Theory & Ledger
Every claim tracked. Every assumption visible. Nothing hidden.
Edge currents split exactly into field-producing cycles and field-free cuts. This is an algebraic identity on K4, not an approximation.
The centroid field operator is orthogonal to the cut projector under T_d symmetry. Guaranteed for regular tetrahedra.
Vertex coil condition number is exactly 1 at the centroid — perfect conditioning, no numerical loss.
Condition degrades to 6.4 at face centers. Still well-conditioned for control.
Severe conditioning at edge midpoints. Vertex coils lose 2+ decades of precision here.
Bundle model recovers filament model exactly as wire radius vanishes. Numerical, not algebraic.
Cross-product of induced current density and applied field. Conjectural regime for K4 geometry.
The 6-dimensional edge current space decomposes exactly as R⁶ = Cycle₃ ⊕ Cut₃ under the graph Laplacian.
The centroid field matrix F₀ annihilates the cut projector G when the tetrahedron has full T_d symmetry.
At the centroid κ²=1, at face centers κ²=32/5, at edge midpoints κ²=108. All exact under T_d.
For each cycle mode there exists a transverse plane supporting a rotating B-field orbit at the drive frequency.
The wire-bundle Biot-Savart integral converges pointwise to the filament result as bundle radius r → 0.
There exist edge current + vertex electrode configurations that produce stable Lorentz-driven flow in conducting fluid filling the tetrahedron.