The Spectral Geometry of Plasma Confinement: A Davis Field Equations Framework for Fusion Stability, Transport Bottlenecks, and Cross-Domain Universality

Summary

This paper applies the Davis Field Equations framework to plasma confinement in magnetic and inertial fusion devices. Thirteen computational experiments, one proved theorem, and zero free parameters validate the framework across three levels of MHD physics, two confinement approaches, a cross-domain universality test, and a direct application of the SUPERFLUID C = τ/K law to ballooning stability.

Core Results

1. The Cheeger Effect — Global stability persisting where local criteria fail at 100% of the plasma radius. The Suydam criterion is violated everywhere, yet the global ballooning eigenvalue is positive.
2. Toroidal Stabilization — 15.6× stabilization from toroidal topology with a quantitative Non-Decoupling prediction (Pearson r = 0.92, p = 3.9 × 10⁻⁴).
3. Cheeger Cut Migration — The Cheeger cut migrates inward from x = 0.80 to 0.27, crossing the q = 2 surface at β_N ≈ 1.2.
4. Varadhan Heat Kernel — ρ = −0.977 in the stable regime, redeeming a failed earlier attempt (ρ = −0.53).
5. Spectral Isolation — Single-mode onset with sub-Poisson spacing statistics (⟨r⟩ = 0.121, cf. Poisson 0.386).
6. ICF Codimension Deficit — Deficit of 3 (7 modes, 4 constraints), explaining NIF's historical difficulty.
7. BEC–Plasma Universality — Spectral capacity agreement within 2.4% for shaped toroidal geometry.
8. Davis Law Universality — C = s/α_crit is universal within each tokamak geometry class (CoV < 4% at s ≥ 0.8), with a four-class hierarchy mirroring the BEC obstacle-type structure.
9. Plateau Maximum Theorem (Proved) — C(s) has a unique maximum; the tangent condition α'_crit(s_peak) = α_crit(s_peak)/s_peak is satisfied to 0.0% precision across all four geometry classes. The plateau [1.3, 3.0] coincides with the tokamak operating range.
10. Troyon Reframing — Self-consistent Grad-Shafranov equilibria remain ballooning-stable past the Troyon limit (β_N = 3.0 > 2.8), identifying Troyon as an external kink mode boundary, not a ballooning boundary.

Cross-Geometry Hierarchy

Instability Hierarchy

Two documented failures and two null results are retained as lessons, reinforcing that spectral-geometric analysis must derive from first-principles physics.

Experiments

Repository Contents

Paper

• theory/plasma_confinement.pdf — Compiled PDF

Experiment Scripts (Python)

• experiment1_v2_newcomb.py — Cylindrical Newcomb / Cheeger effect
• experiment2_ballooning.py — CHT toroidal ballooning
• experiment3_shaped.py — Shaped Miller equilibrium
• experiment4_cheeger_cut.py — Cheeger cut migration with β_N
• experiment5_varadhan.py — Varadhan heat kernel test
• experiment6_spectral.py — Full spectral analysis
• experiment7_helicity.py — Helicity correlation
• experiment9_icf.py — ICF codimension counting
• experiment10_ricci.py — Ollivier-Ricci curvature
• experiment11_universality.py — BEC–plasma universality
• experiment12_superfluid_plasma.py — SUPERFLUID C = τ/K applied to CHT
• plateau_maximum_theorem.py — Plateau Maximum Theorem verification
• experiment13_freegs.py — FreeGS Grad-Shafranov / Troyon limit
• experiment13_troyon.py — Troyon limit analysis (built-in GS solver)

Dependencies

• Python 3.12+
• NumPy, SciPy, Matplotlib
• FreeGS (pip install freegs) for Experiment 13

Related Publications

1. Davis, B. R. (2025). The Field Equations of Semantic Coherence. Zenodo. DOI: 10.5281/zenodo.14784553
2. Davis, B. R. (2025). The Incompressibility of Topological Charge: A Spectral Proof of the Yang-Mills Mass Gap. Zenodo. DOI: 10.5281/zenodo.17846521
3. Davis, B. R. (2026). Holonomy-First Navier-Stokes Regularity. Zenodo. DOI: 10.5281/zenodo.18216597
4. Davis, B. R. (2026). The Davis-Landau Sonic Onset Law: Universality of the Critical Mach Number for Vortex Nucleation. Zenodo. DOI: 10.5281/zenodo.18369500
5. Davis, B. R. (2025). The Geometry of Sameness: SUPERFLUID. Amazon.

Keywords

plasma confinement, magnetohydrodynamic stability, spectral gap, Cheeger constant, ballooning modes, tokamak, Davis Field Equations, Connor-Hastie-Taylor equation, s-alpha diagram, Plateau Maximum Theorem, Troyon limit, Grad-Shafranov, BEC universality, geometric phase transition, fusion energy

License

Creative Commons Attribution 4.0 International (CC BY 4.0)