From Coupled Oscillators to the Fine Structure Constant
Standard lattice field theory treats couplings as fixed parameters tuned by hand to match experiment. The coherence lattice inverts this: couplings are dynamical variables governed by their own equation of motion — the Coherence Learning Rule (CLR).
The CLR is a single principle: every bond coupling evolves to maximize coherence capital — the product of phase alignment and structural richness. This creates a tension: alignment wants all phases equal (trivial), while structural richness wants complex topology (incoherent). The CLR finds the compromise.
The result is a living lattice — a self-organizing network of coupled oscillators where:
- Couplings self-select. Bond strengths converge to a binary field: alive bonds at the BKT critical coupling, dead bonds at zero. No parameter tuning.
- Topology emerges. Phase vortices nucleate spontaneously from the co-evolution of phases and couplings. These are not inserted by hand — they are produced by the dynamics.
- The electron is a vortex. A topological defect on the diamond lattice carries quantized charge, spin-1/2, chirality, Dirac dispersion, and an anomalous magnetic moment — all emergent from lattice mathematics. No fields are added; no particles are postulated.
- α is a byproduct. The fine structure constant emerges at the topologically constrained optimum of the CLR. It measures where the self-optimizing vacuum screens a magnetic perturbation — a property of the living lattice, not of the vortex.
Diamond is not a choice but a consequence. Five physics filters — Bravais rank bound, bipartite structure, octahedral symmetry, 2-site unit cell, and vortex line persistence in d ≥ 3 — uniquely select the diamond lattice among all 3D crystal structures. It is the minimal lattice supporting Dirac fermions with topological protection.
The CLR drives couplings upward. Vortex topology prevents them from exceeding the BKT critical point. The equilibrium is at the constraint boundary: K_bulk = 16/π².
From this single operating point, the entire electromagnetic coupling follows:
α = R₀(2/π)⁴ × (π/4)^(1/√e + α/2π)
A linked-cluster expansion over diamond lattice subgraphs adds vacuum polarization:
1/α = 137.035999 (1.5 ppb from CODATA)
g = 2.002319304355 (11.4 matching digits)
Zero free parameters. The only inputs are π, e, the modified Bessel functions I₀ and I₁, and the coordination number z = 4.
A static lattice at the same BKT critical coupling gives 1/α = 143 — wrong by 4.4%. The difference is how the power-law exponent is evaluated. Standard lattice field theory integrates along the RG trajectory. The CLR's Phase-Locked Mode Lemma proves couplings freeze at a fixed point, so the Debye-Waller factor evaluates at the attractor rather than averaging over the path. This single distinction — endpoint evaluation vs path integration — accounts for the entire gap between 143 and 137.
├── paper.tex # Main paper (LaTeX, revtex4-2)
├── paper.pdf # Compiled PDF
├── paper.bbl # Bibliography
├── references.bib # BibTeX source
├── figures/ # All figures (PNG)
├── scripts/ # Verification scripts (Python)
├── data/ # Precomputed results (JSON)
└── LICENSE # CC BY-NC 4.0 (paper) + AGPL-3.0 (code)
pip install numpy scipy matplotlib
# Core α formula (< 1 second)
python scripts/alpha_137_verification.py
# g-factor via QED series (< 1 second)
python scripts/g_factor_from_lattice.py
# Living vs static comparison (< 1 second)
python scripts/living_vs_static_alpha.py
# Green's function G_diff = 1/z (< 10 seconds)
python scripts/diamond_greens_function.py
# Two-vertex linked-cluster expansion (< 5 seconds)
python scripts/two_vertex_lce.py| Quantity | This Work | Measurement | Precision |
|---|---|---|---|
| 1/α | 137.035999 | 137.035999206 | 1.5 ppb |
| g | 2.002319304355 | 2.002319304361 | 11.4 digits |
| Static lattice (no CLR) | 143.134 | — | Wrong by 4.4% |
- Paper and figures: CC BY-NC 4.0
- Code: AGPL-3.0
See LICENSE for details.
@article{sharpe2026coherence,
author = {Sharpe, Michael},
title = {The Coherence Learning Rule on the Diamond Lattice: From Coupled Oscillators to the Fine Structure Constant},
year = {2026},
note = {Preprint}
}