DFT-28: Winding Number Stability and Energy Splitting
Posted: Thu Dec 04, 2025 9:49 pm
In DFT-27 we established that closed-loop coherence selects discrete global winding classes.
We now take the next conceptual step: not all winding classes are energetically equivalent.
The T-frame curvature associated with a given winding assignment has a measurable S-frame energy consequence.
This mechanism — subtle, geometric, and global — is precisely what is needed to understand tiny spectral shifts such as the Lamb shift.
Unlike conventional field-theoretic renormalization, we do not need particle-like models, virtual excitations, or subtractions of infinities.
The energy differences arise because distinct topological embeddings of the same superconducting (or atomic) phase are not isometric in the T-frame.
The simplicity of the DFT explanation hides a deep structural fact:
energy is not “assigned” or “added” to the winding;
energy is the cost of projecting curvature from the T-frame to the S-frame.
We begin with the geometric object that captures this.
Let the T-frame phase along the loop be denoted exactly as in DFT-27,
,
)
and let curvature refer to the second derivative of the phase along that trajectory,
 \equiv \left| \frac{d^2 \Theta}{d\lambda^2} \right|.
)
This quantity is not “curvature in space.”
It is curvature with respect to the phase manifold, and the S-frame only sees its projection.
A particular winding class 𝑛 corresponds to a distinct curve Θ𝑛(𝜆).
The integrated curvature over the loop establishes a geometric functional,
![\mathcal{K}[n]
=
\oint \left| \frac{d^2 \Theta_n}{d\lambda^2} \right|\, d\lambda.](https://reciprocal.systems/cgi-bin/mathtex.cgi?
\mathcal{K}[n]
=
\oint \left| \frac{d^2 \Theta_n}{d\lambda^2} \right|\, d\lambda.
)
This object is what the S-frame ultimately interprets as energy.
The precise correspondence is not an approximation but a structural one:
![E[n] \;\propto\; \mathcal{K}[n].](https://reciprocal.systems/cgi-bin/mathtex.cgi?
E[n] \;\propto\; \mathcal{K}[n].
)
There is no “force” or “interaction” here.
There is simply the geometric fact that some global phase embeddings require more curvature than others, and the S-frame registers this as a slightly different energy.
To see this in perhaps the simplest intuitive form:
the superconducting (or atomic) phase must traverse a certain amount of “turning” in the T-frame to satisfy its global constraints; different allowed windings distort the T-frame path by different amounts, and the S-frame energetically reflects the magnitude of this distortion.
The actual energy splitting between two winding classes 𝑛 and 𝑚 is then given by the difference of their curvature integrals:
![\Delta E_{n,m}
\;\propto\;
\mathcal{K}[n] - \mathcal{K}[m].](https://reciprocal.systems/cgi-bin/mathtex.cgi?
\Delta E_{n,m}
\;\propto\;
\mathcal{K}[n] - \mathcal{K}[m].
)
This relation is not invoked as an empirical rule.
It is an immediate and geometric consequence of the fact that a smooth T-frame trajectory with different winding counts cannot be globally isomorphic to another.
A striking consequence follows.
If the phase difference between two windings is small, then the energy difference will also be small.
In other words,
Tiny geometric perturbations in T-frame curvature produce tiny spectral splittings in the S-frame.
This statement accomplishes something profound:
it shows that the scale of an energy shift is not mysterious.
There is no hidden particle cloud producing corrections.
The shift is simply the projection of geometric differences between otherwise permissible configurations.
This gives a predictive criterion:
The shifts are geometric from the beginning.
The superconducting loop gives a macroscopic demonstration of this principle:
when two winding classes compete, the resulting SQUID oscillations encode a stable, measurable interference pattern.
The atomic system does exactly the same job, but at much smaller scales.
This understanding now closes the conceptual gap between macroscopic winding differences and microscopic spectral shifts.
In the next post, DFT-29, we apply this to atomic orbitals where the curvature is not purely intrinsic but receives a small, universal modification from the Background Phase Geometry (BPG).
That modification changes the curvature of certain windings asymmetrically, and the S-frame registers the asymmetry as a tiny energy shift.
That shift has the correct sign and the correct splitting behavior known as the Lamb Shift.
The key takeaway of DFT-28 is this:
Energy differences do not arise from dynamical interactions but arise from differences in T-frame geometric curvature among globally allowed winding classes, with the S-frame registering curvature as energetics.
DFT-29 will put the missing piece in place: how the global background geometry slightly modifies curvature, producing the famous spectral shifts without renormalization.
We now take the next conceptual step: not all winding classes are energetically equivalent.
The T-frame curvature associated with a given winding assignment has a measurable S-frame energy consequence.
This mechanism — subtle, geometric, and global — is precisely what is needed to understand tiny spectral shifts such as the Lamb shift.
Unlike conventional field-theoretic renormalization, we do not need particle-like models, virtual excitations, or subtractions of infinities.
The energy differences arise because distinct topological embeddings of the same superconducting (or atomic) phase are not isometric in the T-frame.
The simplicity of the DFT explanation hides a deep structural fact:
energy is not “assigned” or “added” to the winding;
energy is the cost of projecting curvature from the T-frame to the S-frame.
We begin with the geometric object that captures this.
Let the T-frame phase along the loop be denoted exactly as in DFT-27,
and let curvature refer to the second derivative of the phase along that trajectory,
This quantity is not “curvature in space.”
It is curvature with respect to the phase manifold, and the S-frame only sees its projection.
A particular winding class 𝑛 corresponds to a distinct curve Θ𝑛(𝜆).
The integrated curvature over the loop establishes a geometric functional,
This object is what the S-frame ultimately interprets as energy.
The precise correspondence is not an approximation but a structural one:
There is no “force” or “interaction” here.
There is simply the geometric fact that some global phase embeddings require more curvature than others, and the S-frame registers this as a slightly different energy.
To see this in perhaps the simplest intuitive form:
the superconducting (or atomic) phase must traverse a certain amount of “turning” in the T-frame to satisfy its global constraints; different allowed windings distort the T-frame path by different amounts, and the S-frame energetically reflects the magnitude of this distortion.
The actual energy splitting between two winding classes 𝑛 and 𝑚 is then given by the difference of their curvature integrals:
This relation is not invoked as an empirical rule.
It is an immediate and geometric consequence of the fact that a smooth T-frame trajectory with different winding counts cannot be globally isomorphic to another.
A striking consequence follows.
If the phase difference between two windings is small, then the energy difference will also be small.
In other words,
Tiny geometric perturbations in T-frame curvature produce tiny spectral splittings in the S-frame.
This statement accomplishes something profound:
it shows that the scale of an energy shift is not mysterious.
There is no hidden particle cloud producing corrections.
The shift is simply the projection of geometric differences between otherwise permissible configurations.
This gives a predictive criterion:
- If two winding classes differ only slightly in curvature, their energy splitting will be tiny.
- If they differ significantly (as between 1𝑆 and 2𝑃 structures), the energy splitting will be larger.
The shifts are geometric from the beginning.
The superconducting loop gives a macroscopic demonstration of this principle:
when two winding classes compete, the resulting SQUID oscillations encode a stable, measurable interference pattern.
The atomic system does exactly the same job, but at much smaller scales.
This understanding now closes the conceptual gap between macroscopic winding differences and microscopic spectral shifts.
In the next post, DFT-29, we apply this to atomic orbitals where the curvature is not purely intrinsic but receives a small, universal modification from the Background Phase Geometry (BPG).
That modification changes the curvature of certain windings asymmetrically, and the S-frame registers the asymmetry as a tiny energy shift.
That shift has the correct sign and the correct splitting behavior known as the Lamb Shift.
The key takeaway of DFT-28 is this:
Energy differences do not arise from dynamical interactions but arise from differences in T-frame geometric curvature among globally allowed winding classes, with the S-frame registering curvature as energetics.
DFT-29 will put the missing piece in place: how the global background geometry slightly modifies curvature, producing the famous spectral shifts without renormalization.