K4 Forge — research engine
Set a magnetic-field vector
with electricity.
Exact at the centre.
Take the simplest 3D shape — a tetrahedron — and run currents through its six edges. Each edge makes a small magnetic field, and the fields combine. Choose the currents carefully and the combined field at the very centre points wherever you like, with whatever strength you like. The math at the centre isn't approximate. It's exact.
02 — what makes it different
Most field-control systems compute. This one solves and proves.
Why the tetrahedron
A tetrahedron is the smallest 3D shape that can deliver three independent control directions at its centre — exactly matching the three components of a magnetic vector. Smaller graphs can't. Larger graphs introduce redundancy.
Exact, not approximate
The centroid solve uses the Hodge decomposition on K4: edge currents split into cycle currents that produce field at the centre and cut currents that produce zero. The split is algebraic. The residual at the centre is structurally zero, not numerical.
Provenance, not claims
Every result carries a claim class — [A] algebraic, [G] geometric, [M] numerical — plus the frozen-core digest, the field model, and what is not in the model. No silent assumptions.
Open and reproducible
The engine is open source. Each run produces a JSON artifact you can inspect or replay. The 182-check frozen core is independently verifiable; an oracle module (k4_theory) cross-validates from a different codebase.
03 — for specialists
Headline theorems
| Class | Identifier | Statement |
|---|---|---|
| [A] | T1.1 Mᵀ·G = 0 | Hodge orthogonality of cycle / cut bases on K₄. |
| [A] | INT.det_CM_32 det(C·M) = 32 | Full controllability — the cycle space spans ℝ³ at the centroid. |
| [A] | INT.gram_CM κ(F₀·M) = 2 | Bounded condition number — controllability is well-conditioned. |
| [G] | T3.1 F₀·G = 0 | Cut currents produce zero centroid field for regular K₄ + Biot-Savart. |
| [G] | T3.2 rank(F·M) = 3 | Three independent control directions at the centroid. |
Full theorem inventory and claim taxonomy → · Source on GitHub →
a note on scope
K4 Forge is a research engine. It is not a medical device, not a consumer product, and not independently validated against physical measurement. Numerical results beyond the stated claim class are unverified.