From a desired field behavior to six drive signals.

01 · THE OBJECT

Unit-edge K4 geometry

Start with a regular tetrahedron whose edge length is exactly one. A convenient centred embedding is the four sign vectors below, scaled by 1/(2√2). Every later physical length is one multiplier L attached at the end.

v₀=( 1, 1, 1)/(2√2)   v₁=( 1,-1,-1)/(2√2)
v₂=(-1, 1,-1)/(2√2)   v₃=(-1,-1, 1)/(2√2)

|vⱼ-vᵢ| = 1 for every edge ij

The canonical oriented edge order is (01, 02, 03, 12, 13, 23), always low-numbered vertex to high-numbered vertex.

02 · THE GRAPH COORDINATES

Six edge values are two perpendicular groups of three

The incidence matrix D adds signed edge values at the four vertices. Its null space contains closed circulation; the row space of D contains differences produced by vertex values. For a connected K4 both spaces have dimension three and together fill all six edge coordinates.

edge space ℝ⁶ = ker(D) ⊕ im(Dᵀ) = cycle(3) ⊕ cut(3)

i = M w + G u
MᵀG = 0        P_cycle + P_cut = I₆        P_cycle P_cut = 0

The physical control vector remains the six edge currents i. The coordinates w and u are a reversible lens on those same currents: w describes circulation and u describes vertex-difference content. No edge permanently belongs to one group. A connected passive K4 circuit can drive only loops; six independently closed edge drives are what make the whole six-dimensional space physically accessible.

03 · SOURCE TO FIELD

One finite-segment kernel, summed over every source element

For a straight segment A→B carrying current I, the magnetic field at P follows the finite Biot–Savart expression. A curved or wound conductor is represented as a polyline and evaluated as a sum of these segments. Superposition is linear in the oriented current measure I(s)t̂(s)ds—not in arbitrary geometric control points.

d=B-A,  ê=d/|d|,  a=P-A,  s₁=a·ê,  s₂=s₁-|d|
p⊥=a-s₁ê,  ρ=|p⊥|

B(P)= (μ₀I/4πρ) [s₁/√(s₁²+ρ²) - s₂/√(s₂²+ρ²)] (ê×p⊥)/ρ

With edge length normalized to one, the field pattern is a dimensionless shape function. A physical attachment supplies the overall edge scale μ₀I/(4πL), plus any finite-radius, turn-count and return-path corrections. The homepage intentionally displays relative field colour so the attachment is not mistaken for the object.

04 · THE OBSERVER

A “field vector” is three numbers at one declared point

A field is a function over space. A field vector B(P) is only the three components seen at one point P. It says nothing by itself about a null elsewhere, a gradient, a flux-tube-like corridor, force on an object, or field topology. Each of those is a different observer built from multiple points and/or derivatives.

05 · FORWARD MATRIX

Evaluate each actuator once, then combine by multiplication

Choose observer samples P₁…Pₙ. Drive one unit source at a time and record what the observer sees. Those response columns form F. Any simultaneous source vector q then produces y=Fq. A single-point magnetic observer gives a 3×6 matrix for the edges; a dense spatial objective can have hundreds or thousands of rows while keeping the same six columns.

F[:,e] = H( field made by unit source e )

y = F q

for several source families: y = [F_edge  F_vertex  F_face  F_electrode] q_all

At the centroid of the regular ideal edge geometry, symmetry makes F_edge G = 0 exactly and maps the three cycle coordinates isotropically onto the three magnetic components. That statement belongs to this observer and source model; it does not say a cut pattern has no field elsewhere.

06 · OBJECTIVE TO SOURCE

The literal inverse-control sequence

1. Declare the source geometry. Every driven path, dipole, electrode, return and coordinate convention gets an identity.
2. Declare the observer. Name the points, derivatives, path samples, object model and relative weights that define success.
3. Assemble F. Evaluate the declared field primitive for one unit of every actuator and pass it through the observer.
4. Declare limits and costs. Current, voltage, slew, power, bandwidth and any source preferences are separate constraints.
5. Solve. Minimize observer error plus actuator cost: min ‖W(Fq-y*)‖² + λ‖Rq‖², subject to the limits.
6. Forward-verify. Recompute ŷ=Fq, report residuals per objective block, singular values and remaining null-space freedom.
7. Compile time behavior. Turn each qₑ into DC, sine, triangle, square, saw or an arbitrary sampled waveform with amplitude, phase and frequency.
8. Pass through the circuit model. Solve Z(ω)i(ω)=v(ω); voltage is not current when resistance, inductance, coupling and back-EMF matter.
9. Translate to hardware. Apply the measured command→current transfer, channel order, polarity, saturation, dead-time and safety envelope.
10. Measure and compare. Keep requested command, firmware echo, measured current and measured field as four different records.

06A · NULLS BY SOURCE FAMILY

A null is a relationship between a source and an observer

For a signed-edge source—whose sign really is current direction along an oriented edge—the regular centroid map kills the three-dimensional cut space. Such centroid-null patterns can use three, four, five or six edges. But a longitudinal racetrack can instead be dominated by its loop dipole: then the effective sign may be the mounted slot side, and its dark family is opposite-pair balance rather than graph cut.

signed-edge family:       F_centroid P_cut = 0
unsigned dipole family:   x₀₁=x₂₃,  x₀₂=x₁₃,  x₀₃=x₁₂
intersection:             one 2|2 cut pattern

A five-edge cut null is therefore a useful current-family signature: the unsigned dipole family cannot make one. Adding the sixth edge does not “complete” that same null; while the original five still sum to zero, the new edge initially appears exactly as it would alone. All six amplitudes must be solved again.

Two independent finite-wire engines agree on the practical mechanism. Tangential slot splitting produces order-one cut leakage in the tested racetrack model. Radial clocking restores signed-edge behavior to the numerical model floor, while crossover packing, mounting errors and return paths set the remaining hardware residual. Exactness belongs to the symmetry-qualified source family—not automatically to any winding called an “edge coil.”

07 · TIME AND WAVEFORM

Waveform shape matters because the plant responds to its spectrum

In the ideal quasistatic field map, each instantaneous current simply scales its source column. A sine, triangle, square or saw wave therefore produces a different time sequence of source vectors. In real hardware the sharp waveforms also contain harmonics, so the circuit, skin/proximity effects, capacitive coupling and controller bandwidth reshape them differently.

iₑ(t) = offsetₑ + Aₑ · shapeₑ( fₑt + φₑ/2π )

B(P,t) = Σₑ Fₑ(P) iₑ(t)          [prescribed-current quasistatic layer]
V(ω) = Z(ω) I(ω)                 [circuit attachment]

A rotating vector of constant magnitude is the simplest useful example: drive two independent spatial modes in quadrature, a cos(ωt)+b sin(ωt). The identity cos²+sin²=1 explains the constant radius. The waveform monitor on the homepage exposes the six underlying currents and the shared instant used by the field renderer.

08 · VERTICES, FACES AND ELECTRIC FIELD

The graph and physical space coincide through source placement

A graph coordinate becomes physical only when an actuator geometry is attached to that vertex, edge or face. Polar corner values generate edge cuts through Dᵀ. Axial vertex coils and face-normal coils transform differently under reflection: each four-source family splits into a three-dimensional vector-like block plus one symmetric one-dimensional mode. The three-dimensional block can independently address a centroid magnetic vector under the regular ideal source model; the remaining mode is a symmetry null at that observer.

Electrodes are native, not an afterthought. Four corner potentials have one irrelevant common mode, leaving three voltage differences. Their electric response matrix is assembled from the actual finite electrode geometry. The ideal barycentric interior model is useful structure; capacitance, boundaries and energized coil potentials belong to the physical attachment. “Cycle equals B and cut equals E” is not a valid general identity.

09 · FROM INTRINSIC TO REAL

The translation layer is explicit and replaceable

The intrinsic engine stores topology, ratios, normalized paths and observer definitions. A device profile then supplies L, actual polylines/windings, conductor cross-section, turns, terminals, return paths, R/L/C and mutual impedance, amplifier limits, sensor pose, and measured command→current transfer. Changing a build should replace that profile—not silently alter the principles or force every visualization to inherit an old 100 mm assay.

intrinsic pattern q̄(t)
  × family amplitude / current scale
  × physical geometry transform (L, rotation, deformation)
  → circuit solution i(t)
  → field prediction
  → observer prediction
  ↔ measured current and measured field

10 · HONESTY BOUNDARY

What is established, and what is still an engineering question

The K4 incidence algebra and cycle/cut orthogonality are exact graph facts. The centroid zeros and isotropic maps are exact for their named regular source models. Off-centre maps, finite conductors and numerical optimizations are model results. V2 per-edge current transfer, simultaneous measured rank, whole-volume field atlas, vertex/face actuators and calibrated E-field authority are not yet established hardware facts. A residual near machine zero proves the matrix problem was solved; it does not prove the matrix describes the apparatus.