The Wave Coherence Model
Topological Untwisting: A Geometric Mechanism
for Cooper Pair Formation from Toroidal Electrons
Stephen Horton
Independent Researcher
January 14, 2026
Published: Horton, S. (2026). The Wave Coherence Model (1.0). Zenodo. https://doi.org/10.5281/zenodo.18500774
Abstract
We propose a geometric mechanism for Cooper pair formation based on the toroidal photon model of the electron developed by Williamson and van der Mark (1997). In this framework, an electron is a circularly polarized photon confined to a toroidal topology with periodic boundary conditions of one Compton wavelength. The electron’s half-integer spin arises from the requirement of 720° rotation for topological closure. We demonstrate that Cooper pairing can be understood as two such toroidal structures achieving a configuration where their topological twists cancel, effectively untwisting from fermionic (720°) to bosonic (360°) character. This provides a geometric explanation for the fermion-to-boson transition in superconductivity and suggests that phonon-mediated attraction in BCS theory serves primarily to establish phase-locking boundary conditions rather than providing the fundamental pairing mechanism. We derive the mathematical conditions for topological untwisting and propose experimental signatures that could distinguish this mechanism from conventional BCS pairing.
1. Introduction
The microscopic mechanism of Cooper pairing remains one of the most profound puzzles in condensed matter physics. While BCS theory successfully describes superconductivity phenomenologically, it treats the fermion-to-boson transition as a mathematical consequence of wavefunction symmetry rather than explaining the geometric origin of this transformation. Two spin-1/2 fermions combine to form a spin-0 or spin-1 boson, but the physical mechanism by which half-integer angular momentum becomes integer angular momentum is typically left unaddressed.
Simultaneously, the internal structure of the electron itself remains an open question. The Williamson-van der Mark model (1997) proposes that the electron is a photon confined in a toroidal topology, with its half-integer spin emerging from the requirement that the electromagnetic field vectors rotate through 720° before returning to their initial orientation. This model successfully derives the electron charge (≈0.91e), the anomalous magnetic moment (g ≈ 2(1 + α/2π)), and provides a geometric interpretation of spin-1/2.
In this paper, we bridge these two domains by proposing that Cooper pair formation is fundamentally a topological untwisting process: two toroidal electrons achieve a geometric configuration in which their 720° twists cancel, producing a composite structure with 360° (bosonic) topology. This reframes superconductivity as a phase transition in the topology of electron states rather than merely their quantum statistics.
2. The Toroidal Electron Model
2.1 Geometric Structure
Following Williamson and van der Mark, we model the electron as a circularly polarized photon confined to periodic boundary conditions of length λC (the Compton wavelength). The photon traces a double-loop path on a toroidal surface, with the electromagnetic field vectors (E and B) rotating commensurate with the orbital motion.
The key geometric parameters are:
- Mean energy transport radius: r = λC/4π ≈ 1.93 × 10−13 m
- Rotation horizon (object boundary): R = λC/2 ≈ 1.21 × 10−12 m
- Internal frequency: ωs = 2ωC (double the Compton frequency)
2.2 Origin of Half-Integer Spin
The confined photon carries angular momentum ℏ. However, because it must traverse a double loop to complete its path, the effective orbital angular momentum observed externally is:
L_orbit = |r × p| = (λ_C / 4π) · (U_photon / c) = ℏ/2 (1)
The factor of 1/2 arises directly from the toroidal topology: the photon must circulate twice around the torus before returning to its original position with the same field orientation. This is the geometric origin of spin-1/2—the electron is not ‘intrinsically’ half-integer, but appears so because of its topological structure.
Crucially, the field vectors must rotate through 720° for topological closure. This can be visualized as a Möbius-like twist in the photon’s path: one complete circuit produces a 360° rotation of the fields, but only after two circuits does the system return to its initial state.
3. Cooper Pairing as Topological Untwisting
3.1 The Pairing Configuration
Consider two toroidal electrons approaching each other with opposite spin orientations. Each carries a 720° topological twist. We propose three possible geometric configurations for pairing:
Configuration A — Coaxial Nesting: One torus nested inside the other, with counter-rotating poloidal flows. The opposite circulations cancel the net twist.
Configuration B — Hopf Linkage: Two tori interlocked like chain links (topologically linked with linking number 1). The linkage itself neutralizes the half-integer character through topological coupling.
Configuration C — Anti-parallel Stacking: Tori positioned coplanar with opposite toroidal circulation, producing spin cancellation (s = +1/2 and s = −1/2 → S = 0).
In each configuration, the key requirement is that the combined structure exhibits 360° rotational closure rather than 720°. The topological twist is shared between the two electrons such that each effectively contributes 1/2 of a full twist, summing to one complete (bosonic) rotation.
3.2 Mathematical Condition for Untwisting
Let Φ1 and Φ2 represent the topological phase accumulated by electrons 1 and 2 over one complete circuit of their respective tori. For an isolated electron:
Φ_electron = 4π (720° in radians) (2)
For a Cooper pair to exhibit bosonic statistics, the combined phase must satisfy:
Φ_pair = Φ_1 + Φ_2 + Φ_coupling = 2π (mod 2π) (3)
where Φcoupling represents the phase contribution from the geometric linkage between the two toroidal structures. For successful pairing:
Φ_coupling = −6π (or equivalently +2π mod 4π) (4)
This coupling phase arises naturally in the Hopf-linked configuration, where the linking number Lk = 1 contributes a phase of 2π per link, and the mutual threading of flux contributes an additional geometric phase.
3.3 Phase-Locking and the Role of Phonons
In conventional BCS theory, phonon exchange provides the attractive interaction between electrons. In our topological framework, we reinterpret this role: phonons do not cause pairing—they permit it.
The lattice vibrations establish boundary conditions that allow two toroidal electron structures to achieve phase coherence. Specifically, the phonon-mediated interaction creates a potential well in which the relative phase between two electrons becomes locked, satisfying:
δ(Φ_1 − Φ_2) = 0 (phase-locked condition) (5)
This phase-locking is necessary but not sufficient for pairing. The geometric configuration must also satisfy equation (3). The lattice provides the ‘scaffolding’ that holds the two toroidal structures in the correct relative orientation for topological untwisting to occur.
This explains why superconductivity is sensitive to lattice structure: different crystal symmetries provide different boundary conditions for phase-locking, and only certain configurations permit the topological untwisting required for Cooper pair formation.
4. Energy Considerations
4.1 Binding Energy from Topology
The binding energy of a Cooper pair can be understood as the energy difference between two isolated toroidal electrons and the untwisted pair configuration. Using the Williamson-van der Mark framework, the energy of an isolated electron is entirely electromagnetic:
U_electron = ℏω_C = m_e c² (6)
The external field energy (contributing to the anomalous magnetic moment) is:
U_ext = (α′/2π) · U_electron ≈ 0.00116 · m_e c² (7)
When two electrons form a Cooper pair, the external field configurations partially cancel (for anti-aligned spins), reducing the total external field energy. The binding energy is approximately:
Δ_binding ≈ 2U_ext · f(geometry) = (α/π) · m_e c² · f(geometry) (8)
where f(geometry) is a factor of order unity depending on the specific pairing configuration. For f ≈ 10−8 to 10−6, this gives binding energies in the meV range, consistent with observed superconducting gaps.
4.2 Coherence Length
The BCS coherence length ξ represents the spatial extent of a Cooper pair. In our model, this corresponds to the maximum separation at which two toroidal electrons can maintain phase coherence for topological untwisting. Using the Williamson-van der Mark result that the electron’s effective ‘size’ scales with wavelength:
ξ ≈ (λ_C / α) · √(E_F / Δ) (9)
where EF is the Fermi energy and Δ is the superconducting gap. This gives ξ ~ 100-1000 nm for typical superconductors, matching experimental observations.
5. Connection to Prior Work
Our framework connects to several independent lines of research:
Markoulakis (2022) proposed that Cooper pairs can be described as electromagnetic flux vortex rings, noting that ‘a vortex ring of EM flux is topologically a photon, therefore a Cooper electron pair inside a superconductor is effectively a photon (Boson).’ Our work provides the microscopic geometric mechanism underlying this observation.
Hestenes’ Zitterbewegung interpretation describes the electron’s internal motion as a light-speed helical trajectory with radius λC/4π—precisely the energy transport radius in the toroidal model. Cooper pairing in this context becomes the coupling of two Zitterbewegung motions into a single coherent oscillation.
Topological superconductivity research has established that global topological invariants can distinguish superconducting phases. Our framework suggests these invariants may have a more fundamental geometric origin in the toroidal structure of the paired electrons themselves.
6. Experimental Predictions
The topological untwisting mechanism makes several predictions distinguishable from conventional BCS theory:
1. Geometric sensitivity: Cooper pair formation should be sensitive to magnetic field geometry, not just magnitude. Helical or toroidal field configurations may enhance or suppress pairing differently than uniform fields.
2. Polarization effects: Circularly polarized electromagnetic radiation at frequencies near 2ωC may couple directly to the internal rotation of Cooper pairs, potentially providing a new probe of pair structure.
3. Isotope effect modification: The phonon role as ‘phase-locking scaffolding’ rather than direct attraction suggests the isotope effect should have corrections at very low temperatures where lattice structure dominates over thermal vibrations.
4. Spin-polarized pairing: The topological untwisting mechanism predicts specific relationships between electron spin orientation and pairing geometry that could be tested in spin-polarized STM experiments on superconducting surfaces.
7. Conclusions
We have proposed that Cooper pair formation can be understood as a topological untwisting process in which two toroidal electrons—each requiring 720° rotation for closure—achieve a combined configuration requiring only 360°. This geometric transformation explains the fermion-to-boson transition at a fundamental level: the change in quantum statistics reflects an actual change in the topological structure of the paired system.
The framework reinterprets the role of phonons in BCS theory as providing phase-locking boundary conditions rather than the fundamental pairing attraction. This suggests that high-temperature superconductivity may be achievable by engineering materials that provide optimal boundary conditions for topological untwisting, independent of phonon-mediated interactions.
Further development of this model requires explicit calculation of the pairing phase Φcoupling for specific geometric configurations, and numerical simulation of the stability of Hopf-linked toroidal electron structures. The experimental predictions outlined above provide concrete paths for testing the topological untwisting hypothesis against conventional BCS theory.
References
[1] J.G. Williamson and M.B. van der Mark, ‘Is the electron a photon with toroidal topology?’, Annales de la Fondation Louis de Broglie 22, 133-160 (1997).
[2] J. Bardeen, L.N. Cooper, and J.R. Schrieffer, ‘Theory of Superconductivity’, Physical Review 108, 1175 (1957).
[3] E. Markoulakis, ‘Cooper electron pairs mechanism inside a superconductive medium: topological description as vortex rings’, ResearchGate (2022).
[4] D. Hestenes, ‘The Zitterbewegung Interpretation of Quantum Mechanics’, Foundations of Physics 20, 1213 (1990).
[5] A.F. Rañada, ‘Topological electromagnetism’, Journal of Physics A 25, 1621 (1992).
[6] D. Finkelstein and J. Rubinstein, ‘Connection between Spin, Statistics, and Kinks’, Journal of Mathematical Physics 9, 1762 (1968).