomes.

From the DFT point of view, uncertainty is not metaphysical indeterminacy:

Uncertainty is the S-frame manifestation of T-frame underdetermination once a particular projection has been chosen.

What standard quantum mechanics expresses with relations like Δ𝑥Δ𝑝≳ℏ/2, DFT expresses as the geometric fact that:

sharpening one S-frame observable corresponds to a projection that discards information about T-frame components needed to sharpen its conjugate observable.

In this post we do not yet derive the exact numerical constant ℏ/2; we only identify the underlying geometric reason why such complementarity relations must exist.

3. Why coherence and interference follow from T-frame phase relations
3.1 Phase evolution in the T-frame

Each T-frame phase evolves according to a local frequency:

$\frac{d\theta^i}{d\lambda} = \omega^i(\lambda).$

For two different paths or histories labeled by $a$ and $b$ , the T-frame phases accumulate as

$\theta_a^i(\lambda) = \theta_a^i(\lambda_0) + \int_{\lambda_0}^{\lambda} \omega_a^i(\lambda')\, d\lambda',$

$\theta_b^i(\lambda) = \theta_b^i(\lambda_0) + \int_{\lambda_0}^{\lambda} \omega_b^i(\lambda')\, d\lambda'.$

The relative phase along a given T-frame axis is

$\Delta\theta^i(\lambda) = \theta_a^i(\lambda) - \theta_b^i(\lambda).$

Because $T^3$ is compact, only these relative phases matter for interference; absolute phases can be shifted without observable effect.

3.2 Coherence as stable relative phase

Two scalar trajectories are coherent with respect to a given observable when their relevant T-frame relative phase remains approximately constant over an interval:

$\Delta\theta^i(\lambda) \approx \text{constant (mod } 2\pi).$

In this regime:

The S-frame can maintain a stable pattern of constructive or destructive overlap between the contributions of the two trajectories.
interference fringes are stable and reproducible.

When the relative phases drift rapidly and uncontrollably, the corresponding S-frame contributions fluctuate in sign and phase and effectively average out. This is decoherence in DFT terms: loss of stable relative phase structure in the relevant T-frame directions.

3.3 Interference from competing phase histories

Consider two alternative scalar histories that project to the same S-frame region:

Path A with phase set ${\theta_a^i}$
Path B with phase set ${\theta_b^i}$

The S-frame cannot distinguish which particular T-frame path produced the event; it only sees the combined effect of all compatible T-frame configurations.

The effective complex amplitude at an S-frame point x is proportional to a sum over compatible phase contributions:

$A(x) \propto \sum_{\text{paths } a} e^{i\Phi_a(x)}$

where $\Phi_a(x)$ encodes the relevant combination of T-frame phases along path a that project to x.

When two or more contributions arrive with different relative phases, the sum can:

be enhanced (constructive interference), or
be suppressed (destructive interference),

depending on the values of $\Phi_a(x)$ .

Thus:

Coherence and interference are the S-frame appearance of stable or unstable T-frame phase relations among multiple scalar trajectories that project to the same region of spacetime.

No literal waves propagate in the NRS. The “wave-like” behavior lives in the projection: it is how the S-frame encodes the aggregation of phase contributions from T-frame histories.

4. How RS vibration/rotation become wavefunctions

RS talks about:

vibration (linear or rotational oscillation)
rotation (multi-dimensional time-region rotation)

DFT interprets these as specific patterns of T-frame evolution that, when projected and aggregated, look like wavefunctions to an S-frame observer..

4.1 RS vibration as fundamental T-frame oscillation

A one-dimensional RS vibration is a scalar-motion pattern that yields a periodic T-frame phase:

$\theta(\lambda + T) = \theta(\lambda) + 2\pi$

for some fundamental period $T$ in $\lambda$ .

The corresponding T-frame phase factor is:

$e^{i\theta(\lambda)}$

This periodic T-frame evolution is not merely analogous to a complex exponential; it is mathematically the same phenomenon viewed under projection. The RS “vibration” is literally the periodic advancement of T-frame phase, and the complex exponential used in the S-frame wavefunction is exactly the compressed encoding of that periodicity.

4.2 RS rotation as multi-dimensional phase winding

A two- or three-dimensional RS rotation corresponds in DFT to nontrivial winding on multiple T-frame angles:

$\theta^i(\lambda + T_i) = \theta^i(\lambda) + 2\pi n_i, \qquad n_i \in \mathbb{Z},$

with integer winding numbers $n_i$ .

These winding numbers determine things like:

angular momentum
discrete energy levels
internal phase structure of “particles” or atomic states

Each rotational configuration contributes a characteristic phase factor

$e^{i\Phi(\lambda)}$

where Φ(𝜆) is a suitable combination of the θⁱ(λ) determined by the specific configuration and the chosen observable.

4.3 From phase factors to wavefunctions

The S-frame cannot directly represent the full T-frame phase vector. It compresses T-frame information into a function over S-frame configurations.

This function is what we call the wavefunction:

$\psi(x) = A(x)\, e^{i\phi(x)}.$

Here:

$A(x)$ encodes how strongly the family of T-frame configurations that project to 𝑥 contributes (roughly, a square root of the total T-frame measure compatible with 𝑥),
$\phi(x)$ encodes the coarse-grained T-frame phase accumulated along paths that reach $x$ .

In DFT language:

The wavefunction is the S-frame’s compressed representation of all T-frame vibrational and rotational possibilities consistent with a given geometric configuration.

This is how RS vibration/rotation become wavefunctions:

RS vibration provides the basic periodic phase structure in T-frame.
RS rotation provides quantized winding patterns that determine discrete energy levels and phase evolution.
The S-frame bundles these into a single complex-valued function psi(x) that encodes both amplitude and phase.

RS vibration provides the basic periodic phase structure in T-frame; RS rotation provides quantized winding patterns that determine discrete energy levels and phase evolution. The S-frame bundles all of that into a single complex-valued function 𝜓(𝑥) that encodes both amplitude and phase.

4.4 Why probabilities are squared amplitudes

Given an S-frame configuration x, the number of compatible T-frame phase configurations is

$N(x) = \text{(measure of T}^3\text{ states projecting to } x\text{)}.$

If we take the amplitude to scale like

$A(x) \propto \sqrt{N(x)}$

then the probability associated with configuration 𝑥 is

$P(x) = |A(x)|^2 \propto N(x).$

The logic is:

probabilities must count how much T-frame “phase volume” is compatible with the observed S-frame outcome,
amplitudes must add linearly when we superpose contributions from different histories or configurations,
this linearity requirement naturally associates amplitudes with square roots of the underlying combinatorial or measure-theoretic counts.

Probability is therefore quadratic because superposition is linear.

In this post we do not derive a full Hilbert-space formalism or normalize 𝜓 explicitly; we identify the structural reason why:

a complex amplitude language is natural, and
probabilities emerge as squared magnitudes of those amplitudes.

5. Summary

We can now answer the three guiding questions explicitly.

5.1 Why projection ambiguity looks like uncertainty

The T-frame has three rotational components and no preferred axis.
An S-frame measurement must select an effective T-frame direction to align with a chosen spatial or dynamical observable.
This projection is many-to-one and non-invertible; different observables correspond to different, generally non-commuting projections.
Many T-frame states map to the same S-frame outcome; once that outcome is fixed, the underlying T-frame state remains underdetermined relative to other observables.

This underdetermination is what appears, in S-frame language, as uncertainty and incompatible observables.

5.2 Why coherence and interference follow from T-frame phase relations

Coherence corresponds to stable relative T-frame phases in the directions relevant to a given observable.
Interference patterns arise when multiple scalar histories with different phase histories project to the same S-frame region.
The S-frame must combine these contributions; the result is constructive or destructive interference depending on relative phase.

Quantum interference is thus the shadow of T-frame phase structure, not a sign of literal waves in the NRS.

5.3 How RS vibration/rotation become wavefunctions

RS vibration corresponds to periodic phase evolution in T-frame, producing complex exponential factors.
RS rotation corresponds to multi-dimensional phase winding with integer winding numbers (𝑛₁,𝑛₂,𝑛₃).
The S-frame cannot display the full T-frame; it compresses it into a complex-valued function 𝜓(𝑥) whose magnitude encodes how much T-frame structure projects to 𝑥, and whose phase encodes coherent T-frame phase relations.
Squared amplitude gives probability as a count of compatible configurations, while linear addition of amplitudes encodes phase-sensitive superposition.

Thus, wavefunctions are not primary entities in the NRS. They are projection artifacts: the S-frame’s dual-frame-compatible encoding of RS vibration and rotation and their allowed combinations.

At this point in the series, DFT has:

given a geometric explanation for relativistic structure (DFT-13, DFT-15),
clarified nonlocality as projection effect (DFT-14),
derived RS-style shell structure from T-frame topology (DFT-16),
reinterpreted interaction as projection constraint (DFT-17), and
shown how key quantum-like features (uncertainty, interference, wavefunctions, Born rule) arise from dual-frame geometry (DFT-18).

What it has not yet done is:

derive the exact numerical value of ℏ,
produce the full Schrödinger equation with specific Hamiltonians,
compute hydrogen spectra, fine structure, or scattering cross-sections,
or match QFT precision.

Those are the tasks that turn this framework into a full-fledged quantitative theory. The dual-frame picture is designed so that such a reconstruction is possible; the actual calculations and comparisons to experiment are the natural next step beyond DFT-18.