K4 Forge — graph compute+control substrate
One graph. Two orthogonal channels.
Computation and field control
on the same conductors.
The K₄ complete graph splits its edge space into two orthogonal subspaces: cycle (field control) and cut (computation). The algebraic identity MT·G = 0 holds to integer zero — not a numerical approximation, not tunable. Both channels run on the same physical conductors simultaneously without cross-talk.
The electromagnetic realization is proven and live in the browser. A second realization (magnetohydrodynamic) is in progress.
The substrate — three orthogonal channels
Three cycle DOFs control the centroid B-vector exactly — any direction, any magnitude. In the EM realization: edge currents along the three independent cycle directions. Proven by Hodge decomposition on K₄; zero residual is structural, not numerical.
Three cut DOFs implement a static 6-to-4 CONSENSUS gate array: 729 inputs, 81 distinct outputs, 100% output utilization. MT·G = 0 guarantees this channel cannot interfere with field control — not by tuning, by graph topology.
Three vertex potentials set a uniform E field throughout the interior. Principal-angle orthogonal to both cycle and cut channels — E cannot be degraded by any B or computation constraint, even under worst-case overconstrain.
MT·G = 0 to integer zero (norm 0.00e+00, leakage ~1e-17). The three channels are algebraically independent of each other — this is a property of the K₄ graph topology, not of any particular operating condition or frequency.
02 — what makes it different
What makes the K₄ substrate different
Why the K₄ graph
K₄ is the smallest complete graph that gives a non-trivial cycle/cut decomposition with full rank in both subspaces. Its six edges span a 3D cycle space and a 3D cut space; its four vertices add a 3D electrode space. Together: nine independent DOFs, three orthogonal channels, two proven realizations (EM, MHD in progress).
Orthogonal channels, not interference
The Hodge decomposition splits edge currents into cycle (→ B at centre) and cut (→ zero centroid B, shapes off-centre field) components. Vertex potentials form a third channel. The three channels occupy orthogonal subspaces of the 9D input space — they cannot interfere with each other by construction, not by tuning.
E independent of B — structurally
The electric field channel (corner inputs) sits at [90°, 90°, 90°] principal angles from every possible B constraint. This means E is indestructible: even with five simultaneous B targets in catastrophic overlap, the E field achieves machine-precision accuracy (10⁻¹⁴ residual). Not a lucky tuning — a consequence of the hardware topology.
Exact, not approximate
Centroid B uses the algebraic Hodge split — the residual is structurally zero. Interior E uses the analytic Laplace solution in barycentric coordinates — uniform throughout the volume, exact within the model. Neither result is a numerical approximation that happens to be small.
Off-centre B: nulls and gradients
The three cut-coil inputs add zero to the centroid B but fully control one spatial B feature at a time: a field null at a chosen interior point, a gradient tilt at the centre, or far-field steering. Nulls are true nulls — positive-definite Hessian, not saddle points — because the interior is not current-free.
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.
03 — for specialists
Headline results
| Class | Identifier | Statement |
|---|---|---|
| [A] | T1.1 Mᵀ·G = 0 | Substrate orthogonality — cycle and cut spaces are algebraically independent. Zero leakage by graph topology. |
| [A] | INT.det_CM_32 det(C·M) = 32 | Full field controllability — cycle space spans ℝ³ at the centroid. |
| [A] | INT.gram_CM κ(F₀·M) = 2 | Bounded condition number — centroid field control is well-conditioned. |
| [G] | T3.1 F₀·G = 0 | Cut currents produce zero centroid B (EM realization, regular K₄ + Biot-Savart). |
| [G] | GATE 81/81 outputs | Static 6-to-4 CONSENSUS gate array: 100% output utilization across all 729 inputs (cut-space computation). |
| [G] | E.rank rank(E_vol) = 3 | Full 3D E controllability — vertex potentials span all interior E directions. |
| [A] | E.indep [90°,90°,90°] | E and B channels are principal-angle orthogonal — E cannot be degraded by any B constraint. |
Full theorem inventory and claim taxonomy → · Source on GitHub →
a note on scope
K4 Forge is a research engine, not a product. The substrate-level algebraic results ([A] class) hold by graph topology. The EM realization results ([G] class) are computationally verified against the Python engine but not yet anchored to an external EM solver or physical hardware measurement. The MHD realization is in progress (dipole-approximation, [H]); finite-solenoid verification is open. No result has been independently replicated.