DFT-12: Mass, Momentum, and Energy as S-Frame Expressions of T-Frame Rotation

MWells · Post by **MWells** » Wed Dec 03, 2025 4:39 pm

In the last post (DFT-11), we saw how the characteristic RS structures—vibration, inward motion, 1-D and 2-D rotational displacements—arise as two different projections of the same scalar motion in the NRS. That post was concerned primarily with motion itself: how scalar increments divide between S-frame translation and T-frame winding in such a way that the RS integer structure emerges automatically.

In this post we now ask:

What do these rotational patterns look like physically when viewed from the S-frame?

We continue to work in the same normalized, natural-unit setting used in DFT-10 and DFT-11. Each intrinsic scalar step ∣Δ𝜎∣ is taken as unit magnitude, and the quantities we call “K” and “U” below are budget fractions of that unit—not yet energies in SI units.

The conversion from these normalized geometric quantities to laboratory units (joules, kilograms, meters per second) will occur explicitly in DFT-13, where we introduce 𝑐 and ℏ as conversion constants, not as independent laws.

1. Why rotation appears as “mass” in the S-frame

In RS, mass is not substance—it is the S-frame appearance of full 3-D rotational curvature.

DFT clarifies this:

T-frame curvature reduces the amount of scalar motion available to the S-frame.
When all three T-frame directions participate, there is no remaining outward S-frame projection.
What S-frame observers experience is exactly what RS calls mass.

Formally, if

$(n_1, n_2, n_3)$

are the T-frame winding numbers, then in the normalized units:

$|\Delta\theta|^{2} = (n_1^{2} + n_2^{2} + n_3^{2})(\Delta\lambda)^{2}$

and the projection constraint gives

$|\Delta x|^{2} = |\Delta\sigma|^{2} - (n_1^{2} + n_2^{2} + n_3^{2})(\Delta\lambda)^{2}$

Mass appears when the bracketed term saturates the unit budget:

$(n_1^{2} + n_2^{2} + n_3^{2})(\Delta\lambda)^{2} \approx 1$

DFT’s interpretation, stated plainly:

Mass is what the S-frame sees when almost all of the scalar increment has been spent on T-frame rotation.

And this is not metaphor; it is strictly budget accounting.

2. Momentum as the S-frame expression of projection persistence

Larson’s “inertia” arises because reassigning curvature from one direction to another cannot occur smoothly. In DFT:

Changing S-frame velocity requires changing T-frame winding.
But integer winding numbers cannot change continuously.
Therefore, persistent S-frame displacement appears as momentum.

This yields the RS result:

Δn_i ≠ 0 ⇒ discontinuous transition

Thus, resistance to acceleration is not due to mass exchange or “forces,” but:

The geometry of discrete winding assignments.

3. Energy as deviation from natural progression

In RS, energy is outward displacement; potential energy is inward displacement; rest energy is full rotational occupation.

In DFT:

The S-frame receives whatever part of the scalar increment the T-frame does not use.
We define normalized kinetic and rotational components:

$K \equiv |\Delta x|^{2}$
$U \equiv |\Delta\theta|^{2}$

with the fundamental relation:

$K + U = |\Delta\sigma|^{2}$

This is the geometric origin of RS energy.

Explicit normalized energy functional

For a given winding class 𝑛,

$E_{DFT}(n) \equiv \oint \left|\frac{d\Theta_{n}}{d\lambda}\right|^{2}\, d\lambda$

This is dimensionless energy.

Later (in DFT-13), SI energy will be:

$E_{SI} = \hbar\,\Omega$

where

$\Omega \equiv \oint \frac{d\Theta}{d\lambda}\, d\lambda$

The role of ℏ is purely conversion, not new physics.

Hydrogen-scaling consistency

For atomic bound states, curvature per loop decreases with loop size, giving:

$E_{n} ∼ 1/n^{2}$

This matches:

RS magnetic scaling,
RS electric displacement hierarchy,
QM hydrogen spectrum.

We do not yet claim to reproduce numeric hydrogen energies;here we establish only the geometric scaling foundation.

4. Why massless vs. massive is automatic

1-D winding ⇒ photon-like (massless)
2-D winding ⇒ no mass
3-D winding ⇒ mass appears

This is not postulated; it follows from projection:

1 and 2 dimensions do not saturate the budget
3 dimensions inevitably do

Thus three-dimensional T-frame rotation is mass.

5. Summary before the conversion step

DFT-12 has shown:

Mass is geometric saturation of the T-frame budget.
Momentum arises because winding assignments are discrete and persistent.
Energy is the S-frame share of the scalar increment.
Only 3-D rotation yields mass, exactly as RS asserts.
RS energy scaling 𝐸∼1/𝑛2 falls directly from curvature budgets.

Next time (DFT-13) we introduce the conversion to SI units:

𝑐 links S-frame spatial and temporal units,
ℏ links curvature accumulations to measured energy,
𝐸=𝑚𝑐², 𝐸²=(𝑝𝑐)²+(𝑚𝑐²)² emerge from projection geometry

.

Technical Supplement

A. Projection consistency (from DFT-10)

$|\Delta x|^{2} + |\Delta\theta|^{2} = |\Delta\sigma|^{2}$

B. Definition of the mass condition

$|\Delta \theta|^{2} = (n_{1}^{2} + n_{2}^{2} + n_{3}^{2})(\Delta\lambda)^{2}$

Projection then forces:

$|\Delta x|^{2} = |\Delta\sigma|^{2} - (n_{1}^{2} + n_{2}^{2} + n_{3}^{2})(\Delta\lambda)^{2}$

C. Momentum as projection persistence

Δn_i ≠ 0 ⇒ discontinuous jump

D. Energy as geometric budget

Define normalized components:

$K = |\Delta x|^{2}$

$U = |\Delta\theta|^{2}$

Then:

$K + U = |\Delta\sigma|^{2}$

And normalized energy functional:

$E_{DFT}(n) = \oint |\dot{\Theta}_{n}(\lambda)|^{2}\,d\lambda$

This becomes physical energy once multiplied by the appropriate conversion constant involving ℏ, introduced in DFT-13.