Reciprocal System Research Society

Posted: **Wed Dec 03, 2025 4:23 pm**

The preceding posts established the projection machinery:
$x^\mu(\lambda) = F_S(\sigma(\lambda))$

$\theta^i(\lambda) = F_T(\sigma(\lambda))$

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

In this post we apply these rules to the actual RS motions—linear vibration and rotational patterns—and show how they produce:

the radial quantum number n
the angular quantum number ℓ
the magnetic quantum number m
without invoking Schrödinger operators, Hilbert spaces, or probabilistic postulates.

Summary (before derivation):

Under DFT:

n counts radial T-frame cycles
ℓ labels eigenmodes on S² (angular winding class)
m labels azimuthal phase closure around the axis

Quantization is not added.
It emerges from projection geometry and winding topology.

1. Linear vibration as the simplest closed projection loop

In RS, the photon is a spatial oscillation; in DFT:

The S-frame sees alternating displacements

$\Delta x = \pm A$

The T-frame sees uniform phase advance

$\Delta\theta = \omega\Delta\lambda$

Enforced by projection consistency:

$A^2 + (\omega\Delta\lambda)^2 = 1$

The RS “one-dimensional displacement” becomes:

a radial loop in the S-frame,
a single winding loop in the T-frame.

This is the seed of the quantum number:

n = # of closed T-frame phase loops

We will see this generalizes to the atomic series.

2. Two-dimensional rotation and inward displacement

A rotational state carries two independent T-frame angular windings:

$\theta(\lambda) = \begin{pmatrix} n_1\lambda \\ n_2\lambda \end{pmatrix}$

Each component is defined modulo 2𝜋.
Thus:

$n_1,n_2 \in \mathbb{Z}$

and the curvature enters S-frame projection through:

$|\Delta x|^2 = 1 - (n_1^2 + n_2^2)(\Delta\lambda)^2$

This gives the RS rule:

greater T-frame curvature → greater inward S-frame displacement

which Larson identified without a geometric mechanism.

3. The RS rotational triplet as winding numbers

RS writes rotational states as:

a - b - c

In DFT:

$a = n_1,\quad b = n_2,\quad c=n_3$

So:

𝑎,𝑏 are the two-plane rotation windings
𝑐 is the residual winding in the third phase direction

Projection consistency couples them:

$|\Delta x|^2 = 1 - (a^2 + b^2 + c^2)(\Delta\lambda)^2$

Thus RS rotational displacements are not postulates—they are forced integer solutions of the projection map.

4. How RS gives the quantum numbers (n,ℓ,m) in DFT language

4.1 Radial quantum number 𝑛

A bound orbital has radial oscillations corresponding to closed T-frame loops.

Thus:

n = # of full phase loops along the T-frame radial direction

This matches the fact that the energy spacing arises from radial node count.

4.2 Angular quantum number ℓ

States on the sphere have angular harmonic structure.

The S-frame sees solutions of the Laplace-Beltrami operator on S²:

$-\Delta_{S^2}Y_{\ell m} = \ell(\ell+1)Y_{\ell m}$

DFT sees this as:

$\ell(\ell+1) = n_1^2 + n_2^2$

Thus:

Angular T-frame curvature ↔ S² eigenvalue
ℓ labels total 2-D curvature class

So:
$\ell = 0,1,2,\ldots,n-1$

just as in standard quantum mechanics,
but here ℓ is a winding/curvature index, not an operator eigenvalue.

4.3 Magnetic quantum number 𝑚

Rotation about the symmetry axis requires closure of azimuthal phase:

m = # of phase wraps around polar axis

with

$m = -\ell,\ldots,+\ell$

This is the projection of T-frame angular loops into azimuthal closure.

5. State counting emerges automatically

For fixed 𝑛, we get exactly the standard counting:

$\sum_{\ell=0}^{n-1}(2\ell+1)=n^2$

This is:

not imposed
not probabilistic
not operator-based

It follows because:

radial winding gives n
angular curvature gives ℓ
azimuthal closure gives m
and projection invariance restricts them

So standard atomic state multiplicities arise directly.

6. Why RS integer rules agree with standard QM

RS has the empirical decomposition:

Electric displacement grows linearly (∼c)
Magnetic displacement grows quadratically (∼a²+b²)

DFT shows:

$a^2+b^2 = \ell(\ell+1)$

Thus:

RS magnetic displacement ∼ 2ℓ(ℓ+1)
Standard QM angular curvature ∼ ℓ(ℓ+1)

They are two views of the same structure.

7. Why this matters

We now have:

radial quantization (n)
angular quantization (ℓ)
azimuthal quantization (m)

all arising from:

integer T-frame winding
phase closure constraints
projection consistency

No Schrödinger operator was used.
The spectrum is not the result of a chosen differential equation;
it is a topological projection requirement.

8. Preview of DFT-12
DFT-12 will show:

how the curvature budget associated with these integer classes
produces mass, rest energy, and stability

and connects directly to:

$E_{n\ell m} \propto \ell(\ell+1) + f(n)$

(where 𝑓(𝑛) is the radial contribution).

This ties into the DFT-20 fine structure interpretation later.

Reciprocal System Research Society

DFT-11: How Quantum Numbers (n,ℓ,m) Arise From RS Winding Structure under Projection

DFT-11: How Quantum Numbers (n,ℓ,m) Arise From RS Winding Structure under Projection