About K4 Forge
What the project is, what it proves, and what it does not claim.
The core idea
A complete graph on four vertices (K4) has six edges, three independent cycles, and three independent cuts. Driving current through these edges creates a magnetic field whose cycle and cut components are algebraically orthogonal: MTG = 0. This identity is unconditional — it holds for any edge lengths, any current pattern, any field model.
When the tetrahedron is regular and the Biot-Savart model applies, a stronger property emerges: the field matrix F0 applied to the cut basis G produces zero — F0·G = 0 (theorem T3.1, independently verified as integer-exact via INT.CG_zero). This means the centroid magnetic field is entirely determined by the cycle currents. The cut currents generate zero net field at the center. Full 3D controllability follows from det(F0·M) = 32 (INT.det_CM_32), with condition number κ = 2 (INT.gram_CM).
This is the foundation for exact, verifiable electromagnetic field control: you can specify a desired B-field at the centroid and solve for the unique cycle currents that produce it, with a known and bounded error.
Claim taxonomy
Every result carries a claim class that says exactly how strong the guarantee is:
| Class | Meaning |
|---|---|
| [A] | Algebraic exact. Integer identity, unconditional.Backed by: T0.1–T0.15, INT.*, T1.1–T1.8 |
| [G] | Geometry-dependent exact. Requires regular K4 + named model.Backed by: T2.5–T2.7, T3.1–T3.3, SYM.* |
| [G*] | Model-limited. Valid within stated regime only. |
| [M] | Numerical. Never promoted without a named proof path.Backed by: Sel.1–Sel.3 |
| [H] | Heuristic. Engineering guidance, not a proof. |
| [C] | Conjectural. Incomplete proof. |
What this project does not claim
- No medical or therapeutic benefit is claimed or implied.
- No consumer product readiness.
- Numerical results beyond the stated claim class are unverified.
- Physical realization models are approximations — real hardware requires independent validation.